Content-Type: multipart/mixed; boundary="-------------9908160457536" This is a multi-part message in MIME format. ---------------9908160457536 Content-Type: text/plain; name="99-298.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="99-298.keywords" epsilon-entropy, hyperbolic-parabolic PDE, infinite domains ---------------9908160457536 Content-Type: application/postscript; name="export.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="export.ps" %!PS-Adobe-2.0 %%Creator: dvips 5.47 Copyright 1986-91 Radical Eye Software %%Title: submitted.dvi %%Pages: 38 1 %%BoundingBox: 0 0 596 843 %%DocumentFonts: Times-Bold Times-Italic Times-Roman ptmro8a %%EndComments %%BeginProcSet: texc.pro /TeXDict 200 dict def TeXDict begin /N /def load def /B{bind def}N /S /exch load def /X{S N}B /TR /translate load N /isls false N /vsize 10 N /@rigin{ isls{[0 1 -1 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale Resolution VResolution 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Fv(di)n(v)o(er)n(ges)22 b(lik)o(e)h Fp(j)p Ft(x)p Fp(j)3392 1733 y FF(4)3431 1776 y Fv(.)35 b(Using)22 b(the)94 1895 y(bound)1474 2067 y Ft(\021)1527 2024 y FF(2)1567 2067 y Fp(j)p Ft(uv)t Fp(j)44 b(\024)1910 1999 y Ft(\021)1963 1956 y FF(2)2002 1999 y Ft(v)2054 1956 y FF(2)p 1910 2044 185 4 v 1977 2135 a Fv(2)2128 2067 y Fs(+)2240 1999 y Ft(\021)2293 1956 y FF(2)2333 1999 y Ft(u)2390 1956 y FF(2)p 2240 2044 190 4 v 2310 2135 a Fv(2)2466 2067 y Ft(;)94 2285 y Fv(this)24 b(implies)1237 2477 y Ft(V)e Fs(\()p Ft(u)p Fs(\))g(+)h Ft(\021)1627 2434 y FF(2)1666 2477 y Ft(uv)49 b Fp(\025)44 b(\000)2030 2409 y Ft(\021)2083 2367 y FF(2)2124 2409 y Ft(v)2176 2367 y FF(2)p 2031 2454 185 4 v 2098 2545 a Fv(2)2250 2477 y Fs(+)22 b Ft(u)2406 2434 y FF(2)2468 2477 y Fp(\000)h Ft(C)2639 2504 y FF(0)2704 2477 y Ft(;)94 2696 y Fv(for)j(some)e(constant)g Ft(C)899 2723 y FF(0)939 2696 y Fv(.)36 b(Therefore,)25 b(one)g(has)g(the)f(inequality)1321 2939 y Fp(k)p Fs(\()p Ft(u;)17 b(v)t Fs(\))p Fp(k)1653 2896 y FF(2)1653 2965 y FE(h)1700 2983 y Fl(\013)1748 2965 y FE(;)p FF(1)1856 2939 y Fp(\024)45 b Fv(2)p Ft(F)2092 2965 y FF(0)2132 2939 y Fs(\()p Ft(u;)17 b(v)t Fs(\))j(+)j Ft(C)2555 2965 y FF(1)2620 2939 y Ft(:)94 3181 y Fv(Using)h(\(2.9\))h(we)g(conclude)94 3373 y Fu(Lemma)47 b(2.3.)62 b Fh(There)34 b(is)f(a)h(constant)f Ft(C)1630 3399 y FF(2)1703 3373 y Fh(\(independent)g(of)h Fv(0)39 b Ft(<)h(\021)k(<)c Fv(1)p Fh(\))34 b(for)g(which)f(the)g(follo)n(wing)94 3492 y(holds.)j(Assume)24 b(that)h Ft(F)957 3519 y FF(0)997 3492 y Fs(\()p Ft(u)1093 3519 y FF(0)1132 3492 y Ft(;)17 b(v)1225 3519 y FF(0)1265 3492 y Fs(\))27 b Ft(<)i Fp(1)p Fh(.)36 b(Then,)25 b(for)g(all)g Ft(t)j(>)g Fv(0)d Fh(one)g(has)g Fp(k)p Fs(\()p Ft(u)2836 3519 y FE(t)2871 3492 y Ft(;)17 b(v)2964 3519 y FE(t)2998 3492 y Fs(\))p Fp(k)3087 3519 y FE(h)3134 3537 y Fl(\013)3184 3519 y FE(;)p FF(1)3276 3492 y Ft(<)28 b Fp(1)d Fh(and)g(there)94 3612 y(is)h(a)h Ft(T)43 b Fs(=)30 b Ft(T)14 b Fs(\()p Ft(u)634 3639 y FF(0)674 3612 y 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FF(2)2879 3090 y Ft(:)94 3340 y Fv(The)25 b(result)g(follo)n(ws)e(from)h (the)h(standard)g(Sobole)n(v)f(inequality)-6 b(.)33 b(The)25 b(proof)g(of)g(Lemma)f(2.5)h(is)f(complete.)p 3615 3425 85 2 v 3615 3506 2 81 v 3698 3506 V 3615 3508 85 2 v 316 3732 a(Using)40 b(this)g(lemma,)k(and)d(observing)f(that)h(the)g Fp(k)31 b(\001)f(k)2283 3759 y FE(\013;)p FF(loc)p FE(;)p FF(1)2548 3732 y Fv(norm)41 b(is)f(translation)g(in)l(v)n(ariant)g(we)94 3851 y(conclude)25 b(immediately)e(from)i(\(2.12\))f(that)h(there)g(is)f(a)h (constant)f Ft(C)2473 3878 y FF(3)2538 3851 y Fv(for)h(which)1578 4092 y(sup)1625 4174 y FE(x)1733 4092 y Fp(j)p Ft(u)p Fs(\()p Ft(x;)17 b(t)p Fs(\))p Fp(j)43 b(\024)h Ft(C)2297 4119 y FF(3)2362 4092 y Ft(;)1229 b Fs(\()p Fv(2)p Ft(:)p Fv(14)p Fs(\))94 4383 y Fv(for)26 b(all)e Ft(t)k(>)g(T)14 b Fv(.)36 b(The)25 b(proof)f(of)h (Proposition)f(2.4)g(is)g(complete.)p 3615 4468 V 3615 4548 2 81 v 3698 4548 V 3615 4550 85 2 v 316 4775 a(W)-8 b(e)32 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b(=)h Fv(1)14 b Fp(\000)g Fv(3)p Ft(s)998 151 y FF(2)1038 194 y Fv(,)21 b(for)h(the)f Ft(U)32 b Fv(we)21 b(ha)n(v)o(e)g(tak)o(en)g(abo)o(v)o(e.)34 b(W)-8 b(e)21 b(consider)g(initial)f(data)h Fs(\()p Ft(w)3400 221 y FF(0)3440 194 y Ft(;)c(z)3531 221 y FF(0)3571 194 y Fs(\))k Fv(which)94 314 y(will)i(be)g(bounded)g(later)g(and)g(assume)g(\(in)g(vie)n (w)g(of)g(Proposition)f(2.4\))h(that)g Fp(k)p Fs(\()p Ft(u)2879 341 y FF(0)2918 314 y Ft(;)17 b(v)3011 341 y FF(0)3051 314 y Fs(\))p Fp(k)3140 341 y FE(\013;)p FF(loc)o FE(;)p FF(1)3391 314 y Fp(\024)28 b Ft(C)3567 341 y FF(2)3631 314 y Fv(which)94 433 y(implies)c(sup)551 460 y FE(x)618 433 y Fp(j)p Ft(u)p Fs(\()p Ft(x;)17 b(t)p Fs(\))p Fp(j)26 b(\024)i Ft(C)1149 460 y FF(3)1214 433 y Fv(for)d(all)f Ft(t)k(>)g Fv(0.)316 553 y(W)-8 b(e)24 b(are)g(going)f(to)g(bound)g(the)g(gro)n(wth)g(of)h Fs(\()p Ft(w)s(;)17 b(z)t Fs(\))23 b Fv(as)g(a)h(function)f(of)h(time.)34 b(W)-8 b(e)25 b(introduce)e(a)g(positi)n(v)o(e)94 672 y(constant)h Ft(\026)h 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1498 y FF(1)p 389 1514 35 4 v 389 1572 a(2)436 1537 y Ft(@)489 1564 y FE(t)524 1537 y Ft(F)588 1564 y FF(1)628 1537 y Fs(\()p Ft(w)738 1564 y FE(t)773 1537 y Ft(;)17 b(z)864 1564 y FE(t)899 1537 y Fs(\))44 b(=)h Ft(\013)1185 1402 y Fq(Z)1301 1537 y Ft(h)1358 1564 y FE(\013)1415 1397 y Fq(\022)1488 1537 y Ft(\021)1541 1494 y FF(2)1581 1537 y Ft(z)1627 1564 y FE(t)1680 1537 y Fs(_)-45 b Ft(z)1709 1564 y FE(t)1766 1537 y Fs(+)23 b Ft(w)1940 1494 y Fr(0)1937 1564 y FE(t)2004 1537 y Fs(_)-60 b Ft(w)2046 1494 y Fr(0)2043 1564 y FE(t)2101 1537 y Fs(+)2212 1470 y Ft(\021)2265 1427 y FF(2)2305 1470 y Ft(\026)p 2212 1514 153 4 v 2264 1606 a Fv(2)2409 1537 y Fs(_)g Ft(w)2448 1564 y FE(t)2484 1537 y Ft(z)2530 1564 y FE(t)2587 1537 y Fs(+)2699 1470 y Ft(\021)2752 1427 y FF(2)2792 1470 y Ft(\026)p 2699 1514 V 2750 1606 a Fv(2)2864 1537 y Ft(w)2935 1564 y FE(t)2987 1537 y Fs(_)-45 b Ft(z)3016 1564 y FE(t)3051 1397 y Fq(\023)982 1829 y Fs(=)45 b Ft(\013)1185 1693 y Fq(Z)1301 1829 y Ft(h)1358 1856 y FE(\013)1415 1688 y Fq(\022)1510 1829 y Fp(\000)23 b Ft(z)1660 1786 y FF(2)1656 1856 y FE(t)1723 1829 y Fs(+)f Ft(z)1868 1856 y FE(t)1904 1829 y Ft(w)1978 1786 y Fr(00)1975 1856 y FE(t)2051 1829 y Fs(+)g Ft(U)2229 1786 y Fr(00)2280 1829 y Fs(\()p Ft(u)2376 1856 y FE(t)2411 1829 y Fs(\))p Ft(z)2496 1856 y FE(t)2531 1829 y Ft(w)2602 1856 y FE(t)2660 1829 y Fs(+)g Ft(w)2833 1786 y Fr(0)2830 1856 y FE(t)2866 1829 y Ft(z)2916 1786 y Fr(0)2912 1856 y FE(t)2969 1829 y Fs(+)3081 1762 y Ft(\021)3134 1719 y FF(2)3174 1762 y Ft(\026)p 3081 1806 V 3132 1897 a Fv(2)3246 1829 y Ft(z)3296 1786 y FF(2)3292 1856 y FE(t)1101 2108 y Fp(\000)1213 2040 y Ft(\026)p 1213 2085 61 4 v 1218 2176 a Fv(2)1285 2108 y Ft(w)1356 2135 y FE(t)1391 2108 y Ft(z)1437 2135 y FE(t)1495 2108 y Fs(+)1606 2040 y Ft(\026)p 1606 2085 V 1611 2176 a Fv(2)1678 2108 y Ft(w)1749 2135 y FE(t)1785 2108 y Ft(w)1859 2065 y Fr(00)1856 2135 y FE(t)1932 2108 y Fs(+)2043 2040 y Ft(\026)p 2043 2085 V 2048 2176 a Fv(2)2115 2108 y Ft(U)2194 2065 y Fr(00)2245 2108 y Fs(\()p Ft(u)2341 2135 y FE(t)2376 2108 y Fs(\))p Ft(w)2489 2065 y FF(2)2486 2135 y FE(t)2528 1967 y Fq(\023)982 2400 y Fs(=)45 b Ft(\013)1185 2264 y Fq(Z)1301 2400 y Ft(h)1358 2426 y FE(\013)1415 2259 y Fq(\022)1510 2400 y Fp(\000)23 b Ft(z)1660 2357 y FF(2)1656 2426 y FE(t)1701 2400 y Fs(\()p Fv(1)e Fp(\000)1923 2332 y Ft(\021)1976 2289 y FF(2)2016 2332 y Ft(\026)p 1923 2377 153 4 v 1974 2468 a Fv(2)2088 2400 y Fs(\))h(+)g Ft(U)2327 2357 y Fr(00)2378 2400 y Fs(\()p Ft(u)2474 2426 y FE(t)2509 2400 y Fs(\))p Ft(z)2594 2426 y FE(t)2629 2400 y Ft(w)2700 2426 y FE(t)1101 2679 y Fp(\000)1213 2611 y Ft(\026)p 1213 2656 61 4 v 1218 2747 a Fv(2)1285 2679 y Ft(w)1356 2705 y FE(t)1391 2679 y Ft(z)1437 2705 y FE(t)1495 2679 y Fp(\000)1606 2611 y Ft(\026)p 1606 2656 V 1611 2747 a Fv(2)1678 2679 y Fs(\()p Ft(w)1791 2636 y Fr(0)1788 2705 y FE(t)1823 2679 y Fs(\))1862 2636 y FF(2)1924 2679 y Fs(+)2036 2611 y Ft(\026)p 2036 2656 V 2041 2747 a Fv(2)2108 2679 y Ft(U)2187 2636 y Fr(00)2237 2679 y Fs(\()p Ft(u)2333 2705 y FE(t)2368 2679 y Fs(\))p Ft(w)2481 2636 y FF(2)2478 2705 y FE(t)2521 2538 y Fq(\023)1101 2949 y Fp(\000)h Ft(\013)1281 2813 y Fq(Z)1398 2949 y Ft(h)1455 2906 y Fr(0)1455 2976 y FE(\013)1512 2868 y Fq(\000)1557 2949 y Ft(z)1603 2976 y FE(t)1639 2949 y Ft(w)1713 2906 y Fr(0)1710 2976 y FE(t)1767 2949 y Fs(+)1879 2881 y Ft(\026)p 1879 2926 V 1884 3017 a Fv(2)1951 2949 y Ft(w)2022 2976 y FE(t)2057 2949 y Ft(w)2131 2906 y Fr(0)2128 2976 y FE(t)2164 2868 y Fq(\001)2234 2949 y Ft(:)3619 2232 y Fs(\()p Fv(2)p Ft(:)p Fv(17)p Fs(\))94 3196 y Fv(Note)k(no)n(w)e(that)h(by)h(the)f(de\256nition)f(\(2.2\))i(of)f Ft(h)1753 3223 y FE(\013)1837 3196 y Fv(and)h(the)f(restriction)g Ft(\013)k Fp(\024)2800 3157 y FF(1)p 2800 3174 35 4 v 2800 3231 a(2)2847 3196 y Fv(,)d(we)g(\256nd)f(that)g(the)g(quotient)94 3316 y Fp(j)p Ft(h)179 3273 y Fr(0)179 3343 y FE(\013)236 3316 y Ft(=h)343 3343 y FE(\013)400 3316 y Fp(j)f Fv(is)f(bounded)g(by)h(2)p Ft(\013)i Fp(\024)i Fv(1.)35 b(Using)24 b(this,)g(we)h(get)1343 3502 y Fp(j)p Ft(h)1428 3459 y Fr(0)1428 3528 y FE(\013)1485 3502 y Ft(z)1531 3528 y FE(t)1567 3502 y Ft(w)1641 3459 y Fr(0)1638 3528 y FE(t)1673 3502 y Fp(j)44 b(\024)h Ft(\013h)1988 3528 y FE(\013)2045 3421 y Fq(\000)2091 3502 y Ft(z)2141 3459 y FF(2)2137 3528 y FE(t)2203 3502 y Fs(+)23 b(\()p Ft(w)2416 3459 y Fr(0)2413 3528 y FE(t)2448 3502 y Fs(\))2487 3459 y FF(2)2527 3421 y Fq(\001)2597 3502 y Ft(;)94 3687 y Fv(and)1259 3807 y Fp(j)p Ft(\026h)1404 3764 y Fr(0)1404 3833 y FE(\013)1461 3807 y Ft(w)1532 3833 y FE(t)1568 3807 y Ft(w)1642 3764 y Fr(0)1639 3833 y FE(t)1674 3807 y Fp(j)44 b(\024)h Ft(\013\026h)2049 3833 y FE(\013)2106 3726 y Fq(\000)2152 3807 y Ft(w)2226 3764 y FF(2)2223 3833 y FE(t)2288 3807 y Fs(+)22 b(\()p Ft(w)2500 3764 y Fr(0)2497 3833 y FE(t)2532 3807 y Fs(\))2571 3764 y FF(2)2611 3726 y Fq(\001)2682 3807 y Ft(:)94 3987 y Fv(W)-8 b(e)26 b(will)f(also)g(use)h(the)f(bound)g Fp(j)p Ft(w)1297 4014 y FE(t)1332 3987 y Ft(z)1378 4014 y FE(t)1413 3987 y Fp(j)k(\024)g Ft(\013)1640 3944 y Fr(\000)p FF(1)1742 3987 y Ft(w)1816 3944 y FF(2)1813 4014 y FE(t)1879 3987 y Fs(+)23 b Ft(\013z)2093 3944 y FF(2)2089 4014 y FE(t)2133 3987 y Fv(.)38 b(Finally)25 b(note)h(that)f(there)g(is)h(a)f(constant)g Ft(C)3692 4014 y FF(5)3758 3987 y Fv(for)94 4107 y(which)1587 4226 y(sup)1539 4317 y Fr(j)p FE(s)p Fr(j\024)p FE(C)1744 4335 y Fi(3)1790 4226 y Fp(j)p Ft(U)1897 4183 y Fr(00)1948 4226 y Fs(\()p Ft(s)p Fs(\))p Fp(j)43 b(\024)i Ft(C)2337 4253 y FF(5)2401 4226 y Ft(:)94 4468 y Fv(Combining)24 b(these)g(bounds)g(with)g(the)h(last)f (equality)g(of)h(\(2.17\),)f(we)i(get)e(for)h(times)f Ft(t)k(>)g Fv(0:)714 4684 y FF(1)p 714 4700 V 714 4757 a(2)761 4723 y Ft(@)814 4750 y FE(t)849 4723 y Ft(F)913 4750 y FF(1)953 4723 y Fs(\()p Ft(w)1063 4750 y FE(t)1098 4723 y Ft(;)17 b(z)1189 4750 y FE(t)1224 4723 y Fs(\))44 b Fp(\024)g(\000)p Ft(\013)1587 4587 y Fq(Z)1703 4723 y Ft(h)1760 4750 y FE(\013)1817 4723 y Ft(z)1867 4680 y FF(2)1863 4750 y FE(t)1908 4642 y Fq(\000)1953 4723 y Fv(1)22 b Fp(\000)2137 4656 y Ft(\021)2190 4613 y FF(2)2230 4656 y Ft(\026)p 2137 4700 153 4 v 2188 4791 a Fv(2)2324 4723 y Fp(\000)g Ft(\013)h Fp(\000)f Ft(\026=)p Fv(2)g Fp(\000)h Ft(\013C)3026 4750 y FF(5)3066 4642 y Fq(\001)1401 4984 y Fp(\000)f Ft(\013)1581 4849 y Fq(Z)1697 4984 y Ft(h)1754 5011 y FE(\013)1812 4984 y Fs(\()p Ft(w)1925 4941 y Fr(0)1922 5011 y FE(t)1957 4984 y Fs(\))1996 4941 y FF(2)2035 4903 y Fq(\000)2081 4984 y Ft(\026=)p Fv(2)g Fp(\000)g Ft(\013\026=)p Fv(2)g Fp(\000)h Ft(\013)2772 4903 y Fq(\001)1401 5245 y Fs(+)1500 5165 y Fq(\000)1546 5245 y Ft(\013\026)f Fs(+)h Ft(\026=)p Fv(2)f Fs(+)g Ft(\026C)2204 5272 y FF(5)2244 5245 y Ft(=)p Fv(2)g Fs(+)h Ft(\013)2530 5203 y Fr(\000)p FF(1)2632 5245 y Ft(C)2703 5272 y FF(5)2743 5165 y Fq(\001)2789 5245 y Ft(\013)2869 5110 y Fq(Z)2986 5245 y Ft(h)3043 5272 y FE(\013)3100 5245 y Ft(w)3174 5203 y FF(2)3171 5272 y FE(t)3239 5245 y Ft(:)3619 4973 y Fs(\()p Fv(2)p Ft(:)p Fv(18)p Fs(\))p eop %%Page: 8 8 bop 94 -45 a Fo(D)m(amped)38 b(Hyperbolic)f(Equa)-7 b(tions)2180 b Fn(8)94 194 y Fv(It)29 b(is)f(clear)i(that)e(if)h(we)g(choose)f Ft(\013)h Fv(and)g Ft(\026)g Fv(suf)n(\256ciently)e(small)h(\(b)n(ut)h (independent)e(of)i Ft(\021)k Fv(for)c(small)f(enough)94 314 y Ft(\021)t Fv(\),)e(then)e(we)h(get,)g(for)g(some)f(\(lar)n(ge\))i(constant) d Ft(C)1832 341 y FF(6)1873 314 y Fv(,)838 536 y FF(1)p 838 553 35 4 v 838 610 a(2)885 576 y Ft(@)938 602 y FE(t)973 576 y Ft(F)1037 602 y FF(1)1077 576 y Fs(\()p Ft(w)1187 602 y FE(t)1222 576 y Ft(;)17 b(z)1313 602 y FE(t)1348 576 y Fs(\))44 b Fp(\024)g Ft(\013)1633 440 y Fq(Z)1749 576 y Ft(h)1806 602 y FE(\013)1863 435 y Fq(\022)1959 576 y Fp(\000)2070 508 y Ft(\026)p 2070 553 61 4 v 2075 644 a Fv(4)2142 495 y Fq(\000)2188 576 y Ft(z)2238 533 y FF(2)2234 602 y FE(t)2301 576 y Fs(+)22 b(\()p Ft(w)2513 533 y Fr(0)2510 602 y FE(t)2546 576 y Fs(\))2585 533 y FF(2)2624 495 y Fq(\001)2692 576 y Fs(+)h Ft(C)2863 602 y FF(6)2903 576 y Ft(w)2977 533 y FF(2)2974 602 y FE(t)3017 435 y Fq(\023)3115 576 y Ft(:)476 b Fs(\()p Fv(2)p Ft(:)p Fv(19)p Fs(\))94 827 y Fv(Note)25 b(no)n(w)f(that)g(for)h(small)f Ft(\026)k(>)g Fv(0)d(and)g Ft(\021)31 b(>)d Fv(0,)1194 1093 y Ft(\021)1247 1050 y FF(2)1287 1093 y Ft(\026w)s(z)49 b Fp(\024)c Fs(\()p Fv(1)22 b Fp(\000)g Ft(\021)1901 1050 y FF(2)1941 1093 y Fs(\))p Ft(z)2030 1050 y FF(2)2093 1093 y Fs(+)2324 1026 y Ft(\021)2377 983 y FF(2)2416 1026 y Ft(\026)p 2204 1070 392 4 v 2204 1162 a Fv(4)p Fs(\()p Fv(1)g Fp(\000)g Ft(\021)2517 1133 y FF(2)2557 1162 y Fs(\))2608 1093 y Ft(w)2682 1050 y FF(2)2747 1093 y Ft(;)94 1343 y Fv(which)j(is)f(equi)n(v)n(alent)f(to)1135 1612 y Fp(\000)p Ft(z)1262 1569 y FF(2)1347 1612 y Fp(\024)45 b(\000)p Ft(\021)1599 1569 y FF(2)1640 1612 y Ft(z)1690 1569 y FF(2)1752 1612 y Fp(\000)23 b Ft(\021)1905 1569 y FF(2)1945 1612 y Ft(\026w)s(z)k Fs(+)2383 1544 y Ft(\021)2436 1501 y FF(2)2476 1544 y Ft(\026)p 2263 1589 V 2263 1680 a Fv(4)p Fs(\()p Fv(1)22 b Fp(\000)g Ft(\021)2576 1651 y FF(2)2616 1680 y Fs(\))2667 1612 y Ft(w)2741 1569 y FF(2)2806 1612 y Ft(:)94 1862 y Fv(Therefore,)k(\(2.19\))f(leads)f(to) 532 2065 y FF(1)p 532 2082 35 4 v 532 2139 a(2)579 2105 y Ft(@)632 2132 y FE(t)667 2105 y Ft(F)731 2132 y FF(1)771 2105 y Fs(\()p Ft(w)881 2132 y FE(t)916 2105 y Ft(;)17 b(z)1007 2132 y FE(t)1042 2105 y Fs(\))44 b Fp(\024)h(\000)1336 2037 y Ft(\013\026)p 1336 2082 125 4 v 1373 2173 a Fv(4)1489 1969 y Fq(Z)1605 2105 y Ft(h)1662 2132 y FE(\013)1719 2024 y Fq(\000)1765 2105 y Ft(\021)1818 2062 y FF(2)1858 2105 y Ft(z)1908 2062 y FF(2)1904 2132 y FE(t)1970 2105 y Fs(+)23 b(\()p Ft(w)2183 2062 y Fr(0)2180 2132 y FE(t)2215 2105 y Fs(\))2254 2062 y FF(2)2316 2105 y Fs(+)g Ft(\021)2469 2062 y FF(2)2508 2105 y Ft(\026w)2639 2132 y FE(t)2675 2105 y Ft(z)2721 2132 y FE(t)2756 2024 y Fq(\001)2824 2105 y Fs(+)g Ft(C)2995 2132 y FF(7)3051 1969 y Fq(Z)3168 2105 y Ft(h)3225 2132 y FE(\013)3282 2105 y Ft(w)3356 2062 y FF(2)3353 2132 y 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Fs(\()p Fg(u)2340 5318 y FE(t)2374 5291 y Ft(;)17 b Fg(v)2470 5318 y FE(t)2505 5291 y Fs(\))24 b Ft(:)1073 b Fs(\()p Fv(3)p Ft(:)p Fv(5)p Fs(\))p eop %%Page: 10 10 bop 94 -45 a Fo(D)m(amped)38 b(Hyperbolic)f(Equa)-7 b(tions)2130 b Fn(10)316 194 y Fv(W)-8 b(e)33 b(ne)o(xt)e(de\256ne)i(a)f(space)h(in)f (which)g(both)f(the)h(functions)f(and)i(their)f(deri)n(v)n(ati)n(v)o(es)d (are)k(controlled.)94 314 y(This)25 b(is)f(the)h(space)g(in)f(which)h(our)f (\256nal)h(bounds)f(will)g(be)h(spelled)f(out.)35 b(Using)24 b(still)g(the)g(cutof)n(f)h(function)1508 614 y Ft(h)1565 641 y FE(\016)1610 614 y Fs(\()p Ft(x)p Fs(\))43 b(=)2134 546 y Fv(1)p 1922 591 474 4 v 1922 682 a Fs(\()p Fv(1)22 b Fs(+)g Ft(\016)2180 653 y FF(2)2220 682 y Ft(x)2277 653 y FF(2)2317 682 y Fs(\))2356 653 y FF(2)2432 614 y Ft(;)94 918 y Fv(we)k(de\256ne)f(the)g (norm)686 1208 y Fp(k)p Fs(\()p Ft(u;)17 b(v)t Fs(\))p Fp(k)1018 1165 y FF(2)1018 1234 y FE(h)1065 1252 y Fl(\016)1102 1234 y FE(;)p FF(2)1210 1208 y Fs(=)1332 1072 y 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FE(h)2310 2031 y Fl(\016)2347 2013 y FE(;)p FF(2)2439 1986 y Ft(<)28 b Fp(1)2644 1845 y Fq(\033)2743 1986 y Ft(:)898 b Fs(\()p Fv(3)p Ft(:)p Fv(7)p Fs(\))316 2369 y Fv(W)-8 b(e)25 b(also)f(need)h(the)g(translates)f(of)h Ft(h)1582 2395 y FE(\016)1651 2369 y Fv(to)f(de\256ne)i(a)f(translation)e(in)l(v)n (ariant)h(topology)-6 b(.)34 b(Let)24 b Ft(h)3532 2395 y FE(\016)n(;\030)3634 2369 y Fs(\()p Ft(x)p Fs(\))k(=)94 2488 y Ft(h)151 2515 y FE(\016)196 2488 y Fs(\()p Ft(x)21 b Fp(\000)i Ft(\030)5 b Fs(\))p Fv(.)94 2680 y Fu(De\014nition)36 b(3.2.)h Fh(The)25 b(space)g Fm(H)1360 2637 y FF(2)1359 2707 y FE(\016)n(;)p FF(loc)1531 2680 y Fh(is)f(de\256ned)i (by)1057 3009 y Fm(H)1147 2966 y FF(2)1146 3035 y FE(\016)n(;)p FF(loc)1337 3009 y Fs(=)1459 2868 y Fq(\032)1534 3009 y Fs(\()p Ft(u;)17 b(v)t Fs(\))1789 2864 y Fq(\014)1789 2924 y(\014)1789 2984 y(\014)1789 3043 y(\014)1864 3009 y Fv(sup)1914 3095 y FE(\030)2019 3009 y Fp(k)p Fs(\()p Ft(u;)g(v)t Fs(\))p Fp(k)2351 3035 y FE(h)2398 3053 y 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FE(;)p FF(1)3617 3977 y Fv(can)25 b(be)94 4097 y(e)o(xtended)g (immediately)e(to)h(similar)g(bounds)f(on)i Fp(k)p Fs(\()p Fg(u)1983 4124 y FE(t)2017 4097 y Ft(;)17 b Fg(v)2113 4124 y FE(t)2147 4097 y Fs(\))p Fp(k)2236 4124 y FE(h)2283 4142 y Fl(\016)2322 4124 y FE(;)p FF(2)2410 4097 y Fv(and)25 b(we)g(get)g(from)f (\(3.5\):)94 4289 y Fu(Theorem)38 b(3.3.)e Fh(There)25 b(is)g(a)g(constant)f Ft(C)1641 4315 y FF(11)1741 4289 y Fh(such)g(that)h(if)f Fs(\()p Fg(u)2299 4315 y FE(t)2334 4289 y Ft(;)17 b Fg(v)2430 4315 y FE(t)2464 4289 y Fs(\))25 b Fh(solv)o(es)e(\(3.1\))i(then)1230 4542 y Fp(k)p Fs(\()p Fg(u)1371 4569 y FE(t)1405 4542 y Ft(;)17 b Fg(v)1501 4569 y FE(t)1536 4542 y Fs(\))p Fp(k)1625 4499 y FF(2)1625 4569 y FE(h)1672 4587 y Fl(\016)1710 4569 y FE(;)p FF(2)1818 4542 y Fp(\024)45 b Ft(e)1986 4499 y FE(C)2043 4517 y Fi(11)2098 4499 y FE(t)2133 4542 y Fp(k)p Fs(\()p Fg(u)2274 4569 y FF(0)2313 4542 y Ft(;)17 b Fg(v)2409 4569 y FF(0)2448 4542 y Fs(\))p Fp(k)2537 4499 y FF(2)2537 4569 y FE(h)2584 4587 y Fl(\016)2622 4569 y FE(;)p FF(2)2711 4542 y Ft(:)930 b Fs(\()p Fv(3)p Ft(:)p Fv(9)p Fs(\))p eop %%Page: 11 11 bop 94 -45 a Fo(D)m(amped)38 b(Hyperbolic)f(Equa)-7 b(tions)2130 b Fn(11)94 194 y Fw(4.)46 b(Momen)l(tum)f(lo)t(calization)316 503 y Fv(Let)25 b Ft(m)i Fv(be)f(a)g(bounded)e(positi)n(v)o(e)g(function)h (of)g Ft(x)h Fv(which)f(decays)h(f)o(aster)g(than)g(an)o(y)f(in)l(v)o(erse)g (po)n(wer)g(as)94 623 y Fp(j)p Ft(x)p Fp(j)i(!)h(1)p Fv(.)35 b(W)-8 b(e)25 b(de\256ne)h(the)f(con)l(v)n(olution)d(operator)j Ft(M)2041 649 y FE(a)2114 623 y Fv(by)1188 800 y Fq(\000)1233 881 y Ft(M)1330 908 y FE(a)1378 881 y Ft(f)1438 800 y Fq(\001)1483 881 y Fs(\()p Ft(x)p Fs(\))44 b(=)1784 745 y Fq(Z)1880 881 y Fv(d)p Ft(y)20 b(m)2086 800 y Fq(\000)2132 881 y Ft(a)p Fs(\()p Ft(x)h Fp(\000)i Ft(y)t Fs(\))2494 800 y Fq(\001)2539 881 y Ft(f)11 b Fs(\()p Ft(y)t Fs(\))23 b Ft(:)888 b Fs(\()p Fv(4)p Ft(:)p Fv(1)p Fs(\))94 1129 y Fv(Let)25 b(again)1508 1295 y Ft(h)1565 1322 y FE(\016)1610 1295 y Fs(\()p Ft(x)p Fs(\))43 b(=)2134 1228 y Fv(1)p 1922 1272 474 4 v 1922 1364 a Fs(\()p Fv(1)22 b Fs(+)g Ft(\016)2180 1335 y FF(2)2220 1364 y Ft(x)2277 1335 y FF(2)2317 1364 y Fs(\))2356 1335 y FF(2)2432 1295 y Ft(:)1209 b Fs(\()p Fv(4)p Ft(:)p Fv(2)p Fs(\))94 1661 y Fu(Lemma)43 b(4.1.)48 b Fh(There)29 b(is)g(a)g(constant)f Ft(C)1593 1688 y FF(12)1697 1661 y Fh(such)g(that)h(if)g Ft(\016)37 b(>)d Fv(0)28 b Fh(and)h Ft(a)k(>)h Fv(0)p Fh(,)c(then)e(the)h(operator)g Ft(M)3731 1688 y FE(a)3808 1661 y Fh(is)94 1781 y(bounded)c(on)f Fv(L)648 1738 y FF(2)688 1781 y Fs(\()p Ft(h)784 1807 y FE(\016)841 1781 y Fv(d)p Ft(x)16 b Fs(\))p Fh(,)25 b(with)f(norm)g(bounded)g(by)1512 2092 y Fp(k)p Ft(M)1659 2119 y FE(a)1706 2092 y Fp(k)1756 2119 y FE(\016)1844 2092 y Fp(\024)45 b Ft(C)2037 2119 y FF(12)2124 2019 y Fv(1)22 b Fs(+)2310 1979 y FE(\016)2349 1937 y Fi(2)p 2307 1996 74 4 v 2307 2053 a FE(a)2350 2033 y Fi(2)p 2124 2069 269 4 v 2232 2160 a Ft(a)2429 2092 y(:)1212 b Fs(\()p Fv(4)p Ft(:)p 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Fv(0)d(a)g(bound)416 4127 y Fq(Z)471 4353 y Fr(j)p FE(y)s Fr(j)p FE(<)p FF(2)p Fr(j)p FE(x)p Fr(j)603 4262 y Fv(d)p Ft(y)734 4188 y(h)791 4137 y FF(1)p FE(=)p FF(2)791 4218 y FE(\016)907 4188 y Fs(\()p Ft(x)p Fs(\))p Fp(j)p Ft(m)1157 4108 y Fq(\000)1202 4188 y Ft(a)p Fs(\()p Ft(x)d Fp(\000)g Ft(y)t Fs(\))1564 4108 y Fq(\001)1609 4188 y Fp(j)p 734 4239 903 4 v 1012 4332 a Ft(h)1069 4359 y FE(\016)1113 4332 y Fs(\()p Ft(y)t Fs(\))1244 4303 y FF(1)p FE(=)p FF(2)1649 4262 y Fp(j)p Ft(f)11 b Fs(\()p Ft(y)t Fs(\))p Fp(j)42 b(\024)j Fv(4)p Ft(K)2195 4289 y FE(`)2249 4127 y Fq(Z)2304 4353 y Fr(j)p FE(y)s Fr(j)p FE(<)p FF(2)p Fr(j)p FE(x)p Fr(j)2486 4262 y Fv(d)p Ft(y)2905 4195 y Fv(1)p 2617 4239 627 4 v 2617 4330 a Fs(\()p Fv(1)22 b Fs(+)g Ft(a)p Fp(j)p Ft(x)g Fp(\000)g Ft(y)t Fp(j)p Fs(\))3206 4302 y FE(`)3255 4262 y Fp(j)p Ft(f)11 b Fs(\()p Ft(y)t Fs(\))p Fp(j)23 b Ft(:)94 4557 y Fv(When)37 b Ft(`)44 b(>)g Fv(1,)39 b(then)e Ft(a=)p Fs(\()p Fv(1)27 b Fs(+)j Ft(a)p Fp(j)p Ft(x)p Fp(j)p 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FE(\016)3849 618 y Fv(.)94 737 y(Combining)24 b(the)g(tw)o(o)h(pieces)g(completes)f(the)g (proof)h(of)g(Lemma)f(4.1.)p 3615 820 85 2 v 3615 900 2 81 v 3698 900 V 3615 902 85 2 v 316 1117 a(W)-8 b(e)23 b(need)f(later)h(the)f (follo)n(wing)e(v)n(ariant)i(of)h(this)e(result:)34 b(Let)22 b Ft(#)28 b Fs(=)g Ft(#)p Fs(\()p Ft(k)s Fs(\))22 b Fv(be)g(a)h(smooth)e (characteristic)94 1236 y(function)k(which)h(equals)f(1)g(for)h Fp(j)p Ft(k)s Fp(j)j(\024)g Fv(1)c(and)h(0)g(for)g Fp(j)p Ft(k)s Fp(j)i Ft(>)h Fv(2.)38 b(Let)26 b Ft(q)2489 1263 y FE(a)2538 1236 y Fs(\()p Ft(x)p Fs(\))i(=)h Ft(a)2873 1210 y Fs(~)2860 1236 y Ft(#)p Fs(\()p Ft(ax)p Fs(\))p Fv(,)c(with)3372 1210 y Fs(~)3360 1236 y Ft(#)g Fv(the)h(in)l(v)o(erse)94 1356 y(F)o(ourier)f (transform)f(of)h Ft(#)p Fv(.)36 b(F)o(or)25 b Ft(a)i(>)h Fv(0,)d(let)g Fm(Q)1689 1383 y FE(a)1762 1356 y Fv(be)g(the)f(con)l(v)n(olution)f(operator) i(de\256ned)g(by)1272 1550 y Fq(\000)1318 1631 y Fm(Q)1384 1658 y FE(a)1432 1631 y Ft(f)1492 1550 y Fq(\001)1537 1631 y Fs(\()p Ft(x)p Fs(\))44 b(=)1838 1495 y Fq(Z)1934 1631 y Fv(d)p Ft(y)20 b(q)2097 1658 y FE(a)2146 1631 y Fs(\()p Ft(x)h Fp(\000)i Ft(y)t Fs(\))p Ft(f)11 b Fs(\()p Ft(y)t Fs(\))23 b Ft(:)972 b Fs(\()p Fv(4)p Ft(:)p Fv(5)p Fs(\))94 1922 y Fv(This)27 b(operator)h(is)f(a)h(substitute)e(for)h(a)h(projection)f(onto)g(momenta)f (less)h(than)h Ft(a)p Fv(.)43 b(Setting)27 b Ft(m)p Fs(\()p Ft(x)p Fs(\))32 b(=)3669 1896 y(~)3656 1922 y Ft(#)p Fs(\()p Ft(x)p Fs(\))p Fv(,)94 2042 y(we)26 b(get)e(from)h(Lemma)f(4.1:)94 2234 y Fu(Corollary)41 b(4.2.)i Fh(There)27 b(is)g(a)g(constant)f Ft(C)1681 2260 y FF(12)1783 2234 y Fh(such)h(that)f(if)h Ft(\016)35 b(>)c Fv(0)c Fh(and)g Ft(a)j(>)h Fv(0)c Fh(then)g(the)f(operator)h Fm(Q)3732 2260 y FE(a)3808 2234 y Fh(is)94 2353 y(bounded)e(on)f Fv(L)648 2310 y FF(2)688 2353 y Fs(\()p Ft(h)784 2380 y FE(\016)841 2353 y Fv(d)p Ft(x)16 b Fs(\))p Fh(,)25 b(with)f(norm)g(bounded)g(by)1484 2650 y Fp(k)p Fm(Q)1600 2677 y FE(a)1648 2650 y Fp(k)1698 2677 y FE(\016)1785 2650 y Fp(\024)45 b Ft(C)1978 2677 y FF(12)2053 2569 y Fq(\000)2099 2650 y Fv(1)22 b Fs(+)2285 2583 y Ft(\016)2333 2540 y FF(2)p 2282 2627 93 4 v 2282 2718 a Ft(a)2335 2689 y FF(2)2387 2569 y Fq(\001)2457 2650 y Ft(:)1184 b Fs(\()p Fv(4)p Ft(:)p Fv(6)p Fs(\))94 3198 y Fw(5.)46 b(High)f(momen)l(tum)h(b)t(ounds)316 3507 y Fv(W)-8 b(e)24 b(consider)g(again)g(the)f(function)h Ft(h)1631 3534 y FE(\016)1699 3507 y Fv(as)g(de\256ned)h(in)e(\(4.2\),)i(and) f(we)g(study)f(functions)g Ft(u)h Fv(for)h(which)94 3546 y Fq(R)158 3626 y Fv(d)p Ft(x)16 b(h)338 3653 y FE(\016)382 3626 y Fs(\()p Ft(x)p Fs(\))517 3546 y Fq(\000)562 3626 y Fp(j)p Ft(u)p Fs(\()p Ft(x)p Fs(\))p Fp(j)810 3584 y FF(2)866 3626 y Fs(+)i Fp(j)p Ft(u)1046 3584 y Fr(0)1074 3626 y Fs(\()p Ft(x)p Fs(\))p Fp(j)1237 3584 y FF(2)1275 3546 y Fq(\001)1349 3626 y Ft(<)28 b Fp(1)p Fv(.)35 b(Such)23 b(functions)e(ha)n(v)o(e)i(a)f(F)o (ourier)h(transform)28 b Fs(~)-56 b Ft(u)23 b Fv(in)f(the)h(sense)f(of)94 3746 y(tempered)j(distrib)n(utions,)d(and)j(we)g(de\256ne)g(no)n(w)374 4025 y Fp(K)450 4052 y FE(a)542 4025 y Fs(=)664 3884 y Fq(\032)739 4025 y Ft(u)821 3880 y Fq(\014)821 3940 y(\014)821 4000 y(\014)821 4060 y(\014)895 3889 y(Z)992 4025 y Fv(d)p Ft(x)16 b(h)1172 4052 y FE(\016)1216 4025 y Fs(\()p Ft(x)p Fs(\))1351 3944 y Fq(\000)1396 4025 y Fp(j)p Ft(u)p Fs(\()p Ft(x)p Fs(\))p Fp(j)1644 3982 y FF(2)1705 4025 y Fs(+)23 b Fp(j)p Ft(u)1890 3982 y Fr(0)1917 4025 y Fs(\()p Ft(x)p Fs(\))p Fp(j)2080 3982 y FF(2)2119 3944 y Fq(\001)2193 4025 y Ft(<)28 b Fp(1)c Fv(and)h(supp)30 b Fs(~)-56 b Ft(u)28 b Fp(2)g Fu(R)22 b Fp(n)g Fs(\()p Fp(\000)p Ft(a;)17 b(a)p Fs(\))3469 3884 y Fq(\033)3567 4025 y Ft(:)94 4300 y Fv(Thus,)i(apart)g(from)f(not)g(being)g(de\256ned)h(as)f(a)h(function,)f(the) h(F)o(ourier)f(transform)24 b Fs(~)-56 b Ft(u)18 b Fv(of)h(a)f Ft(u)28 b Fp(2)g(K)3357 4327 y FE(a)3424 4300 y Fv(has)18 b(support)94 4420 y(at)26 b(momenta)f(lar)n(ger)h(than)g Ft(a)p Fv(.)38 b(If)27 b Ft(h)1309 4447 y FE(\016)1353 4420 y Fs(\()p Ft(x)p Fs(\))h Fp(\021)i Fv(1)c(and)f Ft(u)1925 4377 y Fr(0)1982 4420 y Fp(2)30 b Fv(L)2139 4377 y FF(2)2179 4420 y Fs(\()p Fv(d)p Ft(x)p Fs(\))p Fv(,)25 b(then,)h(ob)o(viously)-6 b(,)23 b(for)j Ft(u)j Fp(2)g(K)3472 4447 y FE(a)3520 4420 y Fv(,)d(one)g(has)94 4459 y Fq(R)178 4539 y Fp(j)p Ft(u)p Fp(j)291 4497 y FF(2)379 4539 y Fp(\024)49 b Ft(a)558 4497 y Fr(\000)p FF(2)676 4459 y Fq(R)759 4539 y Fp(j)p Ft(u)844 4497 y Fr(0)872 4539 y Fp(j)900 4497 y FF(2)939 4539 y Fv(.)d(The)28 b(follo)n(wing)e(proposition)g(whose)i (ele)o(gant)f(proof)h(w)o(as)g(kindly)f(pro)o(vided)f(by)94 4659 y FG(H.)f(Epstein)p Fv(,)f(sho)n(ws)f(that)i(the)f(cutof)n(f)h(function) f Ft(h)1832 4686 y FE(\016)1901 4659 y Fv(does)g(only)g(moderately)g(change)h (this)f(property)-6 b(.)94 4851 y Fu(Prop)s(osition)40 b(5.1.)h Fh(Assume)25 b(that)h Ft(a)k(>)g Fv(0)c Fh(and)g Ft(\016)34 b(>)d Fv(0)p Fh(.)40 b(There)26 b(is)g(a)h(constant)e Ft(\027)6 b Fs(\()p Ft(a;)17 b(\016)t Fs(\))29 b Ft(<)i Fp(1)26 b Fh(such)g(that)94 4970 y(for)g(all)e Ft(u)k Fp(2)g(K)615 4997 y FE(a)688 4970 y Fh(one)d(has)f(the)h(inequality)972 5110 y Fq(Z)1068 5245 y Fv(d)p Ft(x)16 b(h)1248 5272 y FE(\016)1293 5245 y Fs(\()p Ft(x)p Fs(\))p Fp(j)p Ft(u)p Fs(\()p Ft(x)p Fs(\))p Fp(j)1676 5203 y FF(2)1758 5245 y Fp(\024)45 b Ft(\027)6 b Fs(\()p Ft(a;)17 b(\016)t Fs(\))2176 5110 y Fq(Z)2271 5245 y Fv(d)p Ft(x)f(h)2451 5272 y FE(\016)2495 5245 y Fs(\()p Ft(x)p Fs(\))p Fp(j)p Ft(u)2715 5203 y Fr(0)2742 5245 y Fs(\()p Ft(x)p Fs(\))p Fp(j)2905 5203 y FF(2)2969 5245 y Ft(:)672 b Fs(\()p Fv(5)p Ft(:)p Fv(1)p Fs(\))p eop %%Page: 13 13 bop 94 -45 a Fo(D)m(amped)38 b(Hyperbolic)f(Equa)-7 b(tions)2130 b Fn(13)94 194 y Fh(There)26 b(is)e(a)h(constant)f Ft(C)940 221 y FF(14)1043 194 y Ft(>)k Fv(0)d Fh(such)f(that)h(one)f(can)h(choose)1458 561 y Ft(\027)6 b Fs(\()p Ft(a;)17 b(\016)t Fs(\))43 b(=)i Ft(C)1973 588 y FF(14)2060 488 y Fs(\()p Fv(1)22 b Fs(+)2285 448 y FE(\016)2324 406 y Fi(2)p 2282 465 74 4 v 2282 522 a FE(a)2325 502 y Fi(2)2367 488 y Fs(\))2406 445 y FF(2)p 2060 538 386 4 v 2207 629 a Ft(a)2260 601 y FF(2)2483 561 y Ft(:)1158 b Fs(\()p Fv(5)p Ft(:)p Fv(2)p Fs(\))94 998 y Fu(Remark.)39 b Fv(W)-8 b(e)25 b(will)f(need)h(the)g(result)f(only)g(for)h Ft(\016)32 b(<)c(a)p Fv(,)c(so)h(that)f(we)h(can)g(use)g(the)g(simpler)f (bound)1651 1306 y Ft(\027)6 b Fs(\()p Ft(a;)17 b(\016)t Fs(\))43 b Fp(\024)2107 1239 y Ft(C)2178 1266 y FF(15)p 2107 1283 146 4 v 2134 1374 a Ft(a)2187 1346 y FF(2)2290 1306 y Ft(:)1351 b Fs(\()p Fv(5)p Ft(:)p Fv(3)p Fs(\))94 1686 y Fu(Pro)s(of.)38 b Fv(Let)23 b Ft(#)h Fv(be)f(a)h(smooth)e(characteristic)i(function)e(which)h (equals)g(1)h(for)f Fp(j)p Ft(k)s Fp(j)k(\024)3105 1646 y FF(1)p 3105 1663 35 4 v 3105 1720 a(2)3176 1686 y Fv(and)c(0)g(for)h Fp(j)p Ft(k)s Fp(j)j Ft(>)h Fv(1.)94 1805 y(Let)23 b Ft(u)28 b Fp(2)g(K)505 1832 y FE(a)553 1805 y Fv(.)35 b(Since)24 b(the)f(distrib)n (ution)j Fs(~)-56 b Ft(u)23 b Fv(has)g(support)f(in)g(the)h(complement)f(of)h (the)f(interv)n(al)g Fs(\()p Fp(\000)p Ft(a;)17 b(a)p Fs(\))p Fv(,)22 b(and)94 1939 y(the)j(F)o(ourier)g(transform)985 1914 y Fq(e)970 1939 y Ft(u)1027 1910 y Fr(0)1080 1939 y Fv(of)g(the)g(deri)n(v)n (ati)n(v)o(e)d(is)i Ft(ik)10 b Fs(~)-57 b Ft(u)q Fs(\()p Ft(k)s Fs(\))p Fv(,)24 b(we)h(see)g(that)1417 2257 y Fs(~)-57 b Ft(u)p Fs(\()p Ft(k)s Fs(\))44 b(=)1778 2189 y Fv(1)22 b Fp(\000)g Ft(#)p Fs(\()p Ft(k)s(=a)p Fs(\))p 1778 2234 466 4 v 1966 2325 a Ft(ik)2303 2232 y Fq(e)2288 2257 y Ft(u)2345 2228 y Fr(0)2373 2257 y Fs(\()p Ft(k)s Fs(\))j Ft(:)94 2543 y Fv(De\256ne)h(ne)o(xt)1590 2718 y Fs(~)-68 b Ft(m)p Fs(\()p Ft(k)s Fs(\))44 b Fp(\021)1969 2650 y Fv(1)22 b Fp(\000)h Ft(#)p Fs(\()p Ft(k)s Fs(\))p 1969 2695 363 4 v 2106 2786 a Ft(ik)2369 2718 y(:)94 2962 y Fv(The)e(\(in)l(v)o (erse\))g(F)o(ourier)g(transform,)f Ft(m)p Fv(,)j(of)39 b Fs(~)-69 b Ft(m)22 b Fv(decreases)f(f)o(aster)h(than)e(an)o(y)g(po)n(wer)h(of)g Fp(j)p Ft(x)p Fp(j)f Fv(at)h(in\256nity)-6 b(.)33 b(If)21 b(we)94 3081 y(let)26 b Ft(m)307 3108 y FE(a)356 3081 y Fs(\()p Ft(x)p Fs(\))i(=)i Ft(m)p Fs(\()p Ft(ax)p Fs(\))p Fv(,)c(then)44 b Fs(~)-68 b Ft(m)1237 3108 y FE(a)1285 3081 y Fs(\()p Ft(k)s Fs(\))29 b(=)49 b(~)-69 b Ft(m)p Fs(\()p Ft(k)s(=a)p Fs(\))p Ft(=a:)24 b Fv(Thus,)i(it)f(follo)n(ws)g(with)g(the)h(notation)e(of)i (Section)g(4)94 3201 y(that)1513 3342 y Ft(u)p Fs(\()p Ft(x)p Fs(\))43 b(=)1870 3261 y Fq(\000)1916 3342 y Ft(M)2013 3369 y FE(a)2061 3342 y Fs(\()p Ft(u)2157 3299 y Fr(0)2184 3342 y Fs(\))2223 3261 y Fq(\001)2269 3342 y Fs(\()p Ft(x)p Fs(\))24 b Ft(:)94 3553 y Fv(By)i(Lemma)e(4.1,)g(we)h(conclude)g(that)321 3914 y Fp(k)p Ft(u)p Fp(k)478 3872 y FF(2)478 3941 y FE(\016)565 3914 y Fs(=)687 3779 y Fq(Z)803 3914 y Ft(h)860 3941 y FE(\016)904 3914 y Fp(j)p Ft(u)p Fp(j)1017 3872 y FF(2)1101 3914 y Fs(=)1223 3779 y Fq(Z)1339 3914 y Ft(h)1396 3941 y FE(\016)1440 3914 y Fp(j)p Ft(M)1565 3941 y FE(a)1613 3914 y Fs(\()p Ft(u)1709 3872 y Fr(0)1736 3914 y Fs(\))p Fp(j)1803 3872 y FF(2)1887 3914 y Fs(=)45 b Fp(k)p Ft(M)2156 3941 y FE(a)2203 3914 y Fs(\()p Ft(u)2299 3872 y Fr(0)2327 3914 y Fs(\))p Fp(k)2416 3872 y FF(2)2416 3941 y FE(\016)2504 3914 y Fp(\024)f Ft(C)2703 3872 y FF(2)2696 3941 y(12)2783 3841 y Fs(\()p Fv(1)22 b Fs(+)3008 3802 y FE(\016)3047 3759 y Fi(2)p 3005 3818 74 4 v 3005 3876 a FE(a)3048 3856 y Fi(2)3091 3841 y Fs(\))3130 3798 y FF(2)p 2783 3892 386 4 v 2930 3983 a Ft(a)2983 3954 y FF(2)3198 3779 y Fq(Z)3314 3914 y Ft(h)3371 3941 y FE(\016)3415 3914 y Fp(j)p Ft(u)3500 3872 y Fr(0)3528 3914 y Fp(j)3556 3872 y FF(2)3620 3914 y Ft(;)94 4221 y Fv(and)j(the)g(claim)f(\(5.2\))h(follo)n(ws.)p 3615 4312 85 2 v 3615 4393 2 81 v 3698 4393 V 3615 4395 85 2 v eop %%Page: 14 14 bop 94 -45 a Fo(D)m(amped)38 b(Hyperbolic)f(Equa)-7 b(tions)2130 b Fn(14)94 194 y Fw(6.)46 b(The)e(linear)i(high)f(frequency)g(part)316 512 y Fv(W)-8 b(e)19 b(be)o(gin)f(by)g(de\256ning)g(the)h(projection)e(onto)h (high)g(frequencies,)i(on)f(a)g(space)g(with)e(weight)h Ft(h)3590 539 y FE(\016)3634 512 y Fs(\()p Ft(x)p Fs(\))28 b(=)94 631 y(\()p Fv(1)q Fs(+)q Ft(\016)310 588 y FF(2)351 631 y Ft(x)408 588 y FF(2)448 631 y Fs(\))487 588 y Fr(\000)p FF(2)589 631 y Fv(.)k(W)-8 b(e)16 b(\256rst)g(recall)g(the)g(notion)g(of)g(projection)g (onto)g(lo)m(w)f(frequencies)h(from)f(Section)h(4.)26 b(Denote)94 764 y(by)231 738 y Fq(e)220 764 y Ft(#)h Fv(a)f(smooth)f(characteristic)i (function,)e(equal)i(to)e(1)i(for)f Fp(j)p Ft(k)s Fp(j)j(\024)h Fv(1)d(and)f(v)n(anishing)e(for)j Fp(j)p Ft(k)s Fp(j)i Ft(>)h Fv(2.)40 b(W)-8 b(e)27 b(\256x)94 884 y(no)n(w)d(a)i(\(lar)n(ge\))f(cutof)n (f)g(scale)g Ft(k)1183 911 y Fr(\003)1253 884 y Fv(and)g(we)g(de\256ne)g(as)g (before)1547 1132 y Ft(q)1591 1159 y FE(k)1633 1176 y Fe(\003)1680 1132 y Fs(\()p Ft(x)p Fs(\))44 b(=)h Ft(k)2033 1159 y Fr(\003)2078 1132 y Ft(#)p Fs(\()p Ft(k)2228 1159 y Fr(\003)2274 1132 y Ft(x)p Fs(\))24 b Ft(;)94 1379 y Fv(and)1232 1454 y Fq(\000)1277 1534 y Fm(Q)1343 1561 y FE(k)1385 1579 y Fe(\003)1432 1534 y Ft(f)1492 1454 y Fq(\001)1537 1534 y Fs(\()p Ft(x)p Fs(\))44 b(=)1838 1399 y Fq(Z)1934 1534 y Fv(d)p Ft(y)20 b(q)2097 1561 y FE(k)2139 1579 y Fe(\003)2186 1534 y Fs(\()p Ft(x)i Fp(\000)h Ft(y)t Fs(\))p Ft(f)11 b Fs(\()p Ft(y)t Fs(\))22 b Ft(:)932 b Fs(\()p Fv(6)p Ft(:)p Fv(1)p Fs(\))94 1824 y Fv(In)29 b(Corollary)f(4.2,)g (we)h(sho)n(wed)e(that)h(on)g(L)1636 1781 y FF(2)1675 1824 y Fs(\()p Ft(h)1771 1851 y FE(\016)1829 1824 y Fv(d)p Ft(x)16 b Fs(\))p Fv(,)29 b(the)f(operator)g Fm(Q)2621 1851 y FE(k)2663 1869 y Fe(\003)2738 1824 y Fv(is)g(bounded)f(by)h Ft(C)3403 1851 y FF(12)3478 1744 y Fq(\000)3524 1824 y Fv(1)22 b Fs(+)3715 1785 y FE(\016)3754 1742 y Fi(2)p 3707 1801 85 4 v 3707 1859 a FE(k)3751 1839 y Fi(2)3749 1886 y Fe(\003)3803 1744 y Fq(\001)3849 1824 y Fv(.)94 1963 y(Therefore,)k(the)f(projection)f(onto)g(high)g(momenta) 1634 2210 y Fm(P)1700 2237 y FE(k)1742 2255 y Fe(\003)1833 2210 y Fs(=)45 b Fv(1)22 b Fp(\000)h Fm(Q)2193 2237 y FE(k)2235 2255 y Fe(\003)2307 2210 y Ft(;)1334 b Fs(\()p Fv(6)p Ft(:)p Fv(2)p Fs(\))94 2458 y Fv(is)18 b(also)g(bounded)g(on)g(that)g(space.)34 b(Henceforth,)19 b(we)g(shall)f(assume)f Ft(\016)32 b(<)c(k)2664 2485 y Fr(\003)2710 2458 y Fv(,)19 b(and)g(thus)e(we)i(get)f(immediately)94 2578 y(the)25 b(bound)94 2769 y Fu(Lemma)39 b(6.1.)d Fh(There)25 b(is)g(a)g(constant)f Ft(C)1561 2796 y FF(16)1660 2769 y Fh(such)h(that)f(if) h Ft(k)2180 2796 y Fr(\003)2253 2769 y Ft(>)k(\016)i(>)d Fv(0)p Fh(,)d(then)g(the)f(operator)h Fm(P)3405 2796 y FE(k)3447 2814 y Fe(\003)3519 2769 y Fh(satis\256es)1653 3017 y Fp(k)p Fm(P)1769 3044 y FE(k)1811 3062 y Fe(\003)1858 3017 y Fp(k)1908 3044 y FE(\016)1995 3017 y Fp(\024)45 b Ft(C)2188 3044 y FF(16)2288 3017 y Ft(;)1353 b Fs(\()p Fv(6)p Ft(:)p Fv(3)p Fs(\))94 3290 y Fh(as)25 b(a)h(map)e(on)h Fv(L)654 3247 y FF(2)693 3290 y Fs(\()p Ft(h)789 3317 y FE(\016)846 3290 y Fv(d)p Ft(x)17 b Fs(\))p Fh(.)94 3547 y Fu(Lemma)46 b(6.2.)60 b Fh(There)33 b(is)f(a)h(constant)f Ft(C)1623 3574 y FF(17)1731 3547 y Fh(such)g(that)h (for)g Ft(k)2330 3574 y Fr(\003)2414 3547 y Ft(>)39 b(\016)k(>)c Fv(0)32 b Fh(the)h(operator)g Fm(P)3401 3574 y FE(k)3443 3592 y Fe(\003)3516 3547 y Fp(\010)27 b Fm(P)3686 3574 y FE(k)3728 3592 y Fe(\003)3808 3547 y Fh(is)94 3692 y(bounded)e(in)f(norm)g(by)h Ft(C)996 3718 y FF(17)1096 3692 y Fh(as)f(a)i(map)e(from)h Fm(H)1778 3649 y FF(2)1777 3718 y FE(\016)n(;)p FF(loc)1949 3692 y Fh(to)f(itself.)94 3963 y Fu(Pro)s(of.)42 b Fv(W)-8 b(e)27 b(ha)n(v)o(e)f(already)i(check)o(ed)f(in)f(Lemma)h(6.1)f(that)g Fm(P)2315 3990 y FE(k)2357 4008 y Fe(\003)2431 3963 y Fv(is)h(bounded)f(on)g (L)3082 3920 y FF(2)3122 3963 y Fs(\()p Ft(h)3218 3990 y FE(\016)3275 3963 y Fv(d)p Ft(x)16 b Fs(\))p Fv(.)42 b(Note)27 b(that)94 4082 y Fm(P)160 4109 y FE(k)202 4127 y Fe(\003)281 4082 y Fv(is)j(a)i(con)l (v)n(olution)d(operator)i(and)g(so)g Fm(P)1679 4109 y FE(k)1721 4127 y Fe(\003)1799 4082 y Fv(and)g Ft(@)2027 4109 y FE(x)2108 4082 y Fv(commute,)h(and)f(the)g(e)o(xtension)f(of)h(the)g(result)g(to)94 4227 y Fm(H)184 4184 y FF(2)183 4254 y FE(\016)n(;)p FF(loc)356 4227 y Fv(\(as)25 b(de\256ned)g(in)g(De\256nition)f(3.2\))g(follo)n(ws)f(at)i (once.)p 3615 4314 85 2 v 3615 4395 2 81 v 3698 4395 V 3615 4397 85 2 v 316 4630 a(So)h(f)o(ar)l(,)g(we)g(ha)n(v)o(e)f(ar)n(gued)i(that)e Fm(P)1492 4657 y FE(k)1534 4675 y Fe(\003)1606 4630 y Fv(is)g(bounded.)38 b(W)-8 b(e)26 b(will)f(no)n(w)f(use)i(the)f(high)g(momentum)f(bound)94 4775 y(of)34 b(Section)g(5)g(with)f Ft(a)39 b Fs(=)i Ft(k)1101 4801 y Fr(\003)1147 4775 y Fv(,)36 b(and)d Ft(k)1437 4801 y Fr(\003)1523 4775 y Fp(\024)41 b Ft(\021)1694 4732 y Fr(\000)p FF(1)1830 4775 y Fv(to)33 b(sho)n(w)g(that)g(the)g(semi-group)g(generated)h (by)g(the)f(free)94 4894 y(e)n(v)n(olution)23 b(\(see)i(belo)n(w\))e(is)h(a)h (\(strong\))f(contraction.)35 b(In)24 b(f)o(act,)h(we)g(will)f(sho)n(w)f (that)h(the)g(contraction)g(rate)h(is)1421 5142 y Fm(O)p Fs(\()p Ft(k)1585 5099 y FF(2)1582 5169 y Fr(\003)1628 5142 y Fs(\))f Fv(as)h(long)f(as)h Ft(k)2161 5169 y Fr(\003)2235 5142 y Ft(<)j(\021)2393 5099 y Fr(\000)p FF(1)2520 5142 y Ft(;)1317 5291 y Fm(O)p Fs(\()p Ft(\021)1479 5248 y Fr(\000)p FF(2)1581 5291 y Fs(\))d Fv(for)g(an)o(y)f (cutof)n(f)g Ft(k)2265 5318 y Fr(\003)2339 5291 y Fp(\025)k Ft(\021)2497 5248 y Fr(\000)p FF(1)2624 5291 y Ft(:)p eop %%Page: 15 15 bop 94 -45 a Fo(D)m(amped)38 b(Hyperbolic)f(Equa)-7 b(tions)2130 b Fn(15)94 194 y Fv(This)22 b(beha)n(vior)g(is)f(typical)h(of)g(the)g(mix)o (ed)e(parabolic-hyperbolic)h(problems)g(we)i(consider)e(here,)i(since)f(the) 94 314 y(linearized)j(e)n(v)n(olution,)e(written)h(in)h(momentum)d(space,)j (has)g(the)g(generator)1564 425 y Fq(\022)1793 505 y Fv(0)330 b(1)1654 625 y Fp(\000)p Ft(\021)1784 582 y Fr(\000)p FF(2)1887 625 y Ft(k)1942 582 y FF(2)2081 625 y Fp(\000)p Ft(\021)2211 582 y Fr(\000)p FF(2)2331 425 y Fq(\023)94 812 y Fv(with)25 b(eigen)l(v)n(alues)1390 1008 y Ft(\025)1448 1035 y Fr(\006)1560 1008 y Fs(=)1694 940 y Fp(\000)p Fv(1)d Fp(\006)1943 852 y Fq(p)p 2042 852 472 4 v 2042 940 a Fv(1)g Fp(\000)h Fv(4)p Ft(k)2319 912 y FF(2)2359 940 y Ft(\021)2412 912 y Fr(\000)p FF(2)p 1694 985 821 4 v 2032 1076 a Fv(2)p Ft(\021)2135 1047 y FF(2)2551 1008 y Ft(:)94 1223 y Fv(One)k(can)f(see)g(from)g(the)g(e)o (xpression)e(for)i(the)g(eigen)l(v)n(alues)f(that)h(the)f(real)i(part)f(ne)n (v)o(er)f(gets)h(more)f(ne)o(gati)n(v)o(e)94 1342 y(than)g Fp(\000)p Fm(O)p Fs(\()p Ft(\021)530 1299 y Fr(\000)p FF(2)633 1342 y Fs(\))p Fv(.)35 b(W)-8 b(e)25 b(study)f(no)n(w)f(the)i(properties)f (of)g(the)h(operator)f Fm(L)h Fv(de\256ned)g(as)g(in)f(Eqs.\(3.1\))g(and)g (\(3.2\))94 1462 y(by)1341 1638 y Fm(L)1429 1527 y Fq(\020)1500 1570 y Fg(u)1500 1706 y(v)1563 1527 y Fq(\021)1667 1638 y Fs(=)1789 1497 y Fq(\022)2148 1570 y Fg(v)1874 1706 y Ft(\021)1927 1677 y Fr(\000)p FF(2)2029 1626 y Fq(\000)2075 1706 y Fp(\000)p Fg(v)f Fs(+)f Fg(u)2377 1677 y Fr(00)2428 1626 y Fq(\001)2485 1497 y(\023)2600 1638 y Ft(:)316 1870 y Fv(W)-8 b(e)27 b(introduce)f (parameters)h Ft(\015)36 b(>)31 b Fv(0,)c(and)g Ft(\016)34 b(>)d Fv(0)c(which)f(will)g(be)h(\256x)o(ed)g(later)g(and)g(we)g(consider)f (the)94 1990 y(functional)e Ft(J)10 b Fv(:)323 2212 y Ft(J)54 b Fs(=)45 b Ft(J)609 2239 y FE(h)656 2257 y Fl(\016)695 2239 y FE(;\015)769 2212 y Fs(\()p Fg(u)p Ft(;)17 b Fg(v)p Fs(\))43 b(=)i Fp(k)p Fs(\()p Fg(u)p Ft(;)17 b Fg(v)p Fs(\))p Fp(k)1486 2169 y FF(2)1486 2239 y FE(h)1533 2257 y Fl(\016)1570 2239 y FE(;)p FF(2)1655 2212 y Fs(+)23 b Ft(\021)1808 2169 y FF(2)1848 2212 y Ft(\015)1921 2077 y Fq(Z)2018 2212 y Fv(d)p Ft(x)16 b(h)2198 2239 y FE(\016)2242 2212 y Fs(\()p Ft(x)p Fs(\))2377 2131 y Fq(\000)2422 2212 y Fg(u)p Fs(\()p Ft(x)p Fs(\))p Fg(v)p Fs(\()p Ft(x)p Fs(\))21 b(+)h Fg(u)2967 2169 y Fr(0)2995 2212 y Fs(\()p Ft(x)p Fs(\))p Fg(v)3181 2169 y Fr(0)3208 2212 y Fs(\()p Ft(x)p Fs(\))3343 2131 y Fq(\001)3413 2212 y Ft(;)228 b Fs(\()p Fv(6)p Ft(:)p Fv(4)p Fs(\))94 2458 y Fv(where)26 b(the)e(norm)h Fp(k)p Fs(\()p Fg(u)p Ft(;)17 b Fg(v)p Fs(\))p Fp(k)1071 2485 y FE(h)1118 2503 y Fl(\016)1154 2485 y FE(;)p FF(2)1243 2458 y Fv(w)o(as)25 b(de\256ned)g(in)f(Eq.\(3.6\):)698 2733 y Fp(k)p Fs(\()p Fg(u)p Ft(;)17 b Fg(v)p Fs(\))p Fp(k)1024 2690 y FF(2)1024 2760 y FE(h)1071 2778 y Fl(\016)1108 2760 y FE(;)p FF(2)1216 2733 y Fs(=)1337 2597 y Fq(Z)1454 2733 y Ft(h)1511 2760 y FE(\016)1555 2592 y Fq(\022)1628 2733 y Ft(\021)1681 2690 y FF(2)1721 2652 y Fq(\000)1767 2733 y Fg(v)1818 2690 y FF(2)1880 2733 y Fs(+)22 b(\()p Fg(v)2069 2690 y Fr(0)2097 2733 y Fs(\))2136 2690 y FF(2)2176 2652 y Fq(\001)2243 2733 y Fs(+)h Fg(u)2395 2690 y FF(2)2457 2733 y Fs(+)f Fv(2)p Fs(\()p Fg(u)2697 2690 y Fr(0)2724 2733 y Fs(\))2763 2690 y FF(2)2825 2733 y Fs(+)h(\()p Fg(u)3016 2690 y Fr(00)3066 2733 y Fs(\))3105 2690 y FF(2)3144 2592 y Fq(\023)3243 2733 y Ft(:)94 2987 y Fv(Consider)i(the)g(solution)e Fs(\()p Fg(u)1063 3014 y FE(t)1097 2987 y Ft(;)17 b Fg(v)1193 3014 y FE(t)1227 2987 y Fs(\))25 b Fv(of)g(\(3.2\):)36 b(Then,)24 b(writing)g Ft(J)35 b Fv(for)25 b Ft(J)2507 3014 y FE(h)2554 3032 y Fl(\016)2593 3014 y FE(;\015)2667 2987 y Fv(,)g(we)g(\256nd)189 3223 y FF(1)p 189 3239 35 4 v 189 3297 a(2)236 3262 y Ft(@)289 3289 y FE(t)324 3262 y Ft(J)10 b Fs(\()p Fg(u)480 3289 y FE(t)514 3262 y Ft(;)17 b Fg(v)610 3289 y FE(t)644 3262 y Fs(\))44 b(=)849 3127 y Fq(Z)965 3262 y Ft(h)1022 3289 y FE(\016)1066 3122 y Fq(\022)1140 3262 y Ft(\021)1193 3219 y FF(2)1232 3181 y Fq(\000)1278 3262 y Fg(v)1329 3289 y FE(t)1376 3253 y Fs(_)1364 3262 y Fg(v)1415 3289 y FE(t)1472 3262 y Fs(+)23 b Fg(v)1623 3219 y Fr(0)1623 3289 y FE(t)1670 3253 y Fs(_)1658 3262 y Fg(v)1709 3219 y Fr(0)1709 3289 y FE(t)1744 3181 y Fq(\001)1812 3262 y Fs(+)g Fg(u)1964 3289 y FE(t)2010 3258 y Fs(_)1998 3262 y Fg(u)2050 3289 y FE(t)2107 3262 y Fs(+)g Fv(2)p Fg(u)2309 3219 y Fr(0)2309 3289 y FE(t)2355 3258 y Fs(_)2343 3262 y Fg(u)2395 3219 y Fr(0)2395 3289 y FE(t)2452 3262 y Fs(+)f Fg(u)2603 3219 y Fr(00)2603 3289 y FE(t)2666 3258 y Fs(_)2654 3262 y Fg(u)2706 3219 y Fr(00)2706 3289 y FE(t)2756 3122 y Fq(\023)722 3532 y Fs(+)833 3493 y FF(1)p 833 3509 V 833 3567 a(2)880 3532 y Ft(\021)933 3489 y FF(2)973 3532 y Ft(\015)1047 3397 y Fq(Z)1163 3532 y Ft(h)1220 3559 y FE(\016)1264 3452 y Fq(\000)1310 3532 y Fg(u)1362 3559 y FE(t)1408 3523 y Fs(_)1396 3532 y Fg(v)1447 3559 y FE(t)1504 3532 y Fs(+)h Fg(v)1655 3559 y FE(t)1702 3528 y Fs(_)1690 3532 y Fg(u)1742 3559 y FE(t)1799 3532 y Fs(+)f Fg(u)1950 3489 y Fr(0)1950 3559 y FE(t)1997 3523 y Fs(_)1985 3532 y Fg(v)2036 3489 y Fr(0)2036 3559 y FE(t)2093 3532 y Fs(+)h Fg(v)2244 3489 y Fr(0)2244 3559 y FE(t)2291 3528 y Fs(_)2279 3532 y Fg(u)2331 3489 y Fr(0)2331 3559 y FE(t)2366 3452 y Fq(\001)727 3802 y Fs(=)849 3667 y Fq(Z)965 3802 y Ft(h)1022 3829 y FE(\016)1066 3662 y Fq(\022)1162 3802 y Fp(\000)f Fg(v)1312 3760 y FF(2)1312 3829 y FE(t)1374 3802 y Fs(+)h Fg(v)1525 3829 y FE(t)1560 3802 y Fg(u)1612 3760 y Fr(00)1612 3829 y FE(t)1685 3802 y Fp(\000)f Fs(\()p Fg(v)1874 3760 y Fr(0)1874 3829 y FE(t)1909 3802 y Fs(\))1948 3760 y FF(2)2010 3802 y Fs(+)g Fg(v)2160 3760 y Fr(0)2160 3829 y FE(t)2195 3802 y Fg(u)2247 3760 y Fr(000)2247 3829 y FE(t)2343 3802 y Fs(+)h Fg(u)2495 3829 y FE(t)2529 3802 y Fg(v)2580 3829 y FE(t)2637 3802 y Fs(+)g Fv(2)p Fg(u)2839 3760 y Fr(0)2839 3829 y FE(t)2874 3802 y Fg(v)2925 3760 y Fr(0)2925 3829 y FE(t)2982 3802 y Fs(+)f Fg(u)3133 3760 y Fr(00)3133 3829 y FE(t)3184 3802 y Fg(v)3235 3760 y Fr(00)3235 3829 y FE(t)3286 3662 y Fq(\023)722 4072 y Fs(+)833 4033 y FF(1)p 833 4050 V 833 4107 a(2)880 4072 y Ft(\015)954 3937 y Fq(Z)1070 4072 y Ft(h)1127 4099 y FE(\016)1171 3992 y Fq(\000)1217 4072 y Fp(\000)p Fg(u)1346 4099 y FE(t)1381 4072 y Fg(v)1432 4099 y FE(t)1489 4072 y Fs(+)h Fg(u)1641 4099 y FE(t)1675 4072 y Fg(u)1727 4030 y Fr(00)1727 4099 y FE(t)1800 4072 y Fs(+)f Ft(\021)1952 4030 y FF(2)1992 4072 y Fg(v)2043 4030 y FF(2)2043 4099 y FE(t)2105 4072 y Fp(\000)h Fg(u)2257 4030 y Fr(0)2257 4099 y FE(t)2292 4072 y Fg(v)2343 4030 y Fr(0)2343 4099 y FE(t)2400 4072 y Fs(+)f Fg(u)2551 4030 y Fr(0)2551 4099 y FE(t)2586 4072 y Fg(u)2638 4030 y Fr(000)2638 4099 y FE(t)2734 4072 y Fs(+)g Ft(\021)2886 4030 y FF(2)2926 4072 y Fs(\()p Fg(v)3016 4030 y Fr(0)3016 4099 y FE(t)3051 4072 y Fs(\))3090 4030 y FF(2)3129 3992 y Fq(\001)727 4343 y Fs(=)849 4207 y Fq(Z)965 4343 y Ft(h)1022 4369 y FE(\016)1066 4202 y Fq(\022)1162 4343 y Fp(\000)1261 4262 y Fq(\000)1307 4343 y Fg(v)1358 4300 y FF(2)1358 4369 y FE(t)1420 4343 y Fs(+)h(\()p Fg(v)1610 4300 y Fr(0)1610 4369 y FE(t)1644 4343 y Fs(\))1683 4300 y FF(2)1723 4262 y Fq(\001)1769 4343 y Fs(\()p Fv(1)e Fp(\000)1991 4303 y FF(1)p 1991 4320 V 1991 4377 a(2)2038 4343 y Ft(\021)2091 4300 y FF(2)2131 4343 y Ft(\015)6 b Fs(\))21 b(+)2348 4262 y Fq(\000)2394 4343 y Fg(u)2446 4369 y FE(t)2481 4343 y Fg(v)2532 4369 y FE(t)2589 4343 y Fs(+)h Fg(u)2740 4300 y Fr(0)2740 4369 y FE(t)2775 4343 y Fg(v)2826 4300 y Fr(0)2826 4369 y FE(t)2861 4262 y Fq(\001)2907 4343 y Fs(\()p Fv(1)f Fp(\000)3129 4303 y FF(1)p 3129 4320 V 3129 4377 a(2)3176 4343 y Ft(\015)6 b Fs(\))3273 4202 y Fq(\023)722 4613 y Fp(\000)833 4573 y FF(1)p 833 4590 V 833 4647 a(2)880 4613 y Ft(\015)954 4477 y Fq(Z)1070 4613 y Ft(h)1127 4640 y FE(\016)1171 4532 y Fq(\000)1217 4613 y Fs(\()p Fg(u)1308 4570 y Fr(0)1308 4640 y FE(t)1342 4613 y Fs(\))1381 4570 y FF(2)1443 4613 y Fs(+)22 b(\()p Fg(u)1633 4570 y Fr(00)1633 4640 y FE(t)1684 4613 y Fs(\))1723 4570 y FF(2)1762 4532 y Fq(\001)1830 4613 y Fp(\000)1930 4477 y Fq(Z)2046 4613 y Ft(h)2103 4570 y Fr(0)2103 4640 y FE(\016)2147 4532 y Fq(\000)2193 4613 y Fg(v)2244 4640 y FE(t)2279 4613 y Fg(u)2331 4570 y Fr(0)2331 4640 y FE(t)2387 4613 y Fs(+)h Fg(v)2538 4570 y Fr(0)2538 4640 y FE(t)2573 4613 y Fg(u)2625 4570 y Fr(00)2625 4640 y FE(t)2698 4613 y Fs(+)2809 4573 y FF(1)p 2809 4590 V 2809 4647 a(2)2856 4613 y Ft(\015)2914 4532 y Fq(\000)2959 4613 y Fg(u)3011 4640 y FE(t)3046 4613 y Fg(u)3098 4570 y Fr(0)3098 4640 y FE(t)3154 4613 y Fs(+)g Fg(u)3306 4570 y Fr(0)3306 4640 y FE(t)3341 4613 y Fg(u)3393 4570 y Fr(00)3393 4640 y FE(t)3443 4532 y Fq(\001\001)3559 4613 y Ft(:)3669 3933 y Fs(\()p Fv(6)p Ft(:)p Fv(5)p Fs(\))94 4865 y Fv(By)d(construction,)f(we)h(ha)n(v)o(e)f Fp(j)p Ft(h)1193 4823 y Fr(0)1193 4892 y FE(\016)1237 4865 y Ft(=h)1344 4892 y FE(\016)1388 4865 y Fp(j)27 b(\024)h Ft(C)1619 4892 y FF(18)1694 4865 y Ft(\016)t Fv(,)21 b(and)e(therefore)h(the)g(last)f(inte)o(gral)f(in)h (\(6.5\))g(can)h(be)g(bounded)94 4985 y(\(in)25 b(modulus\))e(by)560 5237 y Ft(C)631 5263 y FF(18)706 5237 y Ft(\016)771 5101 y Fq(Z)887 5237 y Ft(h)944 5263 y FE(\016)988 5096 y Fq(\022)1062 5237 y Fg(v)1113 5194 y FF(2)1113 5263 y FE(t)1175 5237 y Fs(+)f(\()p Fg(u)1365 5194 y Fr(0)1365 5263 y FE(t)1400 5237 y Fs(\))1439 5194 y FF(2)1500 5237 y Fs(+)h(\()p Fg(v)1690 5194 y Fr(0)1690 5263 y FE(t)1725 5237 y Fs(\))1764 5194 y FF(2)1825 5237 y Fs(+)g(\()p Fg(u)2016 5194 y Fr(00)2016 5263 y FE(t)2066 5237 y Fs(\))2105 5194 y FF(2)2167 5237 y Fs(+)2279 5197 y FF(1)p 2279 5214 V 2279 5271 a(2)2325 5237 y Ft(\015)2383 5156 y Fq(\000)2428 5237 y Fg(u)2480 5194 y FF(2)2480 5263 y FE(t)2542 5237 y Fs(+)f Fv(2)p Fs(\()p Fg(u)2782 5194 y Fr(0)2782 5263 y FE(t)2816 5237 y Fs(\))2855 5194 y FF(2)2917 5237 y Fs(+)h(\()p Fg(u)3108 5194 y Fr(00)3108 5263 y FE(t)3158 5237 y Fs(\))3197 5194 y FF(2)3237 5156 y Fq(\001)3282 5096 y(\023)3381 5237 y Ft(:)p eop %%Page: 16 16 bop 94 -45 a Fo(D)m(amped)38 b(Hyperbolic)f(Equa)-7 b(tions)2130 b Fn(16)94 194 y Fv(Thus,)25 b(we)g(\256nd)697 395 y FF(1)p 697 411 35 4 v 697 468 a(2)743 434 y Ft(@)796 461 y FE(t)831 434 y Ft(J)10 b Fs(\()p Fg(u)987 461 y FE(t)1022 434 y Ft(;)17 b Fg(v)1118 461 y FE(t)1152 434 y Fs(\))44 b Fp(\024)39 b(\000)1451 298 y Fq(Z)1567 434 y Ft(h)1624 461 y FE(\016)1668 353 y Fq(\000)1714 434 y Fg(v)1765 391 y FF(2)1765 461 y FE(t)1827 434 y Fs(+)22 b(\()p Fg(v)2016 391 y Fr(0)2016 461 y FE(t)2051 434 y Fs(\))2090 391 y FF(2)2130 353 y Fq(\001\000)2221 434 y Fv(1)g Fp(\000)2405 395 y FF(1)p 2405 411 V 2405 468 a(2)2452 434 y Ft(\021)2505 391 y FF(2)2544 434 y Ft(\015)28 b Fp(\000)22 b Ft(C)2794 461 y FF(18)2869 434 y Ft(\016)2917 353 y Fq(\001)1351 696 y Fp(\000)1463 657 y FF(1)p 1463 673 V 1463 731 a(2)1510 696 y Ft(\015)1583 561 y Fq(Z)1700 696 y Ft(h)1757 723 y FE(\016)1801 615 y Fq(\000)1846 696 y Fs(\()p Fg(u)1937 653 y Fr(0)1937 723 y FE(t)1972 696 y Fs(\))2011 653 y FF(2)2072 696 y Fs(+)h(\()p Fg(u)2263 653 y Fr(00)2263 723 y FE(t)2313 696 y Fs(\))2352 653 y FF(2)2392 615 y Fq(\001)o(\000)2483 696 y Fv(1)f Fp(\000)2667 629 y Ft(C)2738 656 y FF(18)2813 629 y Ft(\016)p 2667 673 194 4 v 2735 764 a(\015)2895 696 y Fp(\000)g Fv(2)p Ft(C)3115 723 y FF(18)3190 696 y Ft(\016)3238 615 y Fq(\001)1351 959 y Fs(+)1463 920 y FF(1)p 1463 936 35 4 v 1463 994 a(2)1510 959 y Ft(C)1581 986 y FF(18)1656 959 y Ft(\016)t(\015)1777 824 y Fq(Z)1894 959 y Ft(h)1951 986 y FE(\016)1995 959 y Fg(u)2047 916 y FF(2)2047 986 y FE(t)1351 1221 y Fs(+)1451 1085 y Fq(Z)1567 1221 y Ft(h)1624 1247 y FE(\016)1668 1140 y Fq(\000)1714 1221 y Fg(u)1766 1247 y FE(t)1800 1221 y Fg(v)1851 1247 y FE(t)1909 1221 y Fs(+)g Fg(u)2060 1178 y Fr(0)2060 1247 y FE(t)2095 1221 y Fg(v)2146 1178 y Fr(0)2146 1247 y FE(t)2181 1140 y Fq(\001)2227 1221 y Fs(\()p Fv(1)f Fp(\000)2449 1181 y FF(1)p 2449 1198 V 2449 1255 a(2)2496 1221 y Ft(\015)6 b Fs(\))23 b Ft(:)3669 827 y Fs(\()p Fv(6)p Ft(:)p Fv(6)p Fs(\))94 1464 y Fv(Recall)33 b(that)f Ft(\021)41 b(>)d Fv(0)32 b(is)g(gi)n(v)o(en,)g(and)f(that)h(we)g(w)o(ant)g (to)f(pro)o(v)o(e)g(results)g(for)i(all)e Ft(\021)42 b(<)c(\021)3147 1491 y FF(0)3187 1464 y Fv(,)c(where)e Ft(\021)3570 1491 y FF(0)3643 1464 y Fv(is)f(our)94 1583 y(\(only\))25 b(small)f(parameter)-5 b(.)316 1703 y(W)d(e)25 b(re)n(write)g(the)f(last)h(inte)o(gral)e(in)i (\(6.6\))g(as)582 1950 y Fp(\000)671 1911 y FF(1)p 672 1928 V 672 1985 a(8)719 1950 y Ft(\021)772 1908 y FF(2)811 1950 y Ft(\015)869 1908 y FF(2)925 1815 y Fq(Z)1041 1950 y Ft(h)1098 1977 y FE(\016)1142 1870 y Fq(\000)1188 1950 y Fg(u)1240 1977 y FE(t)1275 1950 y Fg(v)1326 1977 y FE(t)1383 1950 y Fs(+)d Fg(u)1534 1908 y Fr(0)1534 1977 y FE(t)1569 1950 y Fg(v)1620 1908 y Fr(0)1620 1977 y FE(t)1655 1870 y Fq(\001)1723 1950 y Fs(+)1823 1815 y Fq(Z)1939 1950 y Ft(h)1996 1977 y FE(\016)2040 1870 y Fq(\000)2086 1950 y Fg(u)2138 1977 y FE(t)2172 1950 y Fg(v)2223 1977 y FE(t)2280 1950 y Fs(+)h Fg(u)2432 1908 y Fr(0)2432 1977 y FE(t)2467 1950 y Fg(v)2518 1908 y Fr(0)2518 1977 y FE(t)2553 1870 y Fq(\001)2598 1950 y Fs(\()p Fv(1)f Fs(+)2821 1911 y FF(1)p 2821 1928 V 2821 1985 a(8)2867 1950 y Ft(\021)2920 1908 y FF(2)2960 1950 y Ft(\015)3018 1908 y FF(2)3079 1950 y Fp(\000)3191 1911 y FF(1)p 3191 1928 V 3191 1985 a(2)3238 1950 y Ft(\015)6 b Fs(\))24 b Ft(:)282 b Fs(\()p Fv(6)p Ft(:)p Fv(7)p Fs(\))94 2196 y Fv(W)-8 b(e)26 b(introduce)e(no)n(w)g (the)h(\256rst)f(restrictions)g(on)h Ft(\021)k Fv(and)24 b Ft(k)2062 2223 y Fr(\003)2108 2196 y Fv(:)36 b(Fix)1727 2447 y Ft(\021)1776 2474 y FF(0)1861 2447 y Fp(\024)2061 2380 y Fv(1)p 1994 2424 183 4 v 1994 2444 a Fp(p)p 2077 2444 100 4 v 84 x Fv(40)2214 2447 y Ft(;)1427 b Fs(\()p Fv(6)p Ft(:)p Fv(8)p Fs(\))94 2697 y Fv(and)1656 2817 y Ft(k)1708 2844 y FF(0)1792 2817 y Fp(\025)1914 2732 y Fq(p)p 2014 2732 246 4 v 85 x Fv(40)p Ft(C)2185 2844 y FF(15)2284 2817 y Ft(:)1357 b Fs(\()p Fv(6)p Ft(:)p Fv(9)p Fs(\))94 2989 y Fv(These)25 b(bounds)f(will)g(be)h(made)g(more)f(stringent)g(belo)n(w)-6 b(.)34 b(W)-8 b(e)25 b(shall)g(al)o(w)o(ays)f(require)1179 3189 y(0)44 b Ft(<)h(\021)j(<)d(\021)1663 3216 y FF(0)1728 3189 y Ft(;)116 b Fv(and)100 b Fp(1)44 b Ft(>)g(k)2433 3216 y Fr(\003)2523 3189 y Ft(>)h(k)2697 3216 y FF(0)2762 3189 y Ft(:)94 3389 y Fv(W)-8 b(e)26 b(ne)o(xt)e(de\256ne)1384 3511 y Ft(\015)49 b Fp(\021)c Fv(min)o Fs(\()p Ft(\021)1854 3468 y Fr(\000)p FF(2)1956 3511 y Ft(;)17 b(k)2056 3468 y FF(2)2053 3538 y Fr(\003)2098 3511 y Ft(=C)2219 3538 y FF(15)2294 3511 y Fs(\))p Ft(=)p Fv(320)24 b Ft(;)3619 3508 y Fs(\()p Fv(6)p Ft(:)p Fv(10)p Fs(\))94 3680 y Fv(and)h(we)g(choose)g(the)g(space-cutof)n(f)f (parameter)i Ft(\016)i Fv(as)1420 3880 y Ft(\016)48 b Fs(=)d Fv(min)1789 3799 y Fq(\000)1834 3880 y Fv(1)p Ft(=)p Fv(2)p Ft(;)17 b Fv(1)p Ft(=)p Fs(\()p Fv(40)p Ft(C)2339 3907 y FF(18)2412 3880 y Fs(\))2451 3799 y Fq(\001)2521 3880 y Ft(:)1070 b Fs(\()p Fv(6)p Ft(:)p Fv(11)p Fs(\))316 4080 y Fv(Note)23 b(that)f Ft(\015)28 b Fv(is)22 b(essentially)g(the)h(in)l(v)o(erse)f(of)h(the)g (dispersion)e(la)o(w)i(as)g(e)o(xplained)e(at)i(the)g(be)o(ginning)e(of)94 4199 y(this)j(section.)316 4319 y(W)l(ith)g(the)h(abo)o(v)o(e)f(requirements) g(we)h(\256nd)g Ft(\015)32 b(>)d Fv(2)24 b(and)1490 4519 y Fp(j)p Fv(1)d Fs(+)1701 4480 y FF(1)p 1701 4496 35 4 v 1701 4553 a(8)1748 4519 y Ft(\021)1801 4476 y FF(2)1841 4519 y Ft(\015)1899 4476 y FF(2)1960 4519 y Fp(\000)2071 4480 y FF(1)p 2071 4496 V 2071 4553 a(2)2118 4519 y Ft(\015)6 b Fp(j)43 b(\024)i Ft(\015)30 b(:)94 4719 y Fv(W)-8 b(e)26 b(polarize)f(the)f FG(second)h Fv(inte)o(gral)e(in)i(\(6.7\))g(\(b)n(ut)f(not)g(the)h(\256rst\))g(and)g (bound)f(it)g(\(in)h(modulus\))e(by)627 4966 y Fp(j)p Fv(1)e Fs(+)838 4927 y FF(1)p 838 4944 V 838 5001 a(8)885 4966 y Ft(\021)938 4924 y FF(2)978 4966 y Ft(\015)1036 4924 y FF(2)1097 4966 y Fp(\000)1208 4927 y FF(1)p 1208 4944 V 1208 5001 a(2)1255 4966 y Ft(\015)6 b Fp(j)1357 4831 y Fq(Z)1473 4966 y Ft(h)1530 4993 y FE(\016)1574 4966 y Fs(\()p Fp(j)p Fg(u)1693 4993 y FE(t)1727 4966 y Fg(v)1778 4993 y FE(t)1813 4966 y Fp(j)22 b Fs(+)h Fp(j)p Fg(u)2043 4924 y Fr(0)2043 4993 y FE(t)2077 4966 y Fg(v)2128 4924 y Fr(0)2128 4993 y FE(t)2163 4966 y Fp(j)p Fs(\))44 b Fp(\024)g Ft(\015)2469 4831 y Fq(Z)2585 4966 y Ft(h)2642 4993 y FE(\016)2686 4966 y Fs(\()p Fp(j)p Fg(u)2805 4993 y FE(t)2839 4966 y Fg(v)2890 4993 y FE(t)2925 4966 y Fp(j)22 b Fs(+)h Fp(j)p Fg(u)3155 4924 y Fr(0)3155 4993 y FE(t)3189 4966 y Fg(v)3240 4924 y Fr(0)3240 4993 y FE(t)3275 4966 y Fp(j)p Fs(\))1368 5237 y Fp(\024)1489 5101 y Fq(Z)1606 5237 y Ft(h)1663 5263 y FE(\016)1707 5096 y Fq(\022)1780 5237 y Fv(8)p Ft(\015)1888 5194 y FF(2)1927 5156 y Fq(\000)1973 5237 y Fg(u)2025 5194 y FF(2)2025 5263 y FE(t)2086 5237 y Fs(+)g(\()p Fg(u)2277 5194 y Fr(0)2277 5263 y FE(t)2311 5237 y Fs(\))2350 5194 y FF(2)2390 5156 y Fq(\001)2457 5237 y Fs(+)2569 5197 y FF(1)p 2569 5214 V 2569 5271 a(8)2616 5156 y Fq(\000)2662 5237 y Fg(v)2713 5194 y FF(2)2713 5263 y FE(t)2775 5237 y Fs(+)f(\()p Fg(v)2964 5194 y Fr(0)2964 5263 y FE(t)2999 5237 y Fs(\))3038 5194 y FF(2)3078 5156 y Fq(\001)3123 5096 y(\023)3221 5237 y Ft(:)3619 5106 y Fs(\()p Fv(6)p Ft(:)p Fv(12)p Fs(\))p eop %%Page: 17 17 bop 94 -45 a Fo(D)m(amped)38 b(Hyperbolic)f(Equa)-7 b(tions)2130 b Fn(17)94 194 y Fv(Combining)24 b(\(6.6\))h(with)f(the)g(decomposition)f (\(6.7\))i(and)f(the)h(bound)f(\(6.12\),)h(we)g(\256nd)214 409 y FF(1)p 214 425 35 4 v 214 482 a(2)261 448 y Ft(@)314 475 y FE(t)349 448 y Ft(J)10 b Fs(\()p Fg(u)505 475 y FE(t)539 448 y Ft(;)17 b Fg(v)635 475 y FE(t)670 448 y Fs(\))44 b Fp(\024)39 b(\000)968 312 y Fq(Z)1085 448 y Ft(h)1142 475 y FE(\016)1186 367 y Fq(\000)1231 448 y Fg(v)1282 405 y FF(2)1282 475 y FE(t)1344 448 y Fs(+)23 b(\()p Fg(v)1534 405 y Fr(0)1534 475 y FE(t)1569 448 y Fs(\))1608 405 y FF(2)1647 367 y Fq(\001\000)1739 448 y Fv(1)f Fp(\000)1922 409 y FF(1)p 1922 425 V 1922 482 a(2)1969 448 y Ft(\021)2022 405 y FF(2)2062 448 y Ft(\015)27 b Fp(\000)c Ft(C)2312 475 y FF(18)2387 448 y Ft(\016)j Fp(\000)2569 409 y FF(1)p 2569 425 V 2569 482 a(8)2615 367 y Fq(\001)869 710 y Fp(\000)980 671 y FF(1)p 980 687 V 980 745 a(2)1027 710 y Ft(\015)1101 575 y Fq(Z)1217 710 y Ft(h)1274 737 y FE(\016)1318 630 y Fq(\000)1364 710 y Fg(u)1416 667 y FF(2)1416 737 y FE(t)1477 710 y Fs(+)d Fv(2)p Fs(\()p Fg(u)1718 667 y Fr(0)1718 737 y FE(t)1752 710 y Fs(\))1791 667 y FF(2)1853 710 y Fs(+)g(\()p Fg(u)2044 667 y Fr(00)2044 737 y FE(t)2094 710 y Fs(\))2133 667 y FF(2)2172 630 y Fq(\001\000)2264 710 y Fv(1)f Fp(\000)2447 643 y Ft(C)2518 670 y FF(18)2593 643 y Ft(\016)p 2447 687 194 4 v 2516 779 a(\015)2675 710 y Fp(\000)h Fv(2)p Ft(C)2896 737 y FF(18)2971 710 y Ft(\016)3019 630 y Fq(\001)869 974 y Fs(+)968 839 y Fq(Z)1085 974 y Ft(h)1142 1001 y FE(\016)1186 894 y Fq(\000)1231 974 y Fg(u)1283 932 y FF(2)1283 1001 y FE(t)1345 974 y Fs(+)g(\()p Fg(u)1536 932 y Fr(0)1536 1001 y FE(t)1570 974 y Fs(\))1609 932 y FF(2)1649 894 y Fq(\001)o(\000)1740 974 y Fs(\()p Fv(8)p Ft(\015)1887 932 y FF(2)1948 974 y Fs(+)2059 935 y FF(1)p 2059 951 35 4 v 2059 1009 a(2)2106 974 y Ft(C)2177 1001 y FF(18)2252 974 y Ft(\016)t(\015)6 b Fs(\))21 b(+)2529 935 y FF(1)p 2529 951 V 2529 1009 a(2)2576 974 y Ft(\015)6 b Fs(\()p Fv(1)21 b Fp(\000)2856 907 y Ft(C)2927 934 y FF(18)3002 907 y Ft(\016)p 2856 951 194 4 v 2924 1043 a(\015)3084 974 y Fp(\000)h Fv(2)p Ft(C)3304 1001 y FF(18)3379 974 y Ft(\016)t Fs(\))3466 894 y Fq(\001)869 1238 y Fp(\000)980 1198 y FF(1)p 980 1215 35 4 v 980 1272 a(8)1027 1238 y Ft(\021)1080 1195 y FF(2)1120 1238 y Ft(\015)1178 1195 y FF(2)1234 1102 y Fq(Z)1350 1238 y Ft(h)1407 1264 y FE(\016)1451 1157 y Fq(\000)1497 1238 y Fg(u)1549 1264 y FE(t)1583 1238 y Fg(v)1634 1264 y FE(t)1692 1238 y Fs(+)g Fg(u)1843 1195 y Fr(0)1843 1264 y FE(t)1878 1238 y Fg(v)1929 1195 y Fr(0)1929 1264 y FE(t)1964 1157 y Fq(\001)2034 1238 y Ft(:)3619 843 y Fs(\()p Fv(6)p Ft(:)p Fv(13)p Fs(\))94 1510 y Fv(The)30 b(bizarre)h(decomposition)d(of)i(the)f(terms)h(in)l(v)n (olving)e Fs(\()p Fg(u)2216 1467 y Fr(0)2216 1537 y FE(t)2250 1510 y Fs(\))2289 1467 y FF(2)2359 1510 y Fv(will)h(become)g(clear)i(soon.)50 b(Note)30 b(that)f(by)94 1629 y(our)c(choice)g(of)g(constants,)f(\(6.13\))g (can)h(be)g(simpli\256ed)f(to)g(the)h(slightly)e(less)h(good)g(bound)988 1844 y FF(1)p 988 1860 V 988 1917 a(2)1035 1883 y Ft(@)1088 1910 y FE(t)1123 1883 y Ft(J)10 b Fs(\()p Fg(u)1279 1910 y FE(t)1313 1883 y Ft(;)17 b Fg(v)1409 1910 y FE(t)1444 1883 y Fs(\))44 b Fp(\024)39 b(\000)1754 1844 y FF(1)p 1754 1860 V 1754 1917 a(2)1818 1747 y Fq(Z)1934 1883 y Ft(h)1991 1910 y FE(\016)2035 1802 y Fq(\000)2081 1883 y Fg(v)2132 1840 y FF(2)2132 1910 y FE(t)2194 1883 y Fs(+)23 b(\()p Fg(v)2384 1840 y Fr(0)2384 1910 y FE(t)2418 1883 y Fs(\))2457 1840 y FF(2)2497 1802 y Fq(\001)1643 2144 y Fp(\000)1754 2105 y FF(1)p 1754 2121 V 1754 2179 a(4)1801 2144 y Ft(\015)1875 2009 y Fq(Z)1991 2144 y Ft(h)2048 2171 y FE(\016)2092 2064 y Fq(\000)2138 2144 y Fg(u)2190 2101 y FF(2)2190 2171 y FE(t)2252 2144 y Fs(+)f Fv(2)p Fs(\()p Fg(u)2492 2101 y Fr(0)2492 2171 y FE(t)2526 2144 y Fs(\))2565 2101 y FF(2)2627 2144 y Fs(+)h(\()p Fg(u)2818 2101 y Fr(00)2818 2171 y FE(t)2868 2144 y Fs(\))2907 2101 y FF(2)2946 2064 y Fq(\001)1643 2406 y Fs(+)g Fv(16)p Ft(\015)1901 2363 y FF(2)1956 2270 y Fq(Z)2072 2406 y Ft(h)2129 2432 y FE(\016)2173 2325 y Fq(\000)2219 2406 y Fg(u)2271 2363 y FF(2)2271 2432 y FE(t)2332 2406 y Fs(+)g(\()p Fg(u)2523 2363 y Fr(0)2523 2432 y FE(t)2557 2406 y Fs(\))2596 2363 y FF(2)2636 2325 y Fq(\001)1643 2667 y Fp(\000)1754 2628 y FF(1)p 1754 2644 V 1754 2701 a(8)1801 2667 y Ft(\021)1854 2624 y FF(2)1894 2667 y Ft(\015)1952 2624 y FF(2)2008 2531 y Fq(Z)2124 2667 y Ft(h)2181 2694 y FE(\016)2225 2586 y Fq(\000)2271 2667 y Fg(u)2323 2694 y FE(t)2357 2667 y Fg(v)2408 2694 y FE(t)2466 2667 y Fs(+)f Fg(u)2617 2624 y Fr(0)2617 2694 y FE(t)2652 2667 y Fg(v)2703 2624 y Fr(0)2703 2694 y FE(t)2738 2586 y Fq(\001)2808 2667 y Ft(:)3619 2275 y Fs(\()p Fv(6)p Ft(:)p Fv(14)p Fs(\))94 2917 y Fv(W)-8 b(e)25 b(project)e(onto)g(high)g(momenta,)g(and)h(e)o(xploit)e(the)i(contraction)f (properties:)35 b FG(W)-9 b(e)24 b(assume)f(fr)l(om)f(now)i(on)94 3036 y(that)31 b Fm(Q)347 3063 y FE(k)389 3081 y Fe(\003)436 3036 y Fg(u)488 3063 y FF(0)563 3036 y Fs(=)37 b Fv(0)31 b FG(and)g Fm(Q)1005 3063 y FE(k)1047 3081 y Fe(\003)1094 3036 y Fg(v)1145 3063 y FF(0)1221 3036 y Fs(=)37 b Fv(0)p FG(.)54 b Fv(Note)31 b(that)g(if)g(this)f(property)h(holds)f(at)h(time)f(zero,)j(it)e (holds)f(for)h(all)94 3156 y(times)26 b(for)h(the)f(e)n(v)n(olution)e (de\256ned)j(by)f Fm(L)p Fv(,)h(because)g Fm(L)g Fv(commutes)e(with)g Fm(Q)2740 3183 y FE(k)2782 3200 y Fe(\003)2852 3156 y Fp(\010)f Fm(Q)3019 3183 y 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b(\(6.14\))f(can)h(be)g(impro)o(v)o(ed)e(to)1136 4422 y FF(1)p 1136 4439 V 1136 4496 a(2)1182 4462 y Ft(@)1235 4489 y FE(t)1270 4462 y Ft(J)54 b Fp(\024)40 b(\000)1607 4422 y FF(1)p 1607 4439 V 1607 4496 a(2)1671 4326 y Fq(Z)1787 4462 y Ft(h)1844 4489 y FE(\016)1888 4381 y Fq(\000)1934 4462 y Fg(v)1985 4419 y FF(2)1985 4489 y FE(t)2047 4462 y Fs(+)22 b(\()p Fg(v)2236 4419 y Fr(0)2236 4489 y FE(t)2271 4462 y Fs(\))2310 4419 y FF(2)2350 4381 y Fq(\001)1496 4723 y Fp(\000)1607 4684 y FF(1)p 1607 4700 V 1607 4757 a(4)1654 4723 y Ft(\015)1728 4587 y Fq(Z)1844 4723 y Ft(h)1901 4750 y FE(\016)1945 4642 y Fq(\000)1991 4723 y Fg(u)2043 4680 y FF(2)2043 4750 y FE(t)2104 4723 y Fs(+)h Fv(2)p Fs(\()p Fg(u)2345 4680 y Fr(0)2345 4750 y FE(t)2379 4723 y Fs(\))2418 4680 y FF(2)2480 4723 y Fs(+)f(\()p Fg(u)2670 4680 y Fr(00)2670 4750 y FE(t)2721 4723 y Fs(\))2760 4680 y FF(2)2799 4642 y Fq(\001)1496 4984 y Fs(+)g Fv(16)p Ft(\027)6 b(\015)1808 4941 y FF(2)1864 4849 y Fq(Z)1980 4984 y Ft(h)2037 5011 y FE(\016)2081 4903 y Fq(\000)2127 4984 y Fs(\()p Fg(u)2218 4941 y Fr(0)2218 5011 y FE(t)2252 4984 y Fs(\))2291 4941 y FF(2)2353 4984 y Fs(+)23 b(\()p Fg(u)2544 4941 y Fr(00)2544 5011 y FE(t)2594 4984 y Fs(\))2633 4941 y FF(2)2673 4903 y Fq(\001)1496 5245 y Fp(\000)1607 5206 y FF(1)p 1607 5222 V 1607 5280 a(8)1654 5245 y Ft(\021)1707 5203 y FF(2)1747 5245 y Ft(\015)1805 5203 y FF(2)1861 5110 y Fq(Z)1977 5245 y Ft(h)2034 5272 y FE(\016)2078 5165 y Fq(\000)2124 5245 y Fg(u)2176 5272 y FE(t)2210 5245 y Fg(v)2261 5272 y FE(t)2318 5245 y Fs(+)g Fg(u)2470 5203 y Fr(0)2470 5272 y FE(t)2505 5245 y Fg(v)2556 5203 y Fr(0)2556 5272 y FE(t)2591 5165 y Fq(\001)2661 5245 y Ft(:)3619 4854 y Fs(\()p Fv(6)p Ft(:)p Fv(16)p Fs(\))p eop %%Page: 18 18 bop 94 -45 a Fo(D)m(amped)38 b(Hyperbolic)f(Equa)-7 b(tions)2130 b Fn(18)94 194 y Fv(This)25 b(leads)f(to)h(a)g(bound)f(of)h(the)f(form)1136 415 y FF(1)p 1136 431 35 4 v 1136 488 a(2)1182 454 y Ft(@)1235 481 y FE(t)1270 454 y Ft(J)54 b Fp(\024)40 b(\000)1607 415 y FF(1)p 1607 431 V 1607 488 a(2)1654 454 y Ft(\021)1707 411 y Fr(\000)p FF(2)1826 318 y Fq(Z)1942 454 y Ft(h)1999 481 y FE(\016)2043 454 y Ft(\021)2096 411 y FF(2)2136 373 y Fq(\000)2182 454 y Fg(v)2233 411 y FF(2)2233 481 y FE(t)2295 454 y Fs(+)22 b(\()p Fg(v)2484 411 y Fr(0)2484 481 y FE(t)2519 454 y Fs(\))2558 411 y FF(2)2598 373 y Fq(\001)1496 715 y Fp(\000)1607 676 y FF(1)p 1607 692 V 1607 750 a(8)1654 715 y Ft(\015)1728 580 y Fq(Z)1844 715 y Ft(h)1901 742 y FE(\016)1945 635 y Fq(\000)1991 715 y Fg(u)2043 672 y FF(2)2043 742 y FE(t)2104 715 y Fs(+)h Fv(2)p Fs(\()p Fg(u)2345 672 y Fr(0)2345 742 y FE(t)2379 715 y Fs(\))2418 672 y FF(2)2480 715 y Fs(+)f(\()p Fg(u)2670 672 y Fr(00)2670 742 y FE(t)2721 715 y Fs(\))2760 672 y FF(2)2799 635 y Fq(\001)1496 977 y Fp(\000)1607 937 y FF(1)p 1607 954 V 1607 1011 a(8)1654 977 y Ft(\021)1707 934 y FF(2)1747 977 y Ft(\015)1805 934 y FF(2)1861 841 y Fq(Z)1977 977 y Ft(h)2034 1003 y FE(\016)2078 896 y Fq(\000)2124 977 y Fg(u)2176 1003 y FE(t)2210 977 y Fg(v)2261 1003 y FE(t)2318 977 y Fs(+)h Fg(u)2470 934 y Fr(0)2470 1003 y FE(t)2505 977 y Fg(v)2556 934 y Fr(0)2556 1003 y FE(t)2591 896 y Fq(\001)2661 977 y Ft(:)94 1253 y Fv(T)-8 b(aking)27 b(the)g(least)g(good)f(bounds)g(abo)o(v)o(e,)h(we)g(\256nally)g (get)g(the)g(decay)g(of)h(the)f(high)f(frequenc)o(y)h(part)g(\(since)94 1373 y Ft(\021)147 1330 y Fr(\000)p FF(2)277 1373 y Fp(\025)h Ft(\015)6 b Fv(\):)294 1621 y FF(1)p 294 1637 V 294 1694 a(2)341 1660 y Ft(@)394 1687 y FE(t)429 1660 y Ft(J)54 b Fp(\021)672 1621 y FF(1)p 672 1637 V 672 1694 a(2)719 1660 y Ft(@)772 1687 y FE(t)823 1524 y Fq(Z)940 1660 y Ft(h)997 1687 y FE(\016)1041 1519 y Fq(\022)1114 1660 y Ft(\021)1167 1617 y FF(2)1207 1579 y Fq(\000)1253 1660 y Fg(v)1304 1617 y FF(2)1304 1687 y FE(t)1366 1660 y Fs(+)22 b(\()p Fg(v)1555 1617 y Fr(0)1555 1687 y FE(t)1590 1660 y Fs(\))1629 1617 y FF(2)1669 1579 y Fq(\001)1736 1660 y Fs(+)1836 1579 y Fq(\000)1882 1660 y Fg(u)1934 1617 y FF(2)1934 1687 y FE(t)1995 1660 y Fs(+)h Fv(2)p Fs(\()p Fg(u)2236 1617 y Fr(0)2236 1687 y FE(t)2270 1660 y Fs(\))2309 1617 y FF(2)2371 1660 y Fs(+)f(\()p Fg(u)2561 1617 y Fr(00)2561 1687 y FE(t)2612 1660 y Fs(\))2651 1617 y FF(2)2690 1579 y Fq(\001)2758 1660 y Fs(+)h Ft(\021)2911 1617 y FF(2)2951 1660 y Ft(\015)3009 1579 y Fq(\000)3053 1660 y Fg(u)3105 1687 y FE(t)3140 1660 y Fg(v)3191 1687 y FE(t)3248 1660 y Fs(+)g Fg(u)3400 1617 y Fr(0)3400 1687 y FE(t)3434 1660 y Fg(v)3485 1617 y Fr(0)3485 1687 y FE(t)3520 1579 y Fq(\001)3566 1519 y(\023)538 1939 y Fp(\024)45 b(\000)749 1900 y FF(1)p 750 1916 V 750 1973 a(8)796 1939 y Ft(\015)870 1803 y Fq(Z)986 1939 y Ft(h)1043 1966 y FE(\016)1087 1798 y Fq(\022)1161 1939 y Ft(\021)1214 1896 y FF(2)1254 1858 y Fq(\000)1299 1939 y Fg(v)1350 1896 y FF(2)1350 1966 y FE(t)1412 1939 y Fs(+)23 b(\()p Fg(v)1602 1896 y Fr(0)1602 1966 y FE(t)1637 1939 y Fs(\))1676 1896 y FF(2)1715 1858 y Fq(\001)1783 1939 y Fs(+)1883 1858 y Fq(\000)1928 1939 y Fg(u)1980 1896 y FF(2)1980 1966 y FE(t)2042 1939 y Fs(+)g Fv(2)p Fs(\()p Fg(u)2283 1896 y Fr(0)2283 1966 y FE(t)2317 1939 y Fs(\))2356 1896 y FF(2)2417 1939 y Fs(+)g(\()p Fg(u)2608 1896 y Fr(00)2608 1966 y FE(t)2658 1939 y Fs(\))2697 1896 y FF(2)2737 1858 y Fq(\001)2805 1939 y Fs(+)f Ft(\021)2957 1896 y FF(2)2997 1939 y Ft(\015)3055 1858 y Fq(\000)3100 1939 y Fg(u)3152 1966 y FE(t)3187 1939 y Fg(v)3238 1966 y FE(t)3295 1939 y Fs(+)g Fg(u)3446 1896 y Fr(0)3446 1966 y FE(t)3481 1939 y Fg(v)3532 1896 y Fr(0)3532 1966 y FE(t)3567 1858 y Fq(\001)3613 1798 y(\023)538 2163 y Fs(=)45 b Fp(\000)749 2124 y FF(1)p 750 2140 V 750 2198 a(8)796 2163 y Ft(\015)6 b(J)54 b Fs(=)44 b Fp(\000)1226 2124 y FF(1)p 1174 2140 140 4 v 1174 2198 a(2560)1342 2163 y Fv(min)o Fs(\()p Ft(\021)1589 2120 y Fr(\000)p FF(2)1691 2163 y Ft(;)17 b(k)1791 2120 y FF(2)1788 2190 y Fr(\003)1833 2163 y Ft(=C)1954 2190 y FF(15)2028 2163 y Fs(\))p Ft(J)35 b(:)3619 2283 y Fs(\()p Fv(6)p Ft(:)p Fv(17)p Fs(\))316 2403 y Fv(Thus)24 b(we)h(ha)n(v)o(e)g(sho)n (wn)e(the)94 2594 y Fu(Prop)s(osition)42 b(6.3.)i Fh(There)28 b(is)f(a)h(\(small\))f Ft(\021)1724 2621 y FF(0)1796 2594 y Ft(>)32 b Fv(0)c Fh(such)f(that)g(for)h(all)f Ft(\021)36 b(<)c(\021)2884 2621 y FF(0)2952 2594 y Fh(the)c(follo)n(wing)d(holds)i(for)94 2714 y(the)e(functional)104 2983 y Ft(J)159 3009 y FE(h)206 3027 y Fl(\016)245 3009 y FE(;\015)320 2983 y Fs(\()p Fg(u)411 3009 y FE(t)445 2983 y Ft(;)17 b Fg(v)541 3009 y FE(t)575 2983 y Fs(\))44 b(=)780 2847 y Fq(Z)896 2983 y Ft(h)953 3009 y FE(\016)997 2983 y Fs(\()p Ft(x)p Fs(\))1132 2842 y Fq(\022)1205 2983 y Ft(\021)1258 2940 y FF(2)1298 2902 y Fq(\000)1344 2983 y Fg(v)1395 2940 y FF(2)1395 3009 y FE(t)1457 2983 y Fs(+)22 b(\()p Fg(v)1646 2940 y Fr(0)1646 3009 y FE(t)1681 2983 y Fs(\))1720 2940 y FF(2)1760 2902 y Fq(\001)1827 2983 y Fs(+)h Fg(u)1979 2940 y FF(2)1979 3009 y FE(t)2041 2983 y Fs(+)f Fv(2)p Fs(\()p Fg(u)2281 2940 y Fr(0)2281 3009 y FE(t)2315 2983 y Fs(\))2354 2940 y FF(2)2416 2983 y Fs(+)h(\()p Fg(u)2607 2940 y Fr(00)2607 3009 y FE(t)2657 2983 y Fs(\))2696 2940 y FF(2)2758 2983 y Fs(+)f Ft(\021)2910 2940 y FF(2)2950 2983 y Ft(\015)3008 2902 y Fq(\000)3053 2983 y Fg(u)3105 3009 y FE(t)3140 2983 y Fg(v)3191 3009 y FE(t)3248 2983 y Fs(+)g Fg(u)3399 2940 y Fr(0)3399 3009 y FE(t)3434 2983 y Fg(v)3485 2940 y Fr(0)3485 3009 y FE(t)3520 2983 y Fs(\))3559 2902 y Fq(\001)3604 2842 y(\023)3678 2983 y Fs(\()p Ft(x)p Fs(\))i Ft(:)94 3293 y Fh(Let)h Fs(\()p Fg(u)343 3320 y FE(t)378 3293 y Ft(;)17 b Fg(v)474 3320 y FE(t)508 3293 y Fs(\))27 b(=)h Ft(e)725 3250 y Fd(L)p FE(t)815 3293 y Fs(\()p Fg(u)906 3320 y FF(0)945 3293 y Ft(;)17 b Fg(v)1041 3320 y FF(0)1080 3293 y Fs(\))p Fh(,)24 b(and)h(assume)f Fs(\()p Fg(u)1746 3320 y FF(0)1786 3293 y Ft(;)17 b Fg(v)1882 3320 y FF(0)1921 3293 y Fs(\))27 b Fp(2)h(K)2157 3320 y FE(k)2199 3337 y Fe(\003)2268 3293 y Fp(\010)23 b(K)2444 3320 y FE(k)2486 3337 y Fe(\003)2533 3293 y Fh(.)36 b(Then)1047 3546 y Ft(J)1102 3573 y FE(h)1149 3591 y Fl(\016)1188 3573 y FE(;\015)1263 3546 y Fs(\()p Fg(u)1354 3573 y FE(t)1388 3546 y Ft(;)17 b Fg(v)1484 3573 y FE(t)1519 3546 y Fs(\))44 b Fp(\024)g Fv(e)o(xp)1866 3465 y Fq(\000)1911 3546 y Fp(\000)p Ft(\015)6 b(t=)p Fv(80)2232 3465 y Fq(\001)2299 3546 y Fp(\001)22 b Ft(J)2404 3573 y FE(h)2451 3591 y Fl(\016)2490 3573 y FE(;\015)2565 3546 y Fs(\()p Fg(u)2656 3573 y FF(0)2695 3546 y Ft(;)17 b Fg(v)2791 3573 y FF(0)2830 3546 y Fs(\))25 b Ft(;)697 b Fs(\()p Fv(6)p Ft(:)p Fv(18)p Fs(\))94 3779 y Fh(where)1384 3898 y Ft(\015)49 b Fs(=)c Fv(min)o Fs(\()p Ft(\021)1854 3856 y Fr(\000)p FF(2)1956 3898 y Ft(;)17 b(k)2056 3856 y FF(2)2053 3925 y Fr(\003)2098 3898 y Ft(=C)2219 3925 y FF(15)2294 3898 y Fs(\))p Ft(=)p Fv(320)24 b Ft(:)1034 b Fs(\()p Fv(6)p Ft(:)p Fv(19)p Fs(\))316 4144 y Fv(W)-8 b(e)19 b(come)g(no)n(w)f(back)g(to)h(the)f (de\256nition)g(\(6.18\))h(of)f Ft(J)10 b Fv(,)20 b(and)f(compare)g(it)f(to)g (the)h(norm)f Fp(k)9 b(\001)g(k)3414 4170 y FE(h)3461 4188 y Fl(\016)3498 4170 y FE(;)p FF(2)3581 4144 y Fv(de\256ned)94 4263 y(in)22 b(Eq.\(3.6\).)35 b(These)22 b(tw)o(o)g(quantities)f(de\256ne)i (equi)n(v)n(alent)e(topologies)g(when)h(considered)g(on)g Fp(K)3449 4290 y FE(k)3491 4308 y Fe(\003)3555 4263 y Fp(\010)17 b(K)3725 4290 y FE(k)3767 4308 y Fe(\003)3814 4263 y Fv(.)94 4455 y Fu(Lemma)39 b(6.4.)d Fh(On)25 b Fp(K)938 4482 y FE(k)980 4499 y Fe(\003)1049 4455 y Fp(\010)d(K)1224 4482 y FE(k)1266 4499 y Fe(\003)1338 4455 y Fh(one)j(has)g(the)f(bound)546 4767 y Ft(\021)599 4725 y FF(2)639 4767 y Ft(\015)713 4623 y Fq(\014)713 4683 y(\014)713 4743 y(\014)713 4802 y(\014)746 4632 y(Z)862 4767 y Ft(h)919 4794 y FE(\016)963 4687 y Fq(\000)1009 4767 y Fg(uv)e Fs(+)g Fg(u)1285 4725 y Fr(0)1313 4767 y Fg(v)1364 4725 y Fr(0)1392 4687 y Fq(\001)1437 4623 y(\014)1437 4683 y(\014)1437 4743 y(\014)1437 4802 y(\014)1515 4767 y Fp(\024)1637 4632 y Fq(Z)1753 4767 y Ft(h)1810 4794 y FE(\016)1854 4627 y Fq(\022)1939 4728 y FF(1)p 1939 4745 35 4 v 1939 4802 a(2)1986 4687 y Fq(\000)2032 4767 y Fs(\()p Fg(u)2123 4725 y Fr(0)2150 4767 y Fs(\))2189 4725 y FF(2)2251 4767 y Fs(+)g(\()p Fg(u)2441 4725 y Fr(00)2491 4767 y Fs(\))2530 4725 y FF(2)2570 4687 y Fq(\001)2638 4767 y Fs(+)2749 4728 y FF(1)p 2749 4745 V 2749 4802 a(8)2796 4687 y Fq(\000)2842 4767 y Fg(v)2893 4725 y FF(2)2955 4767 y Fs(+)h(\()p Fg(v)3145 4725 y Fr(0)3172 4767 y Fs(\))3211 4725 y FF(2)3251 4687 y Fq(\001)3296 4627 y(\023)3395 4767 y Ft(:)94 5172 y Fu(Remark.)48 b Fv(This)33 b(lemma)h(eliminates)f(the)h (some)n(what)f(arbitrary)h(quantity)f Ft(\015)39 b Fv(from)34 b(the)g(topology)-6 b(,)35 b(see)94 5291 y(Theorem)25 b(6.5)f(belo)n(w)-6 b(.)p eop %%Page: 19 19 bop 94 -45 a Fo(D)m(amped)38 b(Hyperbolic)f(Equa)-7 b(tions)2130 b Fn(19)94 194 y Fu(Pro)s(of.)43 b Fv(This)28 b(is)f(a)i(combination)e(of)h (earlier)h(estimates.)45 b(Indeed,)29 b(we)g(ha)n(v)o(e)f(already)g(seen)h (in)f(Eq.\(6.12\))94 314 y(that)d(the)f(mix)o(ed)g(terms)g(in)h(Eq.\(6.7\))f (can)h(be)g(bounded)f(by)383 635 y Ft(\021)436 592 y FF(2)476 635 y Ft(\015)550 491 y Fq(\014)550 550 y(\014)550 610 y(\014)550 670 y(\014)583 499 y(Z)699 635 y Ft(h)756 662 y FE(\016)801 554 y Fq(\000)846 635 y Fg(uv)e Fs(+)g Fg(u)1122 592 y Fr(0)1150 635 y Fg(v)1201 592 y Fr(0)1229 554 y Fq(\001)1275 491 y(\014)1275 550 y(\014)1275 610 y(\014)1275 670 y(\014)1352 635 y Fp(\024)45 b Ft(\021)1527 592 y FF(2)1583 499 y Fq(Z)1700 635 y Ft(h)1757 662 y FE(\016)1801 495 y Fq(\022)1874 635 y Fv(8)p Ft(\015)1982 592 y FF(2)2021 554 y Fq(\000)2066 635 y Fg(u)2118 592 y FF(2)2180 635 y Fs(+)23 b(\()p Fg(u)2371 592 y Fr(0)2398 635 y Fs(\))2437 592 y FF(2)2476 554 y Fq(\001)2544 635 y Fs(+)2656 596 y FF(1)p 2656 612 35 4 v 2656 669 a(8)2703 554 y Fq(\000)2748 635 y Fg(v)2799 592 y FF(2)2861 635 y Fs(+)g(\()p Fg(v)3051 592 y Fr(0)3079 635 y Fs(\))3118 592 y FF(2)3157 554 y Fq(\001)3203 495 y(\023)3320 635 y Fp(\021)45 b Ft(X)32 b(:)94 955 y Fv(Furthermore,)25 b(by)g(\(6.15\))f(and)h(the)g(choice)g(of)g Ft(k)1793 982 y Fr(\003)1838 955 y Fv(,)g(we)g(get)713 1281 y Ft(X)52 b Fp(\024)44 b Ft(\021)1022 1238 y FF(2)1062 1281 y Ft(C)1133 1308 y FF(15)1208 1281 y Ft(k)1263 1238 y Fr(\000)p FF(2)1260 1308 y Fr(\003)1382 1146 y Fq(Z)1498 1281 y Ft(h)1555 1308 y FE(\016)1599 1141 y Fq(\022)1672 1281 y Fv(8)p Ft(\015)1780 1238 y FF(2)1819 1201 y Fq(\000)1865 1281 y Fs(\()p Fg(u)1956 1238 y Fr(0)1983 1281 y Fs(\))2022 1238 y FF(2)2084 1281 y Fs(+)23 b(\()p Fg(u)2275 1238 y Fr(00)2325 1281 y Fs(\))2364 1238 y FF(2)2403 1201 y Fq(\001)2471 1281 y Fs(+)2583 1242 y FF(1)p 2583 1258 V 2583 1316 a(8)2630 1201 y Fq(\000)2675 1281 y Fg(v)2726 1238 y FF(2)2788 1281 y Fs(+)g(\()p Fg(v)2978 1238 y Fr(0)3005 1281 y Fs(\))3044 1238 y FF(2)3084 1201 y Fq(\001)3130 1141 y(\023)3228 1281 y Ft(:)94 1621 y Fv(Since)j(we)f(ha)n(v)o(e)f(also)h(chosen)f Ft(\015)50 b Fs(=)44 b Fv(min)o Fs(\()p Ft(\021)1648 1578 y Fr(\000)p FF(2)1750 1621 y Ft(;)17 b(k)1850 1578 y FF(2)1847 1648 y Fr(\003)1892 1621 y Ft(=C)2013 1648 y FF(15)2088 1621 y Fs(\))p Ft(=)p Fv(320,)24 b(we)h(get)g(\256nally)963 1948 y Ft(X)52 b Fp(\024)1220 1813 y Fq(Z)1336 1948 y Ft(h)1393 1975 y FE(\016)1437 1808 y Fq(\022)1522 1909 y FF(1)p 1522 1926 V 1522 1983 a(2)1569 1868 y Fq(\000)1615 1948 y Fs(\()p Fg(u)1706 1906 y Fr(0)1733 1948 y Fs(\))1772 1906 y FF(2)1834 1948 y Fs(+)22 b(\()p Fg(u)2024 1906 y Fr(00)2074 1948 y Fs(\))2113 1906 y FF(2)2153 1868 y Fq(\001)2221 1948 y Fs(+)2332 1909 y FF(1)p 2332 1926 V 2332 1983 a(8)2379 1868 y Fq(\000)2425 1948 y Fg(v)2476 1906 y FF(2)2538 1948 y Fs(+)h(\()p Fg(v)2728 1906 y Fr(0)2755 1948 y Fs(\))2794 1906 y FF(2)2834 1868 y Fq(\001)2879 1808 y(\023)2978 1948 y Ft(;)94 2265 y Fv(which)i(is)f(what)h (we)g(asserted.)p 3615 2358 85 2 v 3615 2439 2 81 v 3698 2439 V 3615 2441 85 2 v 316 2697 a(Recall)40 b(the)g(de\256nition)e(\(6.2\))i(of)g (the)f(projection)g Fm(P)2218 2724 y FE(k)2260 2742 y Fe(\003)2346 2697 y Fv(onto)g(momenta)g(lar)n(ger)h(than)f Ft(k)3507 2724 y Fr(\003)3553 2697 y Fv(.)80 b(From)94 2816 y(Lemma)23 b(6.4)g(and)g 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Fs(\()p Ft(\021)800 4249 y Fr(\000)p FF(2)902 4292 y Ft(;)17 b(k)1002 4249 y FF(2)999 4319 y Fr(\003)1044 4292 y Ft(C)1115 4319 y FF(15)1190 4292 y Fs(\))p Ft(=)p Fv(320)p Fh(.)p eop %%Page: 20 20 bop 94 -45 a Fo(D)m(amped)38 b(Hyperbolic)f(Equa)-7 b(tions)2130 b Fn(20)94 194 y Fw(7.)46 b(The)e(ev)l(olution)j(of)e(di\013erences)316 513 y Fv(In)23 b(this)g(section,)g(we)g(combine)g(the)g(results)g(of)g (Sections)g(3)h(and)f(6)g(into)g(bounds)f(on)h(the)h(e)n(v)n(olution)d(of)94 632 y(the)i(dif)n(ference)g(of)g(solutions)e(to)i(\(2.1\).)35 b(W)-8 b(e)23 b(will)f(\256rst)h(treat)g(the)g(general)g(case,)h(and)f(sho)n (w)e(a)i(bound)g(which)94 752 y(di)n(v)o(er)n(ges)g(e)o(xponentially)e(with)i (time,)g(and)g(then)g(we)h(will)e(treat)i(the)f(high)g(frequenc)o(y)g(case)h (where)g(we)f(ha)n(v)o(e)94 871 y(decay)-6 b(.)46 b(W)-8 b(e)29 b(consider)f(tw)o(o)g(initial)e(conditions,)i(and)g(their)g(respecti)n(v)o(e) f(e)n(v)n(olutions)f(under)i(the)g(semi-\257o)n(w)94 991 y(de\256ned)h(by)e 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b(u)2294 5046 y FF(2)p FE(;t)2387 5019 y Fs(\))p Fp(j)30 b Fv(and)h(its)f(deri)n(v)n(ati)n(v)o(e)e(are)j (bounded)f(and)94 5139 y(then)25 b(the)g(result)f(follo)n(ws)f(at)i(once)g (from)g(the)f(representation)h(\(7.4\))f(and)h(the)g(bound)f(of)h(Theorem)f (3.3.)p 3615 5226 85 2 v 3615 5307 2 81 v 3698 5307 V 3615 5309 85 2 v eop %%Page: 21 21 bop 94 -45 a Fo(D)m(amped)38 b(Hyperbolic)f(Equa)-7 b(tions)2130 b Fn(21)316 194 y Fv(The)25 b(handling)e(of)i(the)g(high)f(frequenc)o(y)h (part)f Fm(P)1993 221 y FE(k)2035 239 y Fe(\003)2082 194 y Fs(\()p Fg(u)2173 221 y FE(t)2208 194 y Ft(;)17 b Fg(v)2304 221 y FE(t)2338 194 y Fs(\))25 b Fv(is)f(similar)-5 b(.)34 b(Instead)25 b(of)g(\(7.2\),)f(we)h(get)893 375 y Fq( )984 475 y Ft(@)1037 501 y FE(t)1071 475 y Fm(P)1137 501 y FE(k)1179 519 y Fe(\003)1226 475 y Fg(u)1278 501 y FE(t)984 617 y Ft(@)1037 644 y FE(t)1072 617 y Fm(P)1138 644 y FE(k)1180 662 y Fe(\003)1227 617 y Fg(v)1278 644 y FE(t)1325 375 y Fq(!)1448 545 y Fs(=)45 b Fm(L)1658 375 y 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FE(t)3038 1715 y Fs(=)k Ft(u)3204 1742 y FF(1)p FE(;t)3322 1715 y Fp(\000)24 b Ft(u)3480 1742 y FF(2)p FE(;t)3602 1715 y Fh(and)k(let)94 1835 y Fg(v)145 1862 y FE(t)208 1835 y Fs(=)g Ft(v)361 1862 y FF(1)p FE(;t)477 1835 y Fp(\000)23 b Ft(v)625 1862 y FF(2)p FE(;t)719 1835 y Fh(.)36 b(There)25 b(are)h(constants)d Ft(C)1650 1862 y FF(21)1725 1835 y Fh(,)i Ft(C)1846 1862 y FF(22)1921 1835 y Fh(,)g(and)g Ft(C)2211 1862 y FF(23)2310 1835 y Fh(such)g(that)f(for)h (all)g Ft(t)j(>)g Fv(0)p Fh(,)633 2165 y Fp(k)p Fs(\()p Fm(P)788 2192 y FE(k)830 2210 y Fe(\003)876 2165 y Fg(u)928 2192 y FE(t)963 2165 y Ft(;)17 b Fm(P)1074 2192 y FE(k)1116 2210 y Fe(\003)1162 2165 y Fg(v)1213 2192 y FE(t)1248 2165 y Fs(\))p Fp(k)1337 2192 y FE(h)1384 2210 y Fl(\016)1422 2192 y FE(;)p FF(2)1530 2165 y Fp(\024)1652 2025 y Fq(\022)1725 2165 y Ft(C)1796 2192 y FF(21)1871 2165 y Ft(e)1917 2122 y Fr(\000)p FE(\015)t(t=)p FF(80)2194 2165 y Fs(+)22 b Ft(C)2364 2192 y FF(22)2451 2098 y Ft(e)2497 2055 y FE(C)2554 2073 y 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Fq(\015)518 5249 y(\015)574 5274 y Fm(P)640 5301 y FE(k)682 5319 y Fe(\003)728 5193 y Fq(\000)774 5274 y Ft(')h Fp(\000)h Ft( )1030 5193 y Fq(\001)1075 5189 y(\015)1075 5249 y(\015)1131 5314 y FE(\016)n(;L)p Fr(\000)p FE(A=";)p FF(2)1552 5274 y Fp(\024)1673 5189 y Fq(\015)1673 5249 y(\015)1729 5274 y Fm(P)1795 5301 y FE(k)1837 5319 y Fe(\003)1884 5193 y Fq(\000)1929 5274 y Fs(\010)2001 5231 y FE(\034)2038 5249 y Fe(\003)2085 5274 y Fs(\()p Ft(')2189 5301 y FF(0)2228 5274 y Fs(\))f Fp(\000)h Fs(\010)2461 5231 y FE(\034)2498 5249 y Fe(\003)2544 5274 y Fs(\()p Ft( )2648 5301 y FF(0)2688 5274 y Fs(\))2727 5193 y Fq(\001)2772 5189 y(\015)2772 5249 y(\015)2827 5314 y FE(\016)n(;L;)p FF(2)3048 5274 y Fp(\024)45 b Ft(\015)3228 5231 y Fr(\000)p Fc({)3351 5274 y Ft(")26 b(:)p eop %%Page: 23 23 bop 94 -45 a Fo(D)m(amped)38 b(Hyperbolic)f(Equa)-7 b(tions)2130 b Fn(23)94 194 y FG(W)-9 b(e)31 b(now)g(\256x)g Ft(\021)614 221 y FF(0)690 194 y Ft(>)36 b Fv(0)31 b FG(so)f(small)g(and)g Ft(k)1482 221 y FF(0)1553 194 y FG(so)g(lar)l(g)o(e)g(\(and)h(at)f(least)g (satisfying)f(Eqs.\(6.8\))h(and)g(\(6.9\)\))h(suc)o(h)94 314 y(that)24 b(for)h(all)f Ft(\021)31 b(<)e(\021)781 341 y FF(0)846 314 y FG(and)24 b(all)g Ft(k)1202 341 y Fr(\003)1276 314 y Ft(>)k(k)1433 341 y FF(0)1497 314 y FG(one)d(has)1094 598 y Ft(\015)1152 555 y Fr(\000)p Fc({)1320 598 y Fs(=)1442 517 y Fq(\000)1487 598 y Fv(min)o Fs(\()p Ft(\021)1734 555 y Fr(\000)p FF(2)1836 598 y Ft(;)17 b(k)1936 555 y FF(2)1933 625 y Fr(\003)1978 598 y Ft(=C)2099 625 y FF(15)2174 598 y Fs(\))p Ft(=)p Fv(320)2413 517 y Fq(\001)2458 555 y Fr(\000)p Fc({)2626 598 y Fp(\024)2760 531 y Fv(1)p 2760 575 50 4 v 2760 666 a(8)2847 598 y Ft(;)794 b Fs(\()p Fv(8)p Ft(:)p Fv(8)p Fs(\))94 857 y Fv(and)25 b(also)g Ft(\015)32 b(>)d(\015)691 884 y FF(0)730 857 y Fv(,)c(see)g(\(8.4\).)36 b(Summarizing)24 b(the)g(bounds)g(for)h(this)f(piece,)h(we)g(get)1356 1117 y Fp(k)p Fm(P)1472 1144 y FE(k)1514 1162 y Fe(\003)1561 1117 y Fs(\()p Ft(')d Fp(\000)g Ft( )t Fs(\))p Fp(k)1944 1144 y FE(\016)n(;L)p Fr(\000)p FE(A=";)p FF(2)2364 1117 y Fp(\024)2500 1050 y Ft(")p 2498 1094 V 2498 1186 a Fv(8)2585 1117 y Ft(:)1056 b Fs(\()p Fv(8)p Ft(:)p Fv(9)p Fs(\))316 1384 y Fv(W)-8 b(e)33 b(bound)g(ne)o(xt)f Fm(Q)1031 1411 y FE(k)1073 1429 y Fe(\003)1120 1384 y Fs(\()p Ft(')26 b Fp(\000)h Ft( )t Fs(\))33 b Fv(by)g(decomposing)e (it)i(into)f(a)h(\256nite)g(sum)f(plus)g(a)i(remainder)-5 b(.)60 b(W)-8 b(e)94 1503 y(will)31 b(w)o(ork)g(with)g(the)h(tw)o(o)f(components)f (of)h Fm(Q)1752 1530 y FE(k)1794 1548 y Fe(\003)1841 1503 y Ft(')h Fv(or)g Fm(Q)2119 1530 y FE(k)2161 1548 y Fe(\003)2208 1503 y Ft( )j Fv(separately)-6 b(.)55 b(Since)32 b(the)f(norm)g(on)h(the)f (\256rst)94 1623 y(component)23 b(has)h(2)g(deri)n(v)n(ati)n(v)o(es)d(and)j (the)g(norm)f(on)g(the)h(second)g(only)f(1,)h(we)g(will)f(deal)h(only)f(with) g(the)h(\256rst)94 1742 y(case)i(and)f(lea)n(v)o(e)f(the)h(other)g(case)g(to) f(the)h(reader)-5 b(.)316 1862 y(W)d(e)25 b(will)f(w)o(ork)h(with)f(the)g (notion)g(of)h(Bernstein)g(class)f Fm(B)2317 1889 y FE(\033)2371 1862 y Fs(\()p Ft(K)7 b Fs(\))p Fv(,)24 b(de\256ned)h(by)819 2156 y Fm(B)895 2182 y FE(\033)949 2156 y Fs(\()p Ft(K)7 b Fs(\))43 b(=)1284 2015 y Fq(\032)1359 2156 y Ft(h)1441 2011 y Fq(\014)1441 2071 y(\014)1441 2131 y(\014)1441 2190 y(\014)1499 2156 y Fp(j)p Ft(h)p Fs(\()p Ft(x)22 b Fs(+)h Ft(iy)t Fs(\))p Fp(j)k(\024)h Ft(K)7 b(e)2226 2113 y FE(\033)s Fr(j)p FE(y)s Fr(j)2395 2156 y Fv(for)25 b(all)f Ft(x;)17 b(y)31 b Fp(2)d Fu(R)3022 2015 y Fq(\033)3121 2156 y Ft(:)94 2447 y Fv(If)i Ft(h)35 b Fp(2)f Fm(B)458 2474 y FE(\033)512 2447 y Fs(\()p Ft(K)7 b Fs(\))p Fv(,)30 b(it)e(can)i(be)g(represented)f(by)h(the)f (Cartwright)g(interpolation)f(formula)h([KT)-7 b(,)29 b(Eq.)49 b(\(191\)])94 2567 y(\(or)26 b([B])f(for)g(a)g(proof\))g(with)f Ft(\033)1140 2524 y Fr(0)1196 2567 y Fs(=)k Ft(\031)t(=)p Fv(2)c(and)g Ft(!)32 b Fs(=)c Ft(\031)t(=)p Fv(4\))c(as)934 2878 y Ft(h)p Fs(\()p Ft(x)p Fs(\))45 b(=)1304 2811 y Fv(sin)p Fs(\()p Fv(2)p Ft(\033)t(x)p Fs(\))p 1304 2855 362 4 v 1460 2947 a Fv(4)1774 2754 y Fr(1)1742 2784 y Fq(X)1694 2996 y FE(j)t Fj(=)p Fr(\0001)1934 2878 y Fs(\()p Fp(\000)p Fv(1)p Fs(\))2139 2835 y FE(j)2192 2805 y Fv(sin)p Fs(\()2360 2766 y FE(\033)s(x)p 2359 2782 94 4 v 2389 2839 a FF(2)2487 2805 y Fp(\000)2599 2761 y FE(\031)s(j)p 2599 2782 87 4 v 2625 2839 a FF(4)2697 2805 y Fs(\))p 2192 2855 544 4 v 2231 2958 a(\()2282 2919 y FE(\033)s(x)p 2281 2935 94 4 v 2311 2992 a FF(2)2409 2958 y Fp(\000)2521 2914 y FE(\031)s(j)p 2521 2935 87 4 v 2546 2992 a FF(4)2619 2958 y Fs(\))2658 2929 y FF(2)2748 2878 y Ft(h)p Fs(\()p Ft(x)2901 2905 y FE(j)2943 2878 y Fs(\))24 b Ft(;)585 b Fs(\()p Fv(8)p Ft(:)p Fv(10)p Fs(\))94 3232 y Fv(where)26 b(the)e Ft(x)566 3259 y FE(j)636 3232 y Fs(=)753 3188 y FE(j)t(\031)p 753 3209 V 754 3267 a FF(2)p FE(\033)876 3232 y Fv(are)i(discrete)e(sampling)g (points.)34 b(This)24 b(class)h(is)f(useful)g(in)h(our)g(conte)o(xt)e (because)j(of)94 3424 y Fu(Lemma)39 b(8.2.)d Fh(There)25 b(is)g(a)g(constant) f Ft(C)1561 3451 y FF(26)1660 3424 y Fh(such)h(that)f(if)h Ft(u)j Fp(2)g Fv(L)2368 3381 y Fr(1)2452 3424 y Fh(,)d(then)1410 3656 y Fm(Q)1476 3682 y FE(k)1518 3700 y Fe(\003)1565 3656 y Ft(u)j Fp(2)g Fm(B)1820 3682 y FF(2)p FE(k)1897 3700 y Fe(\003)1944 3656 y Fs(\()p Ft(C)2054 3682 y FF(26)2128 3656 y Ft(k)2180 3682 y Fr(\003)2226 3656 y Fp(k)p Ft(u)p Fp(k)2383 3682 y Fr(1)2467 3656 y Fs(\))d Ft(:)1060 b Fs(\()p Fv(8)p Ft(:)p Fv(11)p Fs(\))94 4004 y Fu(Pro)s(of.)40 b Fv(This)24 b(amounts)g(to)h(saying)g(that)f(a)i (function)e(with)g(frequenc)o(y)i(support)e(in)g([)p Fp(\000)p Fv(2)p Ft(k)3261 4031 y Fr(\003)3308 4004 y Ft(;)17 b Fv(2)p Ft(k)3455 4031 y Fr(\003)3499 4004 y Fv(])26 b(is)e(in)h(the)94 4123 y(Bernstein)g(class.)36 b(This)23 b(is)i(almost)e(ob)o(vious,)g(e)o (xcept)h(for)h(the)f(smooth)g(cutof)n(f.)35 b(In)25 b(f)o(act,)g(with)e(the)i (function)94 4243 y Ft(#)g Fv(as)g(de\256ned)g(in)g(Section)g(5,)f(we)h (consider)948 4379 y Fq(Z)1045 4514 y Fv(d)p Ft(k)19 b(e)1212 4472 y FE(ik)r Fj(\()p FE(x)p Fj(+)p FE(iy)s Fj(\))1529 4514 y Ft(#)p Fs(\()p Ft(k)s(=k)1784 4541 y Fr(\003)1830 4514 y Fs(\))44 b(=)g Ft(k)2086 4541 y Fr(\003)2149 4379 y Fq(Z)2245 4514 y Fv(d)p Ft(`)16 b(e)2399 4472 y FE(ik)2469 4490 y Fe(\003)2512 4472 y FE(`)p Fj(\()p FE(x)p Fj(+)p FE(iy)s Fj(\))2790 4514 y Ft(#)p Fs(\()p Ft(`)p Fs(\))23 b Ft(;)599 b Fs(\()p Fv(8)p Ft(:)p Fv(12)p Fs(\))94 4812 y Fv(which)25 b(is)f(in)h(L)618 4769 y FF(1)657 4812 y Fs(\()13 b Fv(d)p Ft(x)k Fs(\))24 b Fv(for)h(an)o(y)g Ft(y)31 b Fp(2)d Fu(R)p Fv(.)36 b(And)24 b(the)h(L)1930 4769 y FF(1)1970 4812 y Fs(\()13 b Fv(d)p Ft(x)j Fs(\))25 b Fv(norm)f(is)g(bounded)g(by)1495 5055 y Fm(O)p Fs(\()p Fv(1)p Fs(\))1693 4974 y Fq(\000)1738 5055 y Ft(k)1790 5081 y Fr(\003)1858 5055 y Fs(+)f Ft(k)2013 5012 y Fr(\000)p FF(1)2010 5081 y Fr(\003)2115 4974 y Fq(\001)2161 5055 y Ft(e)2207 5012 y FF(2)p FE(k)2284 5030 y Fe(\003)2326 5012 y Fr(j)p FE(y)s Fr(j)2446 5055 y Ft(:)94 5286 y Fv(Therefore,)j(the)f(con)l(v)n(olution)e (operator)h(de\256ned)i(by)e(\(8.12\))h(maps)f Ft(u)h Fv(to)f Fm(B)2739 5313 y FF(2)p FE(k)2816 5331 y Fe(\003)2863 5286 y Fs(\()p Fm(O)p Fs(\()p Ft(k)3063 5313 y Fr(\003)3109 5286 y Fs(\))p Fp(k)p Ft(u)p Fp(k)3305 5313 y Fr(1)3389 5286 y Fs(\))p Fv(.)p eop %%Page: 24 24 bop 94 -45 a Fo(D)m(amped)38 b(Hyperbolic)f(Equa)-7 b(tions)2130 b Fn(24)p 3615 129 85 2 v 3615 210 2 81 v 3698 210 V 3615 212 85 2 v 316 418 a Fv(W)-8 b(e)25 b(ne)o(xt)f(bound)g(the)h(functions)e (appearing)i(in)g(\(8.10\))f(in)h(our)f(f)o(a)n(v)n(orite)h(topology:)94 610 y Fu(Lemma)39 b(8.3.)d Fh(Let)25 b Ft(\033)31 b(>)d Fv(2)d Fh(and)f(let)h Ft(f)1483 637 y FE(j)1550 610 y Fh(be)g(de\256ned)g(by)443 910 y Ft(f)492 937 y FE(j)533 910 y Fs(\()p Ft(x)p Fs(\))44 b Fp(\021)846 836 y Fv(sin)962 756 y Fq(\000)1008 836 y Fv(4)p Fs(\()1109 797 y FE(\033)s(x)p 1108 813 94 4 v 1138 871 a FF(2)1236 836 y Fp(\000)1348 792 y FE(\031)s(j)p 1348 813 87 4 v 1373 871 a 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b(is)e(a)h(constant)f Ft(C)940 1216 y FF(27)1040 1189 y Fh(independent)g(of)h Ft(j)30 b Fh(and)25 b Ft(\030)5 b Fh(,)24 b(such)h(that)f(for)h(all)g Ft(j)30 b Fh(and)25 b Ft(\030)k Fh(one)c(has:)382 1334 y Fq(Z)478 1469 y Fv(d)p Ft(x)17 b(h)659 1496 y FE(\016)703 1469 y Fs(\()p Ft(x)22 b Fp(\000)g Ft(\030)5 b Fs(\))1008 1329 y Fq(\022)1080 1469 y Ft(f)1140 1426 y FF(2)1129 1496 y FE(j)1180 1469 y Fs(\()p Ft(x)p Fs(\))21 b(+)i Fv(2)1486 1388 y Fq(\000)1531 1469 y Ft(f)1591 1426 y Fr(0)1580 1496 y FE(j)1622 1469 y Fs(\()p Ft(x)p Fs(\))1757 1388 y Fq(\001)1802 1426 y FF(2)1864 1469 y Fs(+)1964 1388 y Fq(\000)2010 1469 y Ft(f)2070 1426 y Fr(00)2059 1496 y FE(j)2120 1469 y Fs(\()p Ft(x)p Fs(\))2255 1388 y Fq(\001)2300 1426 y FF(2)2340 1329 y Fq(\023)2457 1469 y Fp(\024)2806 1402 y Ft(\033)2867 1359 y FF(4)2907 1402 y Ft(C)2978 1429 y FF(27)p 2591 1446 677 4 v 2591 1537 a Fv(1)f Fs(+)h(\()p Fv(2)p Ft(\033)t(\030)i Fp(\000)e Ft(\031)t(j)6 b Fs(\))3229 1509 y FF(4)3304 1469 y Ft(:)287 b Fs(\()p Fv(8)p Ft(:)p Fv(13)p Fs(\))94 1833 y Fu(Remark.)52 b Fv(The)39 b(numerical)f(coef)n(\256cient)h Ft(C)1734 1860 y FF(27)1848 1833 y Fv(depends)f(on)h Ft(\016)t Fv(,)j(b)n(ut)c Ft(\016)43 b Fv(has)c(been)g(\256x)o(ed)g(in)f(Eq.\(6.11\):) 94 1953 y Ft(\016)32 b Fs(=)c Fv(1)p Ft(=)p Fs(\()p Fv(40)p Ft(C)585 1980 y FF(18)659 1953 y Fs(\))p Fv(.)94 2124 y Fu(Pro)s(of.)40 b Fv(The)25 b(function)f Ft(f)1029 2151 y FE(j)1095 2124 y Fv(can)h(be)g(bounded)f(as)653 2435 y Fp(j)p Ft(f)730 2462 y FE(j)772 2435 y Fs(\()p Ft(x)p Fs(\))p Fp(j)43 b(\024)1100 2260 y Fq(\014)1100 2320 y(\014)1100 2380 y(\014)1100 2440 y(\014)1100 2500 y(\014)1145 2361 y Fv(sin)1261 2280 y Fq(\000)1307 2361 y Fv(4)p Fs(\()1408 2322 y FE(\033)s(x)p 1407 2338 94 4 v 1437 2395 a FF(2)1535 2361 y Fp(\000)1647 2317 y FE(\031)s(j)p 1647 2338 87 4 v 1672 2395 a FF(4)1745 2361 y Fs(\))1784 2280 y Fq(\001)1846 2361 y Fv(sin)o Fs(\()2013 2322 y FE(\033)s(x)p 2013 2338 94 4 v 2042 2395 a FF(2)2141 2361 y Fp(\000)2252 2317 y FE(\031)s(j)p 2252 2338 87 4 v 2278 2395 a FF(4)2350 2361 y Fs(\))p 1145 2412 1245 4 v 1509 2514 a Fv(4)p Fs(\()1610 2475 y FE(\033)s(x)p 1609 2492 94 4 v 1639 2549 a FF(2)1737 2514 y Fp(\000)1849 2470 y FE(\031)s(j)p 1849 2492 87 4 v 1874 2549 a FF(4)1947 2514 y Fs(\))1986 2486 y FF(2)2401 2260 y Fq(\014)2401 2320 y(\014)2401 2380 y(\014)2401 2440 y(\014)2401 2500 y(\014)2479 2435 y Fp(\024)2859 2367 y Ft(C)2930 2394 y FF(28)p 2612 2412 639 4 v 2612 2514 a Fv(1)22 b Fs(+)h(\()2835 2475 y FE(\033)s(x)p 2835 2492 94 4 v 2864 2549 a FF(2)2963 2514 y Fp(\000)3074 2470 y FE(\031)s(j)p 3074 2492 87 4 v 3100 2549 a FF(4)3172 2514 y Fs(\))3211 2486 y FF(2)3288 2435 y Ft(;)94 2722 y Fv(since)k(the)f(numerator)h(v)n(anishes)e(simultaneously)f (with)i(the)g(denominator)g(\(and)g(to)g(order)h(2\).)42 b(Similarly)-6 b(,)94 2842 y(the)25 b(deri)n(v)n(ati)n(v)o(e)d(is)j(bounded)f(by)361 2959 y Fq(\014)361 3019 y(\014)361 3078 y(\014)361 3138 y(\014)361 3198 y(\014)395 3133 y Ft(@)454 3090 y FE(`)448 3160 y(x)514 2963 y Fq( )605 3060 y Fv(sin)721 2979 y Fq(\000)767 3060 y Fv(4)p Fs(\()868 3020 y FE(\033)s(x)p 868 3037 94 4 v 897 3094 a FF(2)995 3060 y Fp(\000)1107 3015 y FE(\031)s(j)p 1107 3037 87 4 v 1133 3094 a FF(4)1205 2979 y Fq(\001)1267 3060 y Fv(sin)p Fs(\()1435 3020 y FE(\033)s(x)p 1434 3037 94 4 v 1464 3094 a FF(2)1562 3060 y Fp(\000)1674 3015 y FE(\031)s(j)p 1674 3037 87 4 v 1699 3094 a FF(4)1772 3060 y Fs(\))p 605 3110 1206 4 v 950 3213 a Fv(4)p Fs(\()1051 3174 y FE(\033)s(x)p 1050 3190 94 4 v 1080 3247 a FF(2)1178 3213 y Fp(\000)1290 3168 y FE(\031)s(j)p 1290 3190 87 4 v 1315 3247 a FF(4)1388 3213 y Fs(\))1427 3184 y FF(2)1823 2963 y Fq(!)1901 2959 y(\014)1901 3019 y(\014)1901 3078 y(\014)1901 3138 y(\014)1901 3198 y(\014)1979 3133 y Fp(\024)2246 3066 y Ft(C)2317 3093 y FF(29)2392 3066 y Fs(\()p Ft(\033)t(=)p Fv(2)p Fs(\))2631 3023 y FE(`)p 2113 3110 689 4 v 2113 3213 a Fv(1)e Fs(+)g Fv(4)p Fs(\()2385 3174 y FE(\033)s(x)p 2385 3190 94 4 v 2414 3247 a FF(2)2513 3213 y Fp(\000)2624 3168 y FE(\031)s(j)p 2624 3190 87 4 v 2650 3247 a FF(4)2722 3213 y Fs(\))2761 3184 y FF(2)2838 3133 y Ft(;)116 b(`)27 b Fs(=)h Fv(1)p Ft(;)17 b Fv(2)24 b Ft(;)266 b Fs(\()p Fv(8)p Ft(:)p Fv(14)p Fs(\))94 3420 y Fv(since)25 b Ft(\033)31 b(>)d Fv(2)d(by)g (assumption.)33 b(It)25 b(follo)n(ws)f(that)429 3538 y Fq(Z)525 3673 y Fv(d)p Ft(x)17 b(h)706 3700 y FE(\016)n(;\030)808 3673 y Fs(\()p Ft(x)p Fs(\))p Fp(j)p Ft(f)1020 3700 y FE(j)1061 3673 y Fs(\()p Ft(x)p Fs(\))p Fp(j)1224 3631 y FF(2)1307 3673 y Fp(\024)44 b Ft(C)1499 3700 y FF(30)1591 3538 y Fq(Z)1687 3673 y Fv(d)p Ft(x)2158 3606 y Fv(1)p 1823 3650 722 4 v 1823 3742 a Fs(\()p Fv(1)21 b Fs(+)i Ft(\016)2081 3713 y FF(2)2121 3742 y Fs(\()p Ft(x)f Fp(\000)g Ft(\030)5 b Fs(\))2426 3713 y FF(2)2465 3742 y Fs(\))2504 3713 y FF(2)2594 3673 y Fp(\001)3049 3606 y Fv(1)p 2673 3650 803 4 v 2673 3742 a Fs(\()p Fv(1)21 b Fs(+)i(\()p Fv(2)p Ft(\033)t(x)e Fp(\000)i Ft(\031)t(j)5 b Fs(\))3357 3713 y FF(2)3397 3742 y Fs(\))3436 3713 y FF(2)3512 3673 y Ft(:)94 3937 y Fv(Setting)25 b Ft(\032)i Fs(=)h Fv(min)o Fs(\()p Ft(\016)n(;)17 b Fv(2)p Ft(\033)t Fs(\))p Fv(,)24 b(we)h(\256nd)f (that)h(this)f(is)g(bounded)g(by)966 4057 y Fq(Z)1062 4193 y Fv(d)p Ft(x)16 b(h)1242 4220 y FE(\016)n(;\030)1344 4193 y Fs(\()p Ft(x)p Fs(\))p Fp(j)p Ft(f)1556 4220 y FE(j)1597 4193 y Fs(\()p Ft(x)p Fs(\))p Fp(j)1760 4150 y FF(2)1843 4193 y Fp(\024)1977 4126 y Ft(C)2048 4153 y FF(31)p 1977 4170 146 4 v 2024 4261 a Ft(\032)2518 4126 y Fv(1)p 2147 4170 792 4 v 2147 4192 a Fq(\000)2193 4273 y Fv(1)22 b Fs(+)g Ft(\032)2416 4244 y FF(2)2456 4273 y Fs(\()p Ft(\030)k Fp(\000)2676 4228 y FE(\031)s(j)p 2676 4250 87 4 v 2677 4307 a FF(2)p FE(\033)2774 4273 y Fs(\))2813 4244 y FF(2)2853 4192 y Fq(\001)2899 4231 y FF(2)2975 4193 y Ft(:)94 4470 y Fv(In)f(vie)n(w)f(of)h(\(8.14\))g(one)g (gets)f(a)h(similar)f(bound)g(for)h(the)g(deri)n(v)n(ati)n(v)o(es,)d(and)i (thus)g(\(8.13\))h(follo)n(ws.)p 3615 4551 85 2 v 3615 4631 2 81 v 3698 4631 V 3615 4633 85 2 v 316 4840 a(Consider)41 b(the)h(element)f Fs(\()p Ft(u;)17 b(v)t Fs(\))50 b Fp(2)h(G)6 b Fv(.)87 b(W)-8 b(e)42 b(kno)n(w)e(that)i Fp(k)p Fs(\()p Ft(u;)17 b(v)t Fs(\))p Fp(k)2770 4867 y FE(\016)n(;)p FF(loc)n FE(;)p FF(2)3026 4840 y Fp(\024)51 b Ft(C)3225 4867 y FF(9)3265 4840 y Fv(.)87 b(F)o(or)42 b(the)f(\256rst)94 4960 y(component,)24 b Ft(u)p Fv(,)h(this)f(means)736 5221 y(sup)726 5308 y FE(\030)r Fr(2)p Fk(R)902 5086 y Fq(Z)998 5221 y Fv(d)p Ft(x)16 b(h)1178 5248 y FE(\016)1223 5221 y Fs(\()p Ft(x)21 b Fp(\000)i Ft(\030)5 b Fs(\))1528 5081 y Fq(\022)1600 5221 y Fp(j)p Ft(u)p Fs(\()p Ft(x)p Fs(\))p Fp(j)1848 5178 y FF(2)1909 5221 y Fs(+)23 b Fv(2)p Fp(j)p Ft(u)2144 5178 y Fr(0)2171 5221 y Fs(\()p Ft(x)p Fs(\))p Fp(j)2334 5178 y FF(2)2395 5221 y Fs(+)g Fp(j)p Ft(u)2580 5178 y Fr(00)2630 5221 y Fs(\()p Ft(x)p Fs(\))p Fp(j)2793 5178 y FF(2)2832 5081 y Fq(\023)2950 5221 y Fp(\024)45 b Ft(C)3150 5178 y FF(2)3143 5248 y(9)3215 5221 y Ft(:)p eop %%Page: 25 25 bop 94 -45 a Fo(D)m(amped)38 b(Hyperbolic)f(Equa)-7 b(tions)2130 b Fn(25)94 194 y Fv(From)19 b(this,)g(we)g(conclude)g(using)f(the)g(Sobole)n (v)g(inequality)g(in)g(the)h(form)f(of)h(Lemma)f(2.5)h(that)f Fp(k)p Ft(u)p Fp(k)3511 221 y Fr(1)3623 194 y Fp(\024)28 b Ft(C)3799 221 y FF(32)94 314 y Fv(for)i(some)e(\256nite)h Ft(C)784 341 y FF(32)859 314 y Fv(.)49 b(By)29 b(Lemma)g(8.2)f(we)i(then)e(get)h(that) g Fp(k)p Fm(Q)2357 341 y FE(k)2399 358 y Fe(\003)2446 314 y Ft(u)p Fp(k)2553 341 y Fr(1)2671 314 y Fp(\024)34 b Ft(C)2853 341 y FF(26)2928 314 y Ft(k)2980 341 y Fr(\003)3025 314 y Ft(C)3096 341 y FF(32)3200 314 y Fv(and)29 b(furthermore,)94 433 y Fm(Q)160 460 y FE(k)202 478 y Fe(\003)249 433 y Ft(u)47 b Fp(2)g Fm(B)542 460 y FF(2)p FE(k)619 478 y Fe(\003)665 433 y Fs(\()p Ft(C)775 460 y FF(26)850 433 y Ft(k)902 460 y Fr(\003)948 433 y Ft(C)1019 460 y FF(32)1094 433 y Fs(\))p Fv(.)76 b(Thus,)41 b(we)d(can)h(apply)f(the)g (Cartwright)g(formula)g(to)g Ft(h)46 b Fs(=)h Fm(Q)3484 460 y FE(k)3526 478 y Fe(\003)3573 433 y Ft(u)p Fv(,)42 b(with)94 553 y Ft(\033)32 b Fs(=)c Fv(2)p Ft(k)390 580 y Fr(\003)435 553 y Fv(.)316 672 y(Throughout,)23 b Ft(Lk)956 699 y Fr(\003)1027 672 y Fv(has)h(to)h(be)g(suf)n(\256ciently)e(lar)n(ge.)37 b(W)-8 b(e)25 b(de\256ne)829 969 y Fm(S)883 995 y FE(L)943 969 y Fs(\()p Ft(h)p Fs(\))44 b(=)1255 901 y Fv(sin)p Fs(\()p Fv(4)p Ft(k)1513 928 y Fr(\003)1558 901 y Ft(x)p Fs(\))p 1255 946 399 4 v 1430 1037 a Fv(4)1770 874 y Fq(X)1682 1092 y Fr(j)p FE(j)t Fr(j\024)p FF(2)p FE(Lk)1961 1110 y Fe(\003)2002 969 y Fs(\()p Fp(\000)p Fv(1)p Fs(\))2207 926 y FE(j)2261 895 y Fv(sin)o Fs(\()p Ft(k)2468 922 y Fr(\003)2514 895 y Ft(x)22 b Fp(\000)2704 851 y FE(\031)s(j)p 2704 872 87 4 v 2730 930 a FF(4)2802 895 y Fs(\))p 2261 946 581 4 v 2299 1048 a(\()p Ft(k)2390 1075 y Fr(\003)2436 1048 y Ft(x)g Fp(\000)2626 1004 y FE(\031)s(j)p 2626 1025 87 4 v 2652 1083 a FF(4)2724 1048 y Fs(\))2763 1019 y FF(2)2853 969 y Ft(h)p Fs(\()p Ft(x)3006 995 y FE(j)3048 969 y Fs(\))j Ft(;)3619 987 y Fs(\()p Fv(8)p Ft(:)p Fv(15)p Fs(\))94 1316 y Fv(where)h Ft(x)420 1343 y FE(j)489 1316 y Fs(=)623 1272 y FE(j)t(\031)p 606 1293 119 4 v 606 1351 a FF(4)p FE(k)683 1369 y Fe(\003)762 1316 y Fv(are)g(the)e(discrete)h(sampling)e(points.)35 b(W)-8 b(e)25 b(decompose)1304 1589 y Fm(Q)1370 1616 y FE(k)1412 1634 y Fe(\003)1459 1589 y Ft(u)44 b Fs(=)1682 1478 y Fq(\020)1741 1589 y Ft(h)23 b Fp(\000)f Fm(S)1974 1616 y FE(L)2034 1589 y Fs(\()p Ft(h)p Fs(\))2169 1478 y Fq(\021)2250 1589 y Fs(+)h Fm(S)2404 1616 y FE(L)2464 1508 y Fq(\000)2509 1589 y Ft(h)2566 1508 y Fq(\001)2637 1589 y Ft(:)954 b Fs(\()p Fv(8)p Ft(:)p Fv(16)p Fs(\))94 1834 y Fv(The)23 b(\256rst)g(term)g(in)f(\(8.16\))h(will)f (be)g(small)g(because)h Ft(h)18 b Fp(\000)h Fm(S)2116 1861 y FE(L)2175 1834 y Fs(\()p Ft(h)p Fs(\))k Fv(is)f(the)h(remainder)g(of)f(the) h(con)l(v)o(er)n(ging)f(sum)94 1953 y(in)j(\(8.10\),)f(and)h(for)g(the)g (second)f(one)h(we)g(will)f(use)h(a)g(co)o(v)o(ering)f(ar)n(gument.)316 2073 y(W)-8 b(e)20 b(\256rst)g(sho)n(w)e(that)h Ft(X)1120 2100 y FE(L)1207 2073 y Fp(\021)28 b Ft(h)11 b Fp(\000)g Fm(S)1522 2100 y FE(L)1582 2073 y Fs(\()p Ft(h)p Fs(\))20 b Fv(is)f(small)g(when)g Ft(L)h Fv(is)f(lar)n(ge.)34 b(The)20 b(dif)n(ference)g(can)g(be)g(written)94 2192 y(as)602 2286 y Fq(\000)648 2367 y Ft(h)i Fp(\000)h Fm(S)881 2394 y FE(L)941 2367 y Fs(\()p Ft(h)p Fs(\))1076 2286 y Fq(\001)1121 2367 y Fs(\()p Ft(x)p Fs(\))44 b(=)1510 2272 y Fq(X)1422 2491 y Fr(j)p FE(j)t Fr(j)p FE(>)p FF(2)p FE(Lk)1701 2509 y Fe(\003)1742 2367 y Fs(\()p Fp(\000)p Fv(1)p Fs(\))1947 2324 y FE(j)2001 2294 y Fv(sin)o Fs(\()p Fv(4)p Ft(k)2258 2321 y Fr(\003)2303 2294 y Ft(x)p Fs(\))17 b Fv(sin)o Fs(\()p Ft(k)2623 2321 y Fr(\003)2668 2294 y Ft(x)22 b Fp(\000)2859 2249 y FE(\031)s(j)p 2859 2271 87 4 v 2884 2328 a FF(4)2957 2294 y Fs(\))p 2001 2344 995 4 v 2221 2447 a Fv(4)p Fs(\()p Ft(k)2362 2473 y Fr(\003)2407 2447 y Ft(x)h Fp(\000)2598 2402 y FE(\031)s(j)p 2598 2424 87 4 v 2624 2481 a FF(4)2696 2447 y Fs(\))2735 2418 y FF(2)3007 2367 y Ft(h)p Fs(\()3136 2300 y Ft(j)6 b(\031)p 3115 2344 148 4 v 3115 2435 a Fv(4)p Ft(k)3217 2462 y Fr(\003)3275 2367 y Fs(\))25 b Ft(:)94 2662 y Fv(Using)f(\(8.13\),)h(we)g(get)g(as)g(a)g(bound)f (for)h Ft(X)1611 2689 y FE(L)1695 2662 y FG(when)g Fp(j)p Ft(\030)5 b Fp(j)26 b(\024)i Ft(L)p Fv(:)570 2808 y Fq(\022)660 2813 y(Z)757 2949 y Fv(d)p Ft(x)16 b(h)937 2976 y FE(\016)981 2949 y Fs(\()p Ft(x)22 b Fp(\000)h Ft(\030)5 b Fs(\))1285 2868 y Fq(\000)1331 2949 y Fp(j)p Ft(X)1442 2976 y FE(L)1501 2949 y Fs(\()p Ft(x)p Fs(\))p Fp(j)1664 2906 y FF(2)1725 2949 y Fs(+)23 b Fv(2)p Fp(j)p Ft(X)1994 2906 y Fr(0)1986 2976 y FE(L)2044 2949 y Fs(\()p Ft(x)p Fs(\))p Fp(j)2207 2906 y FF(2)2268 2949 y Fs(+)g Fp(j)p Ft(X)2487 2906 y Fr(00)2479 2976 y FE(L)2538 2949 y Fs(\()p Ft(x)p Fs(\))p Fp(j)2701 2906 y FF(2)2740 2868 y Fq(\001)2786 2808 y(\023)2859 2847 y FF(1)p FE(=)p FF(2)1313 3219 y Fp(\024)45 b Ft(C)1506 3245 y FF(27)1686 3124 y Fq(X)1597 3342 y Fr(j)p FE(j)t Fr(j\025)p 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Fq(\015)144 3893 y(\015)144 3953 y(\015)200 3948 y Fm(Q)266 3975 y FE(k)308 3993 y Fe(\003)355 3948 y Fs(\()p Ft(')13 b Fp(\000)g Ft( )t Fs(\))g Fp(\000)g Fm(S)827 3975 y FE(L)886 3867 y Fq(\000)932 3948 y Fm(Q)998 3975 y FE(k)1040 3993 y Fe(\003)1087 3948 y Fs(\()p Ft(')g Fp(\000)g Ft( )t Fs(\))1402 3867 y Fq(\001)1447 3834 y(\015)1447 3893 y(\015)1447 3953 y(\015)1502 4018 y FE(\016)n(;L)p Fr(\000)p FE(A=";)p FF(2)1923 3948 y Fp(\024)2057 3881 y Ft(C)2128 3908 y FF(34)2203 3881 y Ft(")p 2057 3925 193 4 v 2116 4017 a(A)2288 3948 y Fv(sup)2278 4035 y FE(\030)r Fr(2)p Fk(R)2454 3948 y Fp(k)p Ft(')g Fp(\000)g Ft( )t Fp(k)2791 3975 y FE(h)2838 3993 y Fl(\016)n(;\030)2929 3975 y FE(;)p FF(2)3037 3948 y Fp(\024)3171 3881 y Ft(C)3242 3908 y FF(34)3317 3881 y Ft(")p 3171 3925 V 3230 4017 a(A)3375 3948 y(C)3446 3975 y FF(25)3542 3948 y Ft(;)49 b Fs(\()p Fv(8)p Ft(:)p Fv(17)p Fs(\))94 4232 y Fv(using)24 b(\(8.6\).)36 b(Clearly)-6 b(,)25 b(if)f Ft(A)h 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Fh(There)28 b(is)g(a)h(constant)f Ft(C)1803 4960 y FF(35)1906 4933 y Fh(such)h(that)f(the)g(follo)n(wing)e (holds.)46 b(Let)28 b Ft(B)34 b Fh(be)28 b(a)h(ball)f(of)94 5052 y(radius)j Ft(")h Fh(in)f(the)g(topology)f(de\256ned)i(by)f Fp(k)26 b(\001)f(k)1730 5079 y FE(\016)n(;L;)p FF(2)1906 5052 y Fh(.)56 b(Then)31 b(the)g(set)g Fm(S)2572 5079 y FE(L)2632 5052 y Fs(\()p Ft(B)f Fp(\\)c(G)6 b Fs(\))31 b Fh(can)h(be)g(co)o(v)o(ered)e (by)h(no)94 5172 y(more)25 b(than)1911 5291 y Ft(C)1989 5248 y FE(L)1982 5318 y FF(35)p eop %%Page: 26 26 bop 94 -45 a Fo(D)m(amped)38 b(Hyperbolic)f(Equa)-7 b(tions)2130 b Fn(26)94 194 y Fh(balls)25 b(of)f(radius)h Ft("=)p Fv(8)p Fh(.)94 430 y Fu(Pro)s(of.)40 b Fv(Since)25 b Ft(';)17 b( )31 b Fp(2)d(G)6 b Fv(,)25 b(Lemma)f(8.2)g(implies)g Fm(Q)1967 457 y FE(k)2009 475 y Fe(\003)2056 430 y Fs(\()p Ft(')e Fp(\000)g Ft( )t Fs(\))27 b Fp(2)h Fm(B)2586 457 y FF(2)p FE(k)2663 475 y Fe(\003)2710 430 y Fs(\()p Ft(X)8 b Fs(\))p Fv(,)24 b(where)1477 662 y Ft(X)51 b 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151 y FF(1)p FE(;)p Fr(1)1696 194 y Fv(,)h(de\256ned)g(by)1081 423 y Fp(k)p Ft(f)11 b Fp(k)1241 450 y FE(W)1327 430 y Fi(1)p Fl(;)p Fe(1)1496 423 y Fp(\021)45 b Fv(max)1789 342 y Fq(\000)1849 423 y Fv(sup)1835 509 y FE(x)p Fr(2)p Fk(R)2017 423 y Fp(j)p Ft(f)11 b Fs(\()p Ft(x)p Fs(\))p Fp(j)p Ft(;)52 b Fv(sup)2336 509 y FE(y)s Fr(2)p Fk(R)2516 423 y Fp(j)p Ft(f)2604 380 y Fr(0)2631 423 y Fs(\()p Ft(y)t Fs(\))p Fp(j)2790 342 y Fq(\001)2859 423 y Ft(:)782 b Fs(\()p Fv(9)p Ft(:)p Fv(2)p Fs(\))94 718 y Fv(This)23 b(is)f(a)h(\252good\272)g(topology)e(for)i(our)g(problem,)f (because)h(we)g(can)g(control)f(the)h(e)n(v)n(olution)e(of)i(functions)f(in) 94 838 y Ft(W)202 795 y FF(1)p FE(;)p Fr(1)345 838 y Fv(.)42 b(Ho)n(we)n(v)o(er)l(,)26 b(it)g(is)g(ob)o(vious)f(that)h(the)g (submultiplicati)n(vity)c(property)27 b(is)f(not)g(immediate,)g(since)g(the) 94 957 y(matching)e(of)h(functions)f(has)h(to)f(be)h(continuous)e(and)i(once) g(dif)n(ferentiable.)94 1309 y Fu(9.2.)39 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2604 y FE(I)1698 2577 y Fs(=)44 b Fv(max)1991 2497 y Fq(\000)2037 2577 y Fp(j)p Ft(g)t Fp(j)2145 2604 y FE(I)2189 2577 y Ft(;)17 b Fp(j)p Ft(g)2314 2534 y Fr(0)2340 2577 y Fp(j)2368 2604 y FE(I)2413 2497 y Fq(\001)2484 2577 y Ft(:)94 2782 y Fv(\(Thus,)37 b(comparing)d(with)g(\(9.2\))h(we)g(ha)n(v)o (e)f Fp(k)p Ft(g)t Fp(k)1810 2809 y FE(W)1896 2789 y Fi(1)p Fl(;)p Fe(1)2061 2782 y Fs(=)42 b Fp(k)p Ft(g)t Fp(k)2332 2809 y Fk(R)2403 2782 y Fv(.\))66 b(Assume)34 b Ft(g)2931 2809 y FF(L)3019 2782 y Fp(2)42 b Ft(W)3235 2734 y FF(1)p FE(;)p Fr(1)3221 2811 y FF([)p Fr(\000)p FE(R;)p FF(0])3488 2782 y Fv(and)35 b Ft(g)3715 2809 y FF(R)3808 2782 y Fp(2)94 2930 y Ft(W)202 2882 y FF(1)p FE(;)p Fr(1)188 2959 y FF([0)p FE(;R)p FF(])384 2930 y Fv(and)25 b(let)826 3181 y Ft(E)900 3208 y FE(";G;g)1087 3226 y Fi(L)1121 3208 y FE(;g)1184 3226 y Fi(R)1271 3181 y Fs(=)1392 3101 y Fq(\010)1450 3181 y Ft(u)j Fp(2)g(C)1687 3138 y FF(2)1727 3181 y Fs(\()p Fv([)p Fp(\000)p Ft(R)q(;)17 b(R)q Fv(])p Fs(\))51 b 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b(will)e(\256rst)i (\256nd)g(\256nite)f(constants)g Ft(a)p Fv(,)h Ft(b)p Fv(,)g Ft(c)g Fv(\()p Ft(>)k Fv(1\))c(with)f(the)g(follo)n(wing)94 1216 y(property:)36 b(Fix)24 b Ft(g)701 1242 y FF(L)773 1216 y Fv(and)h Ft(g)990 1242 y FF(R)1066 1216 y Fv(and)f(assume)g Ft(E)33 b Fp(\021)28 b Ft(E)1838 1242 y FE(";G;g)2025 1260 y Fi(L)2059 1242 y FE(;g)2122 1260 y Fi(R)2193 1216 y Fp(6)p Fs(=)g Fp(;)c Fv(\(that)h(is,)f(there)h(is)f(a)h(connecting)f(function)94 1360 y(in)i(an)g Ft(")p Fv(-neighborhood)g(of)g Ft(g)1135 1387 y FF(L)1208 1360 y Fv(and)g Ft(g)1426 1387 y FF(R)1477 1360 y Fv(\).)40 b(W)-8 b(e)27 b(claim)e(one)h(can)h(construct)e(a)i Ft(W)2884 1317 y FF(2)p FE(;)p Fr(1)3053 1360 y Fv(function)e Ft(g)k Fv(for)e(which)94 1480 y(the)e(follo)n(wing)e(inequalities)g(hold:)546 1727 y Fp(k)p Ft(g)i Fp(\000)d Ft(g)816 1754 y FF(L)863 1727 y Fp(k)913 1754 y FF([)p Fr(\000)p FE(R;)p FF(0])1190 1727 y Fp(\024)45 b Ft(a")25 b(;)116 b Fp(k)p Ft(g)25 b Fp(\000)d Ft(g)1850 1754 y FF(R)1901 1727 y 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Ft(x)j Fp(\025)g Ft(R)q Fv(.)36 b(W)-8 b(e)25 b(de\256ne)g Ft(g)j Fv(by)336 3554 y Ft(g)t Fs(\()p Ft(x)p Fs(\))43 b(=)i Ft(u)745 3581 y FF(0)785 3554 y Fs(\()p Ft(x)p Fs(\))21 b Fp(\000)i Ft( )t Fs(\()p Ft(x)p Fs(\))e Fp(\001)1316 3474 y Fq(\000)1362 3554 y Ft(u)1419 3581 y FF(0)1458 3554 y Fs(\()p Ft(R)q Fs(\))h Fp(\000)g Ft(g)1782 3581 y FF(R)1833 3554 y Fs(\()p Ft(R)q Fs(\))1988 3474 y Fq(\001)2055 3554 y Fp(\000)g Ft( )t Fs(\()p Fp(\000)p Ft(x)p Fs(\))g Fp(\001)2507 3474 y Fq(\000)2552 3554 y Ft(u)2609 3581 y FF(0)2649 3554 y Fs(\()p Fp(\000)p Ft(R)q Fs(\))g Fp(\000)h Ft(g)3051 3581 y FF(L)3098 3554 y Fs(\()p Fp(\000)p Ft(R)q Fs(\))3330 3474 y Fq(\001)3400 3554 y Ft(:)241 b Fs(\()p Fv(9)p Ft(:)p Fv(6)p Fs(\))94 3802 y Fv(This)26 b(function)g(is)g (clearly)h(continuously)e(dif)n(ferentiable)h(since)g Ft(u)2439 3829 y FF(0)2506 3802 y Fv(is)g(continuously)f(dif)n(ferentiable.)40 b(Let)94 3921 y Ft(I)c Fs(=)28 b Fv([0)p Ft(;)17 b(R)q Fv(].)35 b(From)24 b(\(9.6\))h(we)g(\256nd)g(for)g 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Fv(1)p Ft(;)17 b(:)g(:)g(:)d(;)j(N)11 b Fh(.)316 2377 y Fv(Using)32 b(Proposition)g(9.3)h(we)g(can)h(complete)e(the)h (proof)g(of)g(Theorem)g(9.1.)61 b(Gi)n(v)o(en)31 b Ft(g)3401 2403 y FF(L)3482 2377 y Fv(and)i Ft(g)3707 2403 y FF(R)3791 2377 y Fv(as)94 2496 y(abo)o(v)o(e,)24 b(we)h(construct)f(\256rst)h(a)g Ft(g)j Fv(as)d(in)f(Eq.\(9.6\).)36 b(When)24 b Ft(u)k Fp(2)g Ft(E)6 b Fv(,)24 b(then)g Ft(f)39 b Fs(=)28 b Ft(u)22 b Fp(\000)g Ft(g)28 b Fv(is)c(in)h Ft(F)3304 2523 y FE(";A;B)s(;C)3628 2496 y Fv(by)f(the)94 2616 y(bounds)i(Eq.\(9.4\))h(and)g(the)f(equality)g (\(9.5\).)42 b(Thus,)27 b(by)g(Proposition)e(9.3)i(the)g Ft(f)37 b Fv(are)28 b Ft(")p Fv(-co)o(v)o(ered)f(by)f(the)h Ft(N)94 2735 y Fv(functions)e Fp(f)p Ft(h)598 2762 y FF(1)638 2735 y Ft(;)17 b(:)g(:)g(:)d(;)j(h)917 2762 y FE(N)992 2735 y Fp(g)p Fv(.)38 b(De\256ne)27 b(no)n(w)e Ft(u)1649 2762 y FE(i)1711 2735 y Fs(=)k Ft(h)1874 2762 y FE(i)1930 2735 y Fs(+)23 b Ft(g)t Fv(,)i Ft(i)k Fs(=)g Fv(1)p Ft(;)17 b(:)g(:)g(:)d(;)j(N)11 b Fv(,)25 b(and)h(then)f(the)g(set)h Ft(E)3439 2762 y FE(";G;g)3626 2780 y Fi(L)3660 2762 y FE(;g)3723 2780 y Fi(R)3791 2735 y Fv(of)94 2855 y(functions)e Ft(u)h Fv(is)f Ft(")p Fv(-co)o(v)o(ered)h(by)g (the)g Ft(u)1409 2882 y FE(i)1442 2855 y Fv(,)g(since)1137 3098 y Ft(u)d Fp(\000)g Ft(u)1372 3125 y FE(i)1450 3098 y Fs(=)45 b(\()p Ft(u)22 b Fp(\000)g Ft(g)t Fs(\))f Fp(\000)i Fs(\()p Ft(u)2097 3125 y FE(i)2152 3098 y Fp(\000)g Ft(g)t Fs(\))43 b(=)h Ft(f)33 b Fp(\000)23 b Ft(h)2746 3125 y FE(i)2804 3098 y Ft(;)94 3342 y Fv(and)29 b(we)f(ha)n(v)o(e)g(just)f(stated)h(that)g(the)g Ft(f)38 b Fv(are)29 b Ft(")p Fv(-co)o(v)o(ered)g(by)f(a)g(\256nite)g(number)g (of)g Ft(u)2977 3369 y FE(i)3010 3342 y Fv(.)46 b(Furthermore,)29 b(the)f Ft(h)3840 3369 y FE(i)94 3461 y Fv(v)n(anish)33 b(at)h(the)f (boundary)g(of)h([)p Fp(\000)p Ft(R)q(;)17 b(R)q Fv(].)61 b(Thus,)35 b(we)f(ha)n(v)o(e)g(interpolated)e(between)i Ft(g)3154 3488 y FF(L)3235 3461 y Fv(and)f Ft(g)3460 3488 y FF(R)3511 3461 y Fv(,)j(with)d Ft(N)94 3581 y Fv(functions)24 b(in)h Ft(W)701 3538 y FF(1)p FE(;)p Fr(1)868 3581 y Fv(which)g Ft(")p Fv(-co)o(v)o(er)g(the) f(original)g(set.)36 b(The)24 b(proof)h(of)g(Theorem)g(9.1)f(is)g(complete.)p 3615 3666 85 2 v 3615 3747 2 81 v 3698 3747 V 3615 3749 85 2 v 316 3977 a(The)g(corollary)g(then)f(follo)n(ws)g(at)h(once)g(since)g(the) g(f)o(actor)g Ft(N)35 b Fv(does)24 b(not)f(depend)h(on)g(the)g(choice)g(of)g Ft(g)3827 4004 y FF(L)94 4096 y Fv(and)h Ft(g)311 4123 y FF(R)387 4096 y Fv(\(e)o(xcept)g(that)f(the)h(bound)f(is)g(too)g(pessimistic)f(in)i (case)g Ft(E)2383 4123 y FE(";G;g)2570 4141 y Fi(L)2604 4123 y FE(;g)2667 4141 y Fi(R)2734 4096 y Fv(happens)g(to)f(be)h(empty\).)94 4278 y Fu(Remark.)47 b Fv(The)34 b(dif)n(\256culty)f(in)g(pro)o(ving)g (Proposition)f(9.3)h(lies)h(in)f(the)h(f)o(act)g(that)f(the)h Ft(h)3279 4305 y FE(j)3355 4278 y Fv(v)n(anish)e(at)i(the)94 4398 y(endpoints)16 b(while)g(the)g(functions)g Ft(f)25 b Fv(in)16 b Ft(F)1495 4425 y FE(";A;B)s(;C)1811 4398 y Fv(may)g(be)g(as)g(lar)n(ge)g (as)g Ft(")g Fv(near)g(the)g(boundary)-6 b(,)21 b Fp(j)p Ft(f)11 b Fs(\()p Fp(\006)p Ft(R)q Fs(\))p Fp(j)25 b Fs(=)k Ft(")p Fv(,)94 4518 y(so)c(there)g(is)f(no)h(space)g(near)g Fp(\006)p Ft(R)h Fv(with)e(which)h(just)f(to)g(construct)g(an)h(open)g(co)o(v)o(er)-5 b(.)316 4637 y(The)26 b(main)g(ingredient)f(to)h(the)g(proof)h(of)f (Proposition)f(9.3)h(is)g(the)g(follo)n(wing)e(local)i(lemma.)40 b(Before)94 4757 y(we)28 b(formulate)f(it,)h(we)g(assume,)g(without)e(loss)h (of)g(generality)-6 b(,)28 b(that)f Ft(C)39 b(>)32 b Fv(1.)44 b(Since)28 b(we)g(are)g(interested)g(in)94 4876 y(small)c Ft(")p Fv(,)i(we)f(shall)f(also)g(assume)g Ft(")k(<)h Fv(1.)316 4996 y(W)-8 b(e)25 b(introduce)f(tw)o(o)h(fundamental)f(scales)g Ft(\030)29 b Fv(and)c Ft(\034)36 b Fv(in)24 b(our)h(analysis:)1372 5262 y Ft(\030)48 b Fs(=)1663 5194 y Ft(")p 1598 5239 178 4 v 1598 5330 a Fv(10)p Ft(C)1813 5262 y(;)115 b Fv(and)100 b Ft(\034)55 b Fs(=)2459 5194 y Ft(")p 2433 5239 100 4 v 2433 5330 a Fv(10)2569 5262 y Ft(:)p eop %%Page: 31 31 bop 94 -45 a Fo(D)m(amped)38 b(Hyperbolic)f(Equa)-7 b(tions)2130 b Fn(31)94 194 y Fv(W)-8 b(e)26 b(will)e(\256rst)h(consider)f(a)h(\(small\))f (interv)n(al)g Ft(J)35 b Fv(whose)24 b(left)h(endpoint)f(is)g(the)h(origin.) 94 375 y Fu(De\014nition)36 b(of)j Ft(R)q Fu(.)e Fv(W)-8 b(e)26 b(can)f(no)n(w)f(\256x)h Ft(R)g Fv(by)g(setting)e(it)i(to)1358 617 y(41)44 b Fp(\025)g Ft(R)h Fp(\025)g Fv(40)24 b Ft(;)116 b(R)45 b Fs(=)g Ft(m)2464 644 y Fr(\003)2510 617 y Ft(\030)29 b(;)1058 b Fs(\()p Fv(9)p Ft(:)p Fv(8)p Fs(\))94 858 y Fv(where)43 b Ft(m)467 885 y Fr(\003)556 858 y Fv(is)e(an)i(inte)o(ger)-5 b(.)86 b(This)42 b(choice)g(is)g(only)f(good)h(for)g Ft(")52 b Fp(\024)h Fv(10)p Ft(C)c Fv(and)42 b(we)g(lea)n(v)o(e)g(the)g(tri)n(vial)94 978 y(modi\256cations)24 b(for)h(arbitrary)g Ft(")g Fv(to)g(the)f(reader)-5 b(.)94 1169 y Fu(Lemma)37 b(9.4.)f Fh(Let)23 b Ft(J)38 b Fs(=)28 b Fv([0)p Ft(;)17 b(\030)5 b Fv(])p 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y Fp(\000)23 b Ft(g)2598 1714 y Fr(0)2594 1784 y FE(j)2636 1757 y Fp(j)2664 1784 y FE(J)2763 1757 y Fp(\024)45 b Ft(")2968 1689 y Fv(3)p 2943 1734 100 4 v 2943 1825 a(10)3080 1757 y Ft(;)94 2025 y Fh(and)25 b(furthermore,)g(at)g(the)f(right)g(endpoint,)g(one)h(has)1204 2267 y Fp(j)p Ft(f)11 b Fs(\()p Ft(\030)5 b Fs(\))20 b Fp(\000)i Ft(g)1586 2294 y FE(j)1628 2267 y Fs(\()p Ft(\030)5 b Fs(\))p Fp(j)42 b(\024)j Fv(max)2119 2186 y Fq(\000)2164 2267 y Ft(\016)27 b Fp(\000)22 b Ft(\026")2440 2224 y FF(2)2480 2267 y Ft(;)17 b(")2571 2224 y FF(2)2611 2267 y Ft(\027)2666 2186 y Fq(\001)2737 2267 y Ft(;)904 b Fs(\()p Fv(9)p Ft(:)p Fv(9)p Fs(\))94 2508 y Fh(where)1423 2657 y Ft(\027)51 b Fs(=)1721 2589 y Fv(1)p 1656 2634 178 4 v 1656 2725 a(40)p Ft(C)1871 2657 y(;)116 b(\026)45 b Fs(=)2342 2589 y Fv(1)p 2253 2634 228 4 v 2253 2725 a(200)p Ft(C)2518 2657 y(:)1073 b Fs(\()p Fv(9)p Ft(:)p Fv(10)p Fs(\))94 3000 y Fu(Pro)s(of)59 b(of)e(Lemma)g(9.4.)h Fv(This)43 b(is)h(just)g(a)h (construction)e(of)h(the)h(\252right\272)f Ft(j)6 b Fv(,)49 b(follo)n(wed)43 b(by)i(some)94 3120 y(v)o(eri\256cations.)35 b(Note)25 b(\256rst)g(that)f(if)h Ft(f)38 b Fp(2)28 b Ft(F)1564 3077 y FF(0)1550 3147 y FE(\016)n(;A;B)s(;C)q(;\030)1904 3120 y Fv(,)d(then)g(we)g(ha)n(v)o(e)1299 3382 y Ft(f)11 b Fs(\()p Ft(x)p Fs(\))44 b(=)g Ft(f)11 b Fs(\()p Fv(0)p Fs(\))21 b(+)i Ft(xf)2085 3339 y Fr(0)2112 3382 y Fs(\()p Fv(0)p Fs(\))f(+)g Ft(x)2418 3339 y FF(2)2458 3382 y Ft(v)t Fs(\()p Ft(x)p Fs(\))i Ft(;)1271 3532 y(f)1331 3489 y Fr(0)1359 3532 y Fs(\()p Ft(x)p Fs(\))44 b(=)g Ft(f)1719 3489 y Fr(0)1747 3532 y Fs(\()p Fv(0)p Fs(\))21 b(+)i Ft(xw)s Fs(\()p Ft(x)p Fs(\))h Ft(;)94 3773 y Fv(with)g Fp(j)p Ft(v)t Fp(j)404 3800 y FF([0)p FE(;\030)r FF(])579 3773 y Fp(\024)k Ft(C)t(=)p Fv(2)d(and)f Fp(j)p Ft(w)s Fp(j)1182 3800 y FF([0)p FE(;\030)r FF(])1358 3773 y Fp(\024)k Ft(C)7 b Fv(.)36 b(W)-8 b(e)24 b(will)g(pursue)f(the)h(proof)g(for)h(the)f (case)h(when)f Ft(f)11 b Fs(\()p Fv(0)p Fs(\))26 b 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Ft(\034)33 b Fs(+)23 b Ft(\034)11 b(\032)24 b(:)94 4930 y Fv(Clearly)-6 b(,)22 b Ft(g)j Fv(equals)c(one)g(of)g(the)h Ft(g)1238 4957 y FE(j)1301 4930 y Fv(of)f(Lemma)g(9.4)g(if)g(we)h(tak)o(e)f (the)g(\256nite)g(set)g(of)h Ft(j)27 b Fv(to)21 b(contain)g Fp(j)p Ft(j)6 b Fp(j)26 b(\024)3617 4891 y FE(B)p 3617 4907 64 4 v 3626 4965 a(\034)3707 4930 y Fs(+)14 b Fv(2.)94 5050 y(Ne)o(xt,)24 b(we)i(estimate)e(the)g(quality)g(of)h(the)f(approximation.)34 b(First)25 b(we)g(ha)n(v)o(e)648 5291 y Ft(f)708 5248 y Fr(0)736 5291 y Fs(\()p Ft(x)p Fs(\))c Fp(\000)i Ft(g)1044 5248 y Fr(0)1071 5291 y Fs(\()p Ft(x)p Fs(\))44 b(=)h Ft(f)1432 5248 y Fr(0)1459 5291 y Fs(\()p Ft(x)p Fs(\))22 b Fp(\000)g Ft(c)45 b Fs(=)f Ft(f)1984 5248 y Fr(0)2012 5291 y Fs(\()p Fv(0)p Fs(\))21 b(+)i Ft(xw)s Fs(\()p Ft(x)p Fs(\))e Fp(\000)i Ft(f)2708 5248 y Fr(0)2735 5291 y Fs(\()p Fv(0)p Fs(\))f Fp(\000)h Ft(\034)33 b Fp(\000)22 b Ft(\034)11 b(\032)24 b(:)p eop %%Page: 32 32 bop 94 -45 a Fo(D)m(amped)38 b(Hyperbolic)f(Equa)-7 b(tions)2130 b Fn(32)94 194 y Fv(This)25 b(leads,)f(for)h Ft(x)j Fp(2)g Fv([0)p Ft(;)17 b(\030)5 b Fv(],)23 b(to)1160 466 y Ft(f)1220 423 y Fr(0)1248 466 y Fs(\()p Ft(x)p Fs(\))e Fp(\000)i Ft(c)44 b Fp(\024)122 b Ft(C)7 b(\030)27 b Fp(\000)c Ft(\034)55 b Fp(\024)45 b Ft(C)7 b(\030)48 b Fp(\024)2591 398 y Ft(")p 2564 443 100 4 v 2564 534 a Fv(10)2701 466 y Ft(;)1160 710 y(f)1220 667 y Fr(0)1248 710 y Fs(\()p Ft(x)p Fs(\))21 b Fp(\000)i Ft(c)44 b Fp(\025)h(\000)p Ft(C)7 b(\030)27 b Fp(\000)c Fv(2)p Ft(\034)55 b Fp(\025)44 b(\000)2425 642 y Ft(")p 2398 687 V 2398 778 a Fv(10)2533 710 y Fp(\000)2646 642 y Fv(2)p Ft(")p 2644 687 V 2644 778 a Fv(10)2781 710 y Ft(:)94 988 y Fv(W)-8 b(e)26 b(conclude)e(that)1585 1146 y Fp(j)p Ft(f)1673 1103 y Fr(0)1722 1146 y Fp(\000)f Ft(g)1874 1103 y Fr(0)1870 1173 y FE(j)1911 1146 y Fp(j)1939 1173 y FE(J)2039 1146 y Fp(\024)2198 1079 y Fv(3)p 2173 1123 V 2173 1214 a(10)2284 1146 y Ft(")j(:)94 1382 y Fv(W)-8 b(e)26 b(consider)e(ne)o(xt)g Ft(f)11 b Fs(\()p 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b(we)g(get)g(in)g(this)e(case)j(from)e(Eq.\(9.12\),)1187 4498 y Ft(f)11 b Fs(\()p Ft(\030)5 b Fs(\))20 b Fp(\000)j Ft(c\030)49 b Fp(\024)44 b Ft(\016)27 b Fp(\000)1933 4431 y Fv(1)p 1933 4475 50 4 v 1933 4566 a(2)2078 4431 y Ft(")2124 4388 y FF(2)p 2007 4475 228 4 v 2007 4566 a Fv(100)p Ft(C)2291 4498 y Fp(\021)45 b Ft(\016)26 b Fp(\000)c Ft(\026")2688 4455 y FF(2)2754 4498 y Ft(:)837 b Fs(\()p Fv(9)p Ft(:)p Fv(16)p Fs(\))94 4779 y Fv(The)25 b(assertion)f(Eq.\(9.9\))h(follo)n(ws)e(by)i(combining)e (Eq.\(9.16\))h(with)g(Eq.\(9.15\).)316 4899 y(It)g(remains)f(to)g(see)h(that) f(the)h(set)f(of)h(possible)e Ft(j)30 b Fv(is)23 b(\256nite.)35 b(Considering)23 b(Eq.\(9.11\))g(and)h(the)f(f)o(act)i(that)94 5018 y Ft(f)45 b Fp(2)35 b Ft(F)367 4975 y FF(0)353 5045 y FE(\016)n(;A;B)s(;C)q(;\030)736 5018 y Fv(we)30 b(see)g(that)f Ft(j)36 b Fv(can)29 b(tak)o(e)h(at)g(most)e(2)p Fs(\()p Ft(B)5 b(=\034)35 b Fs(+)25 b Fv(2)p Fs(\))f(+)h Fv(1)30 b(possible)e(v)n(alues.)49 b(The)30 b(proof)f(of)94 5138 y(Lemma)c(9.4)f(is)h(complete.)p 3615 5226 85 2 v 3615 5307 2 81 v 3698 5307 V 3615 5309 85 2 v eop %%Page: 33 33 bop 94 -45 a Fo(D)m(amped)38 b(Hyperbolic)f(Equa)-7 b(tions)2130 b Fn(33)94 194 y Fu(Pro)s(of)44 b(of)e(Prop)s(osition)g(9.3.)g Fv(This)27 b(proof)i(is)f(a)h(repeated)g(application)e(of)i(Lemma)f(9.4.)46 b(W)-8 b(e)29 b(retain)94 341 y(the)g(assumptions)e(and)i(notations)f(from)g (that)h(proof.)48 b(Let)29 b Ft(\021)38 b Fs(=)c Ft(\030)5 b(\034)44 b Fs(=)2694 302 y FE(")2732 259 y Fi(2)p 2645 318 167 4 v 2645 376 a FF(100)p FE(C)2823 341 y Fv(.)49 b(W)-8 b(e)29 b(consider)g(the)g(grid)f(\(in)94 461 y(the)d Fs(\()p Ft(x;)17 b(y)t Fs(\))p Fv(-plane\):)637 592 y Fq(\010\000)741 673 y Ft(m\030)5 b(;)17 b(n\021)1035 592 y Fq(\001)1132 673 y Fv(:)52 b Ft(m)28 b Fs(=)g Fp(\000)p Ft(m)1596 700 y Fr(\003)1643 673 y Ft(;)17 b Fp(\000)p Ft(m)1852 700 y Fr(\003)1920 673 y Fs(+)23 b Fv(1)p Ft(;)17 b(:)g(:)g(:)d(;)j(m)2379 700 y Fr(\003)2449 673 y Fv(;)41 b Ft(n)28 b Fs(=)g Fp(\000)p Ft(n)2848 700 y Fr(\003)2894 673 y Ft(;)17 b(:)g(:)g(:)d(;)j(n)3176 700 y Fr(\003)3221 592 y Fq(\011)3304 673 y Ft(;)94 886 y Fv(where)1293 1005 y Ft(m)1380 1032 y Fr(\003)1471 1005 y Fs(=)44 b Ft(R)q(=\030)29 b(;)116 b(n)1996 1032 y Fr(\003)2086 1005 y Fs(=)44 b Fv([)p Ft(A=\021)t Fv(])22 b Fs(+)h Fv(1)i Ft(;)94 1183 y Fv(recalling)k(that)g Ft(R)q(=\030)j Fv(is)d(an)g(inte)o(ger)-5 b(.)47 b(In)29 b(other)g(w)o(ords,) g(we)h(co)o(v)o(er)e(the)h(range)g(of)g(possible)f(ar)n(guments)g(\(in)94 1302 y([)p Fp(\000)p Ft(R)q(;)17 b(R)q Fv(]\))34 b(and)h(v)n(alues)f(\(in)g ([)p Fp(\000)p Ft(A;)17 b(A)p Fv(]\))34 b(of)h Ft(f)52 b Fp(2)41 b Ft(F)1909 1329 y FE(";A;B)s(;C)2243 1302 y Fv(by)34 b(a)h(\256ne)g(grid.)65 b(Consider)35 b(no)n(w)e(the)i(set)f(of)94 1422 y(all)i(continuous,)i(piece)n (wise)d(linear)h(functions)f Ft(h)p Fs(\()p Ft(x)p Fs(\))p Fv(,)k(connecting)c(linearly)h(successi)n(v)o(e)f(lattice)g(points)94 1541 y Fs(\()p Ft(m\030)5 b(;)17 b(n\021)t Fs(\))25 b Fv(with)g Fs(\(\()p Ft(m)d Fs(+)i Fv(1)p Fs(\))p Ft(\030)5 b(;)17 b(n)1225 1498 y Fr(0)1250 1541 y Ft(\021)t Fs(\))p Fv(,)26 b(with)f Fp(\000)p Ft(m)1760 1568 y Fr(\003)1836 1541 y Fp(\024)30 b Ft(m)f(<)h(m)2253 1568 y Fr(\003)2299 1541 y Fv(,)c Fp(j)p Ft(n)p Fp(j)j(\024)g Ft(n)2661 1568 y Fr(\003)2733 1541 y Fv(and)d Fp(j)p Ft(n)2991 1498 y Fr(0)3018 1541 y Fp(j)j(\024)g Ft(n)3241 1568 y Fr(\003)3287 1541 y Fv(.)39 b(Furthermore,)94 1661 y(we)31 b(require)g(that)g Ft(h)p Fs(\()p Fp(\000)p Ft(R)q Fs(\))k(=)i Ft(h)p Fs(\()p Ft(R)q Fs(\))e(=)h Fv(0.)53 b(There)32 b(are)f(a)g(\256nite)g (number)f(of)h(such)f(functions,)h(namely)f(at)94 1780 y(most)24 b Fs(\()p Fv(2)p Ft(n)462 1807 y Fr(\003)529 1780 y Fs(+)f Fv(1)p Fs(\))718 1738 y FF(2)p FE(m)824 1755 y Fe(\003)865 1738 y Fr(\000)p FF(1)967 1780 y Fv(.)316 1900 y(Note)29 b(that)g Ft(\021)k Fv(has)d(been)f(chosen)h(in)e(such)i(a)f(w)o(ay)h(that)f(the)g (slopes)f(of)i(the)f(straight)g(pieces)g(of)g Ft(h)h Fv(are)94 2019 y(inte)o(ger)f(multiples)f(of)h Ft(\034)11 b Fv(.)50 b(W)-8 b(e)30 b(sho)n(w)e(ne)o(xt)h(that)g(e)n(v)o(ery)f Ft(f)45 b Fp(2)34 b Ft(F)2322 2046 y FE(";A;B)s(;C)2651 2019 y Fv(is,)c(together)f (with)g(its)g(deri)n(v)n(ati)n(v)o(e,)94 2139 y Ft(")p Fv(-close)d(to)e(one)h (of)g(the)f Ft(h)p Fv(.)316 2259 y(W)-8 b(e)37 b(be)o(gin)f(by)g (constructing)f(the)i(piece)n(wise)f(linear)h(approximation)e Ft(h)p Fv(.)71 b(W)-8 b(e)38 b(start)e(at)h(the)f(point)94 2378 y Ft(x)28 b Fs(=)g Fp(\000)p Ft(R)q Fv(,)d Ft(y)31 b Fs(=)d Fv(0,)d(and)g(shift)f(the)g(origin)g(to)h(this)f(point)f(by)i(de\256ning:) 1567 2591 y Ft(f)1616 2617 y FF(0)1656 2591 y Fs(\()p Ft(x)p Fs(\))44 b(=)g Ft(f)11 b Fs(\()p Ft(x)22 b Fs(+)g Ft(R)q Fs(\))i Ft(:)94 2803 y Fv(Then)32 b Ft(f)380 2830 y FF(0)452 2803 y Fv(is)f(in)g Ft(F)737 2760 y FF(0)723 2830 y FE(";A;B)s(;C)q(;)p FF(2)p FE(R)1175 2803 y Fp(\033)38 b Ft(F)1368 2760 y FF(0)1354 2830 y FE(\016)n(;A;B)s(;C)q(;\030)1708 2803 y Fv(,)33 b(with)e Ft(\016)42 b Fs(=)c Ft(")p Fv(,)c(and)d(by)h(Lemma)f(9.4,)i Ft(f)3156 2830 y FF(0)3228 2803 y Fv(is)e(approximated)94 2935 y(by)c(one)g(of)h(the)e(linear)i(functions,)e(say)h Ft(n)1549 2962 y FF(0)1589 2935 y Ft(\034)11 b(x)p Fv(,)27 b(with)f Ft(n)2017 2962 y FF(0)2088 2935 y Fs(=)31 b Fv([2)24 b Fs(+)g Ft(f)2464 2892 y Fr(0)2453 2962 y FF(0)2492 2935 y Fs(\()p Fv(0)p Fs(\))p Ft(=\034)11 b Fv(])26 b(on)h(the)g(interv)n(al)f([0)p Ft(;)17 b(\030)5 b Fv(])26 b(\(when)94 3055 y Ft(f)11 b Fs(\()p Fv(0)p Fs(\))27 b Ft(>)h Fv(0\).)36 b(Note)25 b(that)f(we)h(also)f(ha)n(v)o(e)h (\(when)g Ft(")g Fv(is)f(small\))g Fp(j)p Ft(f)2274 3081 y FF(0)2314 3055 y Fs(\()p Ft(\030)5 b Fs(\))21 b Fp(\000)h Ft(n)2621 3081 y FF(0)2661 3055 y Ft(\034)11 b(\030)5 b Fp(j)26 b(\024)i Ft(")23 b Fp(\000)g Ft(\026")3199 3012 y FF(2)3239 3055 y Fv(.)36 b(W)-8 b(e)25 b(de\256ne)1034 3267 y Ft(h)p Fs(\()p Ft(x)p Fs(\))44 b(=)h Ft(n)1452 3294 y FF(0)1492 3267 y Ft(\034)11 b Fs(\()p Ft(x)22 b Fs(+)g Ft(R)q Fs(\))i Ft(;)41 b Fv(for)25 b Ft(x)j Fp(2)g Fv([)p Fp(\000)p Ft(R)q(;)17 b Fp(\000)p Ft(R)23 b Fs(+)f Ft(\030)5 b Fv(])24 b Ft(:)94 3479 y Fv(Ne)o(xt)h(shift)f(the)g (origin)g(of)h(the)g Fs(\()p Ft(x;)17 b(y)t Fs(\))p Fv(-plane)23 b(to)h Fs(\()p Fp(\000)p Ft(R)f Fs(+)g Ft(\030)5 b(;)17 b(n)2254 3506 y FF(0)2292 3479 y Ft(\021)t Fs(\))27 b(=)2516 3399 y Fq(\000)2562 3479 y Fp(\000)p Ft(R)c Fs(+)g Ft(\030)5 b(;)17 b(h)p Fs(\()p Ft(\030)5 b Fs(\))3116 3399 y Fq(\001)3159 3479 y Ft(;)24 b Fv(and)h(de\256ne)640 3692 y Ft(f)689 3719 y FF(1)729 3692 y Fs(\()p Ft(x)p Fs(\))44 b(=)g Ft(f)11 b Fs(\()p Ft(x)22 b Fs(+)g Ft(R)h Fp(\000)f Ft(\030)5 b Fs(\))22 b Fp(\000)g Ft(h)p Fs(\()p Ft(x)g Fs(+)h Ft(R)g Fp(\000)f Ft(\030)5 b Fs(\))43 b(=)i Ft(f)11 b Fs(\()p Ft(x)21 b Fs(+)i Ft(R)g Fp(\000)f Ft(\030)5 b Fs(\))21 b Fp(\000)i Ft(n)3183 3719 y FF(0)3223 3692 y Ft(\021)29 b(:)94 3904 y Fv(The)c(de\256nition)f(of)h(the)g(\256rst)g(se)o(gment)e(of)i Ft(h)g Fv(and)g(the)g(bounds)e(on)i Ft(f)2463 3931 y FF(0)2528 3904 y Fv(sho)n(w)e(that)1359 4117 y Ft(f)1408 4144 y FF(1)1475 4117 y Fp(2)28 b Ft(F)1647 4074 y FF(0)1633 4144 y FE(")p Fr(\000)p FE(\026")1819 4124 y Fi(2)1849 4144 y FE(;A)p Fj(+)p Fr(j)p FE(n)2067 4162 y Fi(0)2097 4144 y Fr(j)p FE(\021)s(;B)s(;C)q(;)p FF(2)p FE(R)p Fr(\000)p FE(\030)2582 4117 y Ft(:)316 4355 y Fv(W)-8 b(e)31 b(no)n(w)f(apply)g(Lemma)g(9.4)g(to)g Ft(f)1576 4382 y FF(1)1616 4355 y Fv(.)53 b(Note)31 b(that)f Ft(f)2148 4382 y FF(1)2218 4355 y Fv(is)h(not)f(in)g Ft(F)2660 4313 y FF(0)2646 4382 y FE(\016)n(;A;B)s(;C)q(;\030)3031 4355 y Fv(b)n(ut)g(in)g(a)h (space)g(with)f(a)94 4475 y(w)o(orse)c(bound)f(on)g(the)g(absolute)g(v)n (alue.)38 b(Ho)n(we)n(v)o(er)l(,)24 b(the)h(v)n(alue)g(of)h Ft(A)f Fv(does)h FG(not)f Fv(enter)g(the)h(construction)e(of)94 4595 y(the)30 b(proof)f(of)g(Lemma)g(9.4)g(and)g(hence)h(is)f(irrele)n(v)n (ant)f(for)h(our)g(inducti)n(v)o(e)f(construction)g(of)h Ft(h)p Fv(.)50 b(Applying)94 4714 y(Lemma)27 b(9.4)f(to)h Ft(f)731 4741 y FF(1)770 4714 y Fv(,)h(we)f(\256nd)g(the)f(second)h(linear)g(piece)g (of)g(the)g(function)f Ft(h)p Fv(,)h(and)g(get)g(a)g(piece)n(wise)f(linear)l (,)94 4834 y(continuous)g(approximation)g(of)h Ft(f)38 b Fv(on)27 b([)p Fp(\000)p Ft(R)q(;)17 b Fp(\000)p Ft(R)24 b Fs(+)g Fv(2)p Ft(\030)5 b Fv(].)43 b(The)27 b(\256nal)h(point)e(of)i(the)f(approximation)f (by)h Ft(h)94 4953 y Fv(is)h(no)n(w)f Fs(\()p Fp(\000)p Ft(R)e Fs(+)f Fv(2)p Ft(\030)5 b(;)17 b(n)908 4980 y FF(1)946 4953 y Ft(\021)t Fs(\))p Fv(,)29 b(and)f(we)g(construct)g Ft(f)1851 4980 y FF(2)1918 4953 y Fv(by)g(translating)f(the)h(origin)f(to)h(that)f (point.)45 b(Assuming)94 5073 y(that)25 b Ft(")d Fp(\000)h Fv(2)p Ft(\026")593 5030 y FF(2)661 5073 y Ft(>)28 b(\027)6 b(")867 5030 y FF(2)908 5073 y Fv(,)25 b(we)g(conclude)f(that)1254 5285 y Ft(f)1303 5312 y FF(2)1370 5285 y Fp(2)k Ft(F)1542 5242 y FF(0)1528 5313 y FE(")p Fr(\000)p FF(2)p FE(\026")1749 5293 y Fi(2)1779 5313 y FE(;A)p Fj(+)p Fr(j)p FE(n)1997 5331 y Fi(0)2026 5313 y Fj(+)p FE(n)2136 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Ft(h)2041 4824 y FE(f)2093 4797 y Fs(\()p Ft(x)p Fs(\))p Fv(.)89 b(It)42 b(is)g(clearly)h(one)f(of)h(the)f(functions)g(we)94 4929 y(constructed.)37 b(It)25 b(approximates)f Ft(f)36 b Fv(and)25 b Ft(f)1578 4886 y Fr(0)1631 4929 y Fv(on)g(all)g(of)g([)p Fp(\000)p Ft(R)q(;)17 b(R)q Fv(].)36 b(W)-8 b(e)26 b(ha)n(v)o(e)f(thus)f (found)h(a)g(\256nite)h(f)o(amily)e(of)94 5048 y(piece)n(wise)h(linear)f (functions)g(which)h Ft(")p Fv(-co)o(v)o(ers)f Ft(F)1851 5075 y FE(";A;B)s(;C)2150 5048 y Fv(.)36 b(The)25 b(proof)g(of)g(Proposition)e (9.3)h(is)h(complete.)p 3615 5124 85 2 v 3615 5205 2 81 v 3698 5205 V 3615 5207 85 2 v eop %%Page: 35 35 bop 94 -45 a Fo(D)m(amped)38 b(Hyperbolic)f(Equa)-7 b(tions)2130 b Fn(35)94 194 y Fu(9.3.)39 b(The)e Ft(")p Fu(-en)m(trop)m(y)h(of)g (Kolmogoro)m(v)i(and)e(Tikhomiro)m(v)94 383 y Fv(W)-8 b(e)31 b(proceed)f(as)g(in)g([CE1],)h(b)n(ut)e(with)h(a)g(change)g(of)g(topology)e (as)i(e)o(xplained)f(abo)o(v)o(e.)50 b(W)-8 b(e)30 b(ha)n(v)o(e)g(de\256ned) 94 503 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1380 y FE(L)p Fj(+)p FF(2)p FE(A=")1208 1460 y FF(24)1526 1431 y Fp(\001)17 b(\001)g(\001)d Ft(C)1736 1380 y FE(L)p Fj(+)p FE(mA=")1729 1460 y FF(24)2067 1431 y Ft(N)2147 1458 y FE(L)p Fj(+)p FE(A=")p Fj(+)p FF(2)p FE(A=")p Fj(+)p Fr(\001\001\001)m Fj(+)p FE(mA=")3043 1431 y Fs(\()p Fv(2)3132 1389 y FE(m)3207 1431 y Ft(")p Fs(\))25 b Ft(:)274 b Fs(\()p Fv(9)p Ft(:)p Fv(21)p Fs(\))94 1672 y Fv(In)31 b(\(2.20\))g(we)f (ha)n(v)o(e)h(sho)n(wn)e(that)h(there)h(is)f(a)h(constant)f Ft(C)2153 1629 y Fr(0)2146 1699 y FF(9)2217 1672 y Fv(which)g(bounds)g(the)g (radius)g(of)h Fp(G)37 b Fv(in)30 b Fm(H)3643 1699 y FE(\016)n(;)p FF(loc)p FE(;)p FF(2)3849 1672 y Fv(.)94 1804 y(\(The)i(bound)e Ft(C)665 1831 y FF(9)737 1804 y Fv(in)g(\(2.20\))h(w)o(as)h(for)f Fm(H)1539 1831 y FE(\013;)p FF(loc)o FE(;)p FF(2)1763 1804 y Fv(.\))55 b(Therefore,)33 b FG(one)e Fv(ball)g(of)h(radius)e Ft(C)3149 1761 y Fr(0)3142 1831 y FF(9)3214 1804 y Fv(suf)n(\256ces)h(to)g (co)o(v)o(er)94 1936 y Fp(G)6 b(j)187 1963 y FF([)p Fr(\000)p FE(L;L)p FF(])434 1936 y Fv(.)35 b(Choosing)23 b Ft(m)28 b Fs(=)h Ft(m)p Fs(\()p Ft(")p Fs(\))24 b Fv(in)f(such)g(a)h(w)o(ay)g(that)f(2) 2143 1893 y FE(m)2219 1936 y Ft(")28 b(>)g(C)2476 1893 y Fr(0)2469 1963 y FF(9)2509 1936 y Fv(,)c(we)g(conclude)f(that)h Fp(G)6 b(j)3348 1963 y FF([)p Fr(\000)p FE(L;L)p FF(])3618 1936 y Fv(can)24 b(be)94 2076 y(co)o(v)o(ered)h(by)f(a)h FG(\256nite)g Fv(number)f(of)h(balls)f(in)g Fm(H)1691 2103 y FE(\016)n(;)p FF(loc)p FE(;)p FF(2)1897 2076 y Fv(.)1922 2034 y FF(1)94 2252 y Fu(Remark.)40 b Fv(This)25 b(ar)n(gument)h(does)f(not)h(use)g(compactness)f (of)h Fp(G)2391 2279 y FF([)p Fr(\000)p FE(L;L)p FF(])2664 2252 y Fv(and)g(does)g(not)f(pro)o(v)o(e)g(it)g(either)-5 b(.)39 b(It)94 2372 y(is)33 b(here)g(that)g(our)g(method)e(dif)n(fers)i(from)g(that) f(of)h(Feireisl[F].)h(He)f(sho)n(ws)e(that)i(the)f(intersection)g(of)h(the)94 2492 y Fs(\010)166 2449 y FE(T)229 2492 y Fs(\()p Fp(G)6 b(j)361 2518 y FF([)p Fr(\000)p FE(L;L)p FF(])607 2492 y Fs(\))25 b Fv(is)g(compact,)f(whereas)i(our)f(approach)g(sho)n(ws)f(that)g Fp(G)6 b(j)2589 2518 y FF([)p Fr(\000)p FE(L;L)p FF(])2861 2492 y Fv(itself)24 b(can)i(be)f(co)o(v)o(ered)f(by)h(a)94 2611 y(\256nite)g(number)f(of)h(balls.)316 2731 y(W)-8 b(e)21 b(de\256ne)h(similarly)e(for)h(an)o(y)g(interv)n(al)f Ft(I)8 b Fv(,)21 b(the)g(minimal)f(number)g Ft(M)2749 2758 y FE(I)2795 2731 y Fs(\()p Ft(")p Fs(\))h Fv(of)g(balls)g(needed)g(to)g(co)o(v)o(er)94 2850 y Fp(G)32 b Fv(in)24 b(the)h(topology)e Fp(k)f(\001)g(k)985 2883 y FE(W)1071 2849 y Fi(1)p Fl(;)p Fe(1)1060 2909 y Fl(I)1196 2850 y Fv(.)35 b(By)26 b(the)e(Sobole)n(v)g(inequality)g(from)g(Lemma)g(2.5,) h(we)g(see)g(that)1430 3099 y Fp(k)p Ft(u)p Fp(k)1587 3132 y FE(W)1673 3098 y Fi(1)p Fl(;)p Fe(1)1662 3157 y Fi([)p Fe(\000)p Fl(L;L)p Fi(])1921 3099 y Fp(\024)44 b Ft(C)2113 3126 y FF(4)2153 3099 y Fp(k)p Ft(u)p Fp(k)2310 3126 y FE(\016)n(;L;)p FF(2)2511 3099 y Ft(:)94 3357 y Fv(Therefore,)29 b(if)e Fp(G)34 b Fv(can)27 b(be)h(co)o(v)o(ered)f(by)g Ft(N)1555 3384 y FE(L)1614 3357 y Fs(\()p Ft("=C)1820 3384 y FF(4)1860 3357 y Fs(\))g Fv(balls)g(of)g(radius) g Ft("=C)2690 3384 y FF(4)2758 3357 y Fv(in)g(the)g(norm)g Fp(k)c(\001)g(k)3424 3384 y FE(\016)n(;L;)p FF(2)3600 3357 y Fv(,)28 b(it)f(can)94 3476 y(ob)o(viously)21 b(be)i(co)o(v)o(ered)f(by)h (the)g(same)g(number)f(of)h(balls)f(of)h(radius)g Ft(")g Fv(in)g(the)g(norm)f Fp(k)c(\001)f(k)3185 3509 y FE(W)3271 3475 y Fi(1)p Fl(;)p Fe(1)3260 3535 y Fi([)p Fe(\000)p Fl(L;L)p Fi(])3475 3476 y Fv(.)36 b(Thus)22 b(we)94 3622 y(ha)n(v)o(e)1429 3741 y Ft(M)1526 3768 y FF([)p Fr(\000)p FE(L;L)p FF(])1772 3741 y Fs(\()p Ft(")p Fs(\))45 b Fp(\024)f Ft(N)2142 3768 y FE(L)2202 3741 y Fs(\()p Ft("=C)2408 3768 y FF(4)2448 3741 y Fs(\))25 b Ft(:)1079 b Fs(\()p Fv(9)p Ft(:)p Fv(22)p Fs(\))94 3924 y Fv(W)-8 b(e)19 b(no)n(w)f(apply)g(Corollary)g(9.2.)34 b(W)-8 b(e)19 b(\256rst)f(note)g(that) g(by)g(Eqs.\(2.22\))g(and)h(\(2.23\))f(it)g(is)g(adequate)g(to)g(consider)94 4044 y(functions)34 b(with)f FG(bounded)h Fv(second)g(deri)n(v)n(ati)n(v)o (e.)61 b(\(In)35 b(f)o(act)f(this)g(is)g(the)g(only)f(place)i(where)f(these)g (higher)94 4163 y(deri)n(v)n(ati)n(v)o(es)17 b(are)j(needed.\))35 b(Thus,)20 b(we)f(can)h(apply)f(Corollary)g(9.2)g(and)h(we)g(conclude)f(that) g(for)g(tw)o(o)g(interv)n(als)94 4283 y Ft(I)138 4310 y FF(1)203 4283 y Fv(and)25 b Ft(I)416 4310 y FF(2)455 4283 y Fv(,)g(one)g(has)1316 4402 y Ft(M)1413 4429 y FE(I)1448 4447 y Fi(1)1478 4429 y Fr([)p FE(I)1567 4447 y Fi(2)1601 4402 y Fs(\()p Ft(")p Fs(\))45 b Fp(\024)f Ft(M)1988 4429 y FE(I)2023 4447 y Fi(1)2058 4402 y Fs(\()p Ft(")p Fs(\))p Ft(M)2279 4429 y FE(I)2314 4447 y Fi(2)2349 4402 y Fs(\()p Ft(")p Fs(\))p Ft(K)2558 4429 y FE(")2625 4402 y Ft(:)966 b Fs(\()p Fv(9)p Ft(:)p Fv(23)p Fs(\))94 4585 y Fv(Thus,)21 b(we)f(ha)n(v)o(e)f(established)g(submultiplicati)n(vity)c (\(in)k Ft(I)8 b Fv(\))20 b(and)g(\256niteness)f(of)h Ft(M)2908 4612 y FE(I)2953 4585 y Fs(\()p Ft(")p Fs(\))p Fv(.)35 b(Furthermore,)20 b(from)94 4705 y(the)25 b(construction)f(of)g Ft(m)i Fv(in)e(\(9.21\))h(with) f(2)1599 4662 y FE(m)1674 4705 y Ft(")29 b(>)f(C)1932 4662 y Fr(0)1925 4732 y FF(9)1965 4705 y Fv(,)d(we)g(\256nd)g(by)f(choosing)g(the) g(minimal)f(such)i Ft(m)p Fv(:)1143 4977 y Ft(M)1240 5004 y FE(I)1286 4977 y Fs(\()p Ft(")p Fs(\))44 b Fp(\024)h Ft(C)1654 4924 y FE(C)1711 4942 y Fi(40)1779 4924 y FF(log)q Fj(\()p FE(")1938 4881 y Fe(\000)p Fi(1)2021 4924 y Fj(\))p Fr(j)p FE(I)5 b Fr(j)p Fj(+)p FF(log)q Fj(\()p FE(")2360 4881 y Fe(\000)p Fi(1)2443 4924 y Fj(\))p FE(")2512 4881 y Fe(\000)p Fi(1)2596 4924 y FE(C)2653 4942 y Fi(41)2708 4924 y FE(A)1647 5006 y FF(24)2798 4977 y Ft(:)p 94 5093 1200 4 v 241 5159 a FD(1)316 5202 y FC(The)16 b(ar)o(gument)f(used)h(here)g(is)g(more)g(ele)o(gant)g(than) g(the)g(one)g(used)g(in)g([CE1].)26 b(W)-6 b(e)16 b(thank)g(Y)-10 b(.)15 b(Colin)h(de)g(V)-9 b(erdi)3315 5201 y(\301)3311 5202 y(ere)16 b(for)f(suggesting)94 5291 y(it.)p eop %%Page: 36 36 bop 94 -45 a Fo(D)m(amped)38 b(Hyperbolic)f(Equa)-7 b(tions)2130 b Fn(36)94 194 y Fv(Using)24 b(this)g(bound)g(and)h(\(9.23\),)g(we)g(get)f (con)l(v)o(er)n(gence)h(and)g(a)g(bound)f(on)h(the)f Ft(")p Fv(-entrop)o(y)h Ft(H)3342 221 y FE(")3385 194 y Fs(\()p Fp(G)6 b Fs(\))p Fv(:)94 386 y Fu(Theorem)38 b(9.5.)e Fh(The)25 b Ft(")p Fh(-entrop)o(y)g(per)g(unit)f(length)g(of)h Fp(G)31 b Fh(in)24 b Ft(W)2397 343 y FF(1)p FE(;)p Fr(1)2565 386 y Fh(e)o(xists)f(and)i(is)g(bounded)f(by)889 678 y Ft(H)972 705 y FE(")1014 678 y Fs(\()p Fp(G)6 b Fs(\))44 b(=)85 b Fv(lim)1323 743 y FE(L)p Fr(!1)1574 611 y Fv(1)p 1565 655 68 4 v 1565 747 a Ft(L)1662 678 y Fv(log)1789 598 y Fq(\000)1835 678 y Ft(M)1932 705 y FF([)p Fr(\000)p FE(L;L)p FF(])2178 678 y Fs(\()p Ft(")p Fs(\))2302 598 y Fq(\001)2392 678 y Fp(\024)45 b Ft(C)2585 705 y FF(42)2676 678 y Fv(log)o Fs(\()p Fv(1)p Ft(=")p Fs(\))25 b Ft(:)539 b Fs(\()p Fv(9)p Ft(:)p Fv(24)p Fs(\))94 1262 y Fu(9.4.)39 b(Existence)e(of)h(the)f(top)s(ological)h(en)m(trop)m(y)f(p)s(er)h (unit)f(length)94 1458 y Fv(This)21 b(material)f(is)h(tak)o(en)f(from)h ([CE2],)h(and)f(we)g(introduce)g(it)f(without)f(proofs,)j(just)e(to)g(sho)n (w)g(what)h(follo)n(ws)94 1577 y(from)k(the)g(bounds)e(of)i(the)g(pre)n (vious)f(sections.)316 1697 y(F)o(or)29 b(an)o(y)g Ft(")35 b(>)f Fv(0)c(and)f(an)o(y)g(interv)n(al)f Ft(I)38 b Fv(in)29 b Fu(R)p Fv(,)h(we)g(de\256ne)g Fp(W)2457 1654 y FE(")2449 1724 y(I)2530 1697 y Fv(as)f(the)g(set)h(of)f(all)g(\256nite)h(co)o(v)o(ers)e (of)i Fp(G)94 1829 y Fv(by)e(open)f(sets)g(in)g Ft(W)833 1781 y FF(1)p FE(;)p Fr(1)819 1858 y FE(I)1004 1829 y Fv(of)h(diameter)f(at)g (most)g Ft(")p Fv(.)44 b(Note)28 b(that)f(by)g(the)g(ar)n(gument)h(of)f (Section)h(9.3,)g(such)f(a)94 1963 y(\256nite)j(co)o(v)o(er)f(e)o(xists.)50 b(Note)29 b(also)h(that)f(elements)g(of)h Ft(W)2077 1915 y FF(1)p FE(;)p Fr(1)2063 1992 y FE(I)2250 1963 y Fv(are)g(pairs)g(of)g (functions)f Fs(\()p Ft(u;)17 b(v)t Fs(\))28 b Fv(and)i(that)f(the)94 2097 y(topology)24 b(is)g Ft(W)673 2049 y FF(1)p FE(;)p Fr(1)659 2126 y FE(I)841 2097 y Fv(on)g(the)h Ft(u)p Fv(-component)f(and)h(L)1900 2054 y Fr(1)1984 2097 y Fs(\()p Ft(I)8 b Fs(\))24 b Fv(of)h(the)f Ft(v)t Fv(-component.)316 2216 y(Let)h Ft(\034)38 b(>)28 b Fv(0)d(be)g(a)g(\256x)o(ed)g(time)f(step,)g(and)h(let)f Ft(T)42 b Fs(=)28 b Ft(n\034)36 b Fv(with)24 b Ft(n)j Fp(2)h Fu(Z)p Fv(.)94 2400 y Fu(De\014nitions.)37 b Fv(Let)24 b Fp(U)39 b(2)28 b(W)1165 2357 y FE(")1157 2427 y(I)1208 2400 y Fv(.)36 b(W)-8 b(e)26 b(say)f(that)f(tw)o(o)h(elements)f Ft(A)2385 2427 y FF(1)2449 2400 y Fv(and)h Ft(A)2693 2427 y FF(2)2758 2400 y Fv(in)g Fp(G)31 b Fv(are)25 b Fs(\()p Fp(U)10 b Ft(;)17 b(T)d Fs(\))p Fv(-)p FG(separ)o(ated)23 b Fv(if)94 2520 y(there)28 b(is)g(at)f(least)h(one)g Ft(k)34 b Fp(2)e(f)p Fv(0)p Ft(;)17 b(:)g(:)g(:)d(;)j(n)p Fp(g)27 b Fv(for)h(which)f(the)h(points)e Fs(\010)2449 2477 y FE(k)r(\034)2543 2520 y Fs(\()p Ft(A)2657 2547 y FF(1)2696 2520 y Fs(\))i Fv(and)g Fs(\010)3007 2477 y FE(k)r(\034)3100 2520 y Fs(\()p Ft(A)3214 2547 y FF(2)3254 2520 y Fs(\))f Fv(do)h(not)f(belong)94 2639 y(to)e(the)g(same)f(atom)g(of)h Fp(U)10 b Fv(.)36 b(W)-8 b(e)26 b(de\256ne)1806 2766 y Ft(N)1886 2793 y FE(T)6 b(;\034)2013 2766 y Fs(\()p Fp(U)k Fs(\))94 2962 y Fv(to)24 b(be)h(the)f(lar)n(gest)g(number)g(of)g(elements)g(which)g(are)h (pairwise)f Fs(\()p Fp(U)10 b Ft(;)17 b(T)d Fs(\))p Fv(-separated)23 b(\(and)i(considered)f(with)94 3092 y(time-step)f Ft(\034)11 b Fv(.\))36 b(Note)23 b(that)h(this)f(number)g(is)h(\256nite)f(since)h(it)f (is)h(at)g(most)e Fs(\()p Fv(Card)j Fp(U)10 b Fs(\))2924 3049 y FF(2)p FE(T)f(=\034)3105 3092 y Fv(.)35 b(Finally)-6 b(,)23 b(we)i(de\256ne)1424 3338 y Ft(N)1504 3365 y FE(I)5 b(;T)h(;\034)t(;")1780 3338 y Fs(=)87 b Fv(inf)1885 3402 y Fr(U)7 b(2W)2081 3378 y Fl(")2075 3428 y(I)2136 3338 y Ft(N)2216 3365 y FE(T)f(;\034)2343 3338 y Fs(\()p Fp(U)k Fs(\))24 b Ft(:)94 3776 y Fu(Lemma)43 b(9.6.)50 b Fh(\(Lemma)29 b(2.1.)49 b(of)29 b([CE2]\).)51 b(Let)29 b Ft(I)1969 3802 y FF(1)2038 3776 y Fh(and)g Ft(I)2255 3802 y FF(2)2324 3776 y Fh(be)h(tw)o(o)f(disjoint)e(interv)n(als)h(\(perhaps)i (with)94 3895 y(common)e(boundary\))h(and)f(let)h Ft(I)41 b Fs(=)34 b Ft(I)1452 3922 y FF(1)1516 3895 y Fp([)25 b Ft(I)1651 3922 y FF(2)1690 3895 y Fh(.)49 b(The)29 b(functions)e Ft(N)2427 3922 y FE(I)5 b(;T)h(;\034)t(;")2705 3895 y Fh(satisfy)28 b(the)h(follo)n (wing)d(bounds:)94 4015 y(There)g(is)e(a)h(constant)f Ft(C)35 b Fs(=)28 b Ft(C)7 b Fs(\()p Ft(")p Fs(\))26 b Fh(such)e(that:)150 4134 y(i\))50 b Ft(N)341 4161 y FE(I)5 b(;T)h(;\034)t(;")614 4134 y Fh(is)24 b(non-increasing)g(in)h Ft(")p Fh(.)122 4254 y(ii\))50 b Ft(N)341 4281 y FE(I)5 b(;T)452 4299 y Fi(1)482 4281 y Fj(+)p FE(T)590 4299 y Fi(2)620 4281 y FE(;\034)t(;")778 4254 y Fp(\024)28 b Ft(N)963 4281 y FE(I)5 b(;T)1074 4299 y Fi(1)1105 4281 y FE(;\034)t(;")1260 4254 y Ft(N)1340 4281 y FE(I)g(;T)1451 4299 y Fi(2)1481 4281 y FE(;\034)t(;")1612 4254 y Fh(.)94 4373 y(iii\))50 b Ft(N)341 4400 y FE(I)376 4418 y Fi(1)406 4400 y Fr([)p FE(I)495 4418 y Fi(2)525 4400 y FE(;T)6 b(;\034)t(;")760 4373 y Fp(\024)28 b Ft(C)c(N)1040 4400 y FE(I)1075 4418 y Fi(1)1105 4400 y FE(;T)6 b(;\034)t(;")1338 4373 y Ft(N)1418 4400 y FE(I)1453 4418 y Fi(2)1483 4400 y FE(;T)g(;\034)t(;")1691 4373 y Fh(.)94 4622 y Fu(Remark.)39 b Fv(It)25 b(is)f(important)f(here)j (that)e Ft(C)7 b Fs(\()p Ft(")p Fs(\))25 b Fv(does)g FG(not)f Fv(depend)h(on)f(the)h(lengths)e(of)i Ft(I)3157 4649 y FF(1)3222 4622 y Fv(and)g Ft(I)3435 4649 y FF(2)3474 4622 y Fv(.)94 4806 y Fu(Pro)s(of.)46 b Fv(The)32 b(properties)f(i\))g(and)h(ii\))f(are)h(sho)n (wn)f(e)o(xactly)g(as)g(in)g([CE2].)57 b(Ho)n(we)n(v)o(er)l(,)32 b(the)f(proof)h(and)f(the)94 4925 y(statement)24 b(of)h(iii\))f(are)i(no)n(w) e(modi\256ed)g(since)h(we)g(consider)f(the)h(topology)e(of)i Ft(W)2945 4882 y FF(1)p FE(;)p Fr(1)3088 4925 y Fv(.)316 5045 y(In)f(order)f(to)h(pro)o(v)o(e)e(iii\),)h(we)h(consider)f Fp(U)1733 5072 y FF(1)1801 5045 y Fp(2)28 b(W)2001 5002 y FE(")1993 5072 y(I)2028 5090 y Fi(1)2087 5045 y Fv(and)c Fp(U)2317 5072 y FF(2)2385 5045 y Fp(2)k(W)2585 5002 y FE(")2577 5072 y(I)2612 5090 y Fi(2)2648 5045 y Fv(.)35 b(Since)24 b(we)g(are)g(using)f(the)g Ft(W)3731 5002 y FF(1)p FE(;)p Fr(1)94 5164 y Fv(norm)i(we)g(ha)n(v)o(e)1176 5291 y Ft(N)1256 5318 y FE(T)6 b(;\034)1383 5291 y Fs(\()p Fp(U)1484 5318 y FF(1)1546 5291 y Fp(\\)23 b(U)1697 5318 y FF(2)1737 5291 y Fs(\))44 b Fp(\024)h Ft(N)2022 5318 y FE(T)6 b(;\034)2149 5291 y Fs(\()p Fp(U)2250 5318 y FF(1)2290 5291 y Fs(\))24 b Ft(N)2433 5318 y FE(T)6 b(;\034)2560 5291 y Fs(\()p Fp(U)2661 5318 y FF(2)2701 5291 y Fs(\))25 b Ft(:)p eop %%Page: 37 37 bop 94 -45 a Fo(D)m(amped)38 b(Hyperbolic)f(Equa)-7 b(tions)2130 b Fn(37)94 194 y Fv(W)-8 b(e)26 b(also)e(ha)n(v)o(e)h(easily)1524 314 y Fp(W)1630 271 y FE(")1622 341 y(I)1657 358 y Fi(1)1715 314 y Fp(\\)d(W)1909 271 y FE(")1901 341 y(I)1936 358 y Fi(2)1999 314 y Fp(\032)29 b(W)2211 271 y FE(")2203 341 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b(a)g(quick)f(inspection)f(of) i(properties)f(of)94 5150 y(the)i(Bernstein)f(class)h(sho)n(ws)e(that)h(we)h (ha)n(v)o(e)f(no)g(reasonable)h(bound)f(on)g Fm(S)2688 5177 y FE(L)2748 5150 y Fs(\()p Fm(Q)2853 5177 y FE(k)2895 5195 y Fe(\003)2941 5150 y Ft(f)11 b Fs(\))21 b Fp(\000)h Fm(S)3214 5177 y FE(L)3274 5150 y Fs(\()p Ft(f)11 b Fs(\))23 b Fv(in)i Ft(W)3646 5107 y FF(1)p FE(;)p Fr(1)3813 5150 y Fv(if)94 5270 y(we)h(only)e(ha)n(v)o(e)g(information)f(about)i(the)f(function)g(and)h(not)f (about)h(its)f(deri)n(v)n(ati)n(v)o(es.)p eop %%Page: 38 38 bop 94 -45 a Fo(D)m(amped)38 b(Hyperbolic)f(Equa)-7 b(tions)2130 b Fn(38)94 194 y Fu(Ac)m(kno)m(wledgmen)m(ts)94 383 y Fv(W)-8 b(e)32 b(ha)n(v)o(e)e(pro\256ted)h(from)f(useful)g(discussions)f(with)h(Th.)g (Gallay)g(and)h(J.)f(Rougemont.)53 b(This)30 b(w)o(ork)g(w)o(as)94 503 y(partially)24 b(supported)g(by)h(the)f(F)o(onds)g(National)g(Suisse.)94 862 y Fu(References)94 1170 y Fv([B])367 b(Boas,)25 b(R.P)-11 b(.:)36 b FG(Entir)l(e)24 b(Functions)p Fv(,)g(Ne)n(w)g(Y)-11 b(ork,)25 b(Academic)f(Press)i(\(1954\).)94 1290 y([CE1])256 b(Collet,)37 b(P)-11 b(.)35 b(and)g(Eckmann,)i(J.-P)-11 b(.:)56 b(Extensi)n(v)o(e)33 b(properties)i(of)g(e)o(xtended)f(systems.)65 b(Com-)593 1410 y(mun.Math.Phys.)22 b Fu(200)p Fv(,)k(699\261722)d(\(1999\).) 94 1529 y([CE2])256 b(Collet,)36 b(P)-11 b(.)34 b(and)g(Eckmann,)i(J.-P)-11 b(.:)54 b(T)-8 b(opological)33 b(entrop)o(y)g(per)h(unit)g(v)n(olume)f(in)g (parabolic)593 1649 y(PDE')-5 b(s.)35 b(Nonlinearity)24 b Fu(12)p Fv(,)h(451\261473)f(\(1999\).)94 1768 y([F])378 b(Feireisl,)42 b(E.:)64 b(Bounded,)42 b(locally)d(compact)f(global)g(attractors)h(for)g (semilinear)g(damped)593 1888 y(w)o(a)n(v)o(e)24 b(equations)g(on)h Fu(R)1441 1845 y FE(N)1516 1888 y Fv(.)36 b(Dif)n(fer)-5 b(.)24 b(Inte)o(gral)g(Equ.)h Fu(9)p Fv(,)g(1147\2611156)e(\(1996\).)94 2007 y([KT])300 b(K)m(olmogoro)o(v)41 b(A.N.)j(and)g(T)m(ikhomiro)o(v)-6 b(,)45 b(V)-13 b(.M.:)74 b Ft(")p Fv(-entrop)o(y)44 b(and)g Ft(")p 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Fv(,)g(Ne)n(w)i(Y)-11 b(ork:)35 b(Benjamin)25 b(\(1963\).)p 94 5182 1200 4 v 241 5248 a FD(1)316 5291 y FC(The)20 b(v)o(ersion)f(in)h (this)g(collection)h(is)f(more)g(complete)h(than)f(the)h(original)e(paper)h (of)g(Uspekhi)f(Mat.)h(Nauk,)g FB(14)p FC(,)f(3\26186)h(\(1959\).)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------9908160457536--