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(e)n(v)o(entual)g(attractor)l(,)g(by)g(sho)n(wing)f(that)h(periodic)h (stationary)e(solutions)-128 3573 y(and)i(pro\002les)g(di)n(vide)e (naturally)i(the)f(phase)h(space)g(into)f(mutually)f(unattainable)h(re) o(gions.)19 3694 y(Our)29 b(proof)f(of)h(\(1.2\))g(is)g(based)f(on)h (the)g(follo)n(wing)e(simple)g(idea.)43 b(First)29 b(of)g(all,)h (constants)d(are)j(clearly)-128 3814 y(stationary)h(solution)e(of)j(KS) 891 3836 y Fs(1)966 3814 y FC(.)50 b(Furthermore,)33 b(\(1.1\))e(has)g(a)h(one)f(parameter)h(f)o(amily)e(of)i(e)o(xplicit)d (\(albeit)-128 3934 y(unbounded\))24 b(solutions)f(of)i(the)g(form)1400 4187 y Fu(\026)p FC(\()p Fu(x;)17 b(t)p FC(\))28 b Ft(=)1875 4120 y Fu(bx)p 1800 4164 246 4 v 1800 4256 a Ft(1)22 b(+)g Fu(bt)2081 4187 y(;)-128 4431 y FC(with)35 b Fu(b)48 b(>)g Ft(0)p FC(,)38 b(sho)n(wing)c(that)h(positi)n(v)o(e)e(constant)i (slopes)g(are)h(rotated)f(clockwise.)63 b(Our)36 b(starting)e(point) -128 4552 y(consists)24 b(in)g(combining)f(these)i(tw)o(o)f(special)h 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b(O)t(N)3073 b Fw(3)109 350 y FC(is)29 b(a)g(good)g(approximate)g (pro\002le)h(for)f Fu(t)37 b(>)f Ft(0)p FC(.)44 b(In)30 b(f)o(act,)h(while)e(the)g(de\002nition)g(of)g Fu( )3078 373 y Fr(a)3120 350 y FC(\()p Fu(x;)17 b(t)p FC(\))31 b(is)e(suggesti)n(v)o(e,)109 470 y(as)f(long)f(as)h(we)g(only)f(ha)n(v) o(e)h(bounds,)f(and)h(not)g(small)f(bounds)f(in)i(L)2493 435 y Fy(2)2532 470 y FC(,)h(we)f(may)-6 b(,)28 b(and)g(will,)g(w)o (ork)f(with)g(the)109 591 y Fk(\002xed)e FC(function)g Fu( )743 613 y Fr(a)813 591 y Ft(=)k Fu( )981 613 y Fr(a)1023 591 y FC(\()p Fu(x)p FC(\))d(that)f(is,)g(with)g(a)g(pro\002le)h(which) f(is)g(not)g(changing)f(in)h(time.)32 b(Our)25 b(main)g(result)109 711 y(is)k(that)g Fk(if)g(the)g(initial)f(condition)g Fu(\026)1353 734 y Fy(0)1422 711 y Fk(is)h FC(L)1579 676 y Fy(2)1618 711 y Fk(-close)h(to)f Fu( )2056 734 y Fr(a)2098 711 y FC(\()p Fu(x)p FC(\))h Fk(and)f(the)h(dif)n(fer)l (ence)f(decays)h(at)f(in\002nity)g(then)109 832 y(the)f(solution)g(r)l (emains)g FC(L)1017 796 y Fy(2)1056 832 y Fk(-close)h(to)g Fu( )1493 854 y Fr(a)1535 832 y FC(\()p Fu(x)p FC(\))p Fk(,)h(and)f(the)g(dif)n(fer)l(ence)f(still)g(decays)h(at)g(in\002nity) e(for)i(all)f Fu(t)35 b(>)h Ft(0)p Fk(.)109 952 y FC(This)e(result)h (also)g(holds)g(if)g Fu( )1185 974 y Fr(a)1263 952 y FC(is)g(replaced)h(by)f Fu( )1936 974 y Fr(per)2042 952 y FC(,)j(with)d Fu( )2381 974 y Fr(per)2523 952 y FC(a)h(periodic)f (and)g(analytic)g(stationary)109 1072 y(solution)21 b(of)h(\(1.1\))h (as)g(constructed)f(in)g([10].)30 b(Therefore,)24 b(there)f(e)o(xist)f (solutions)e(of)j(KS)3169 1095 y Fs(1)3267 1072 y FC(which)f(stay)h (near)109 1193 y Fv(\006)p Fu(Y)50 b FC(at)28 b(in\002nity)f(for)h(all) f(times,)h(and)g(thus)f(we)i(ha)n(v)o(e)e(found)h(a)g(f)o(amily)f(of)h (lar)n(ge)h(initial)d(conditions)h(whose)109 1313 y(e)n(v)n(olution)e (does)i(not)f(get)h(smaller)f(in)h(L)1507 1277 y Fs(1)1609 1313 y FC(as)g(time)f(goes)h(to)g Fv(1)p FC(.)2363 1277 y Fp(3)2437 1313 y FC(On)g(the)g(other)g(hand,)g(initial)f(conditions) 109 1434 y(which)20 b(beha)n(v)o(e)h(asymptotically)d(lik)o(e)j (periodic)f(stationary)g(solutions)f(apart)i(from)g(L)3088 1398 y Fy(2)3148 1434 y FC(corrections)g(remain)109 1554 y(so)28 b(for)h(all)g(times.)42 b(Since)30 b(the)e(dif)n(ference)i(of)f (tw)o(o)f(periodic)h(functions)f(with)g(dif)n(ferent)g(periods)h(is)f Fk(not)h FC(in)109 1674 y(L)170 1639 y Fy(2)209 1674 y FC(,)37 b(this)d(sho)n(ws)g(that)g(the)h(phase)g(space)g(naturally)f (splits)f(into)h(disconnected)g(components.)60 b(This)34 b(last)109 1795 y(result)24 b(is)g(an)h(e)o(xtension)f(of)g([6].)255 1915 y(The)i(discussion)f(abo)o(v)o(e)g(suggests)g(to)g(consider)h(the) g(equation)f(for)i Fu(\027)6 b FC(\()p Fu(x;)17 b(t)p FC(\))31 b Ft(=)f Fu(\026)p FC(\()p Fu(x;)17 b(t)p FC(\))24 b Fv(\000)f Fu( )t FC(\()p Fu(x)p FC(\),)28 b(which)109 2035 y(reads)868 2248 y Ft(_)-44 b Fu(\027)34 b Ft(=)28 b Fv(\000)p Fu(\027)1168 2212 y Fs(0)q(000)1272 2248 y Fv(\000)22 b Fu(\027)1425 2212 y Fs(0)q(0)1490 2248 y Fv(\000)1600 2209 y Fy(1)p 1600 2225 36 4 v 1600 2282 a(2)1645 2248 y FC(\()p Fu(\027)1732 2212 y Fy(2)1772 2248 y FC(\))1805 2212 y Fs(0)1851 2248 y Fv(\000)h FC(\()p Fu(\027)6 b( )t FC(\))2138 2212 y Fs(0)2184 2248 y Ft(+)22 b Fu(\037)j(;)141 b(\027)6 b FC(\()p Fu(x;)17 b Ft(0)p FC(\))29 b Ft(=)e Fu(\027)2984 2270 y Fy(0)3024 2248 y FC(\()p Fu(x)p FC(\))552 b(\(1.3\))109 2460 y(where)1499 2673 y Fu(\037)28 b Fv(\021)g(\000)p Fu( )1837 2637 y Fs(0)q(000)1941 2673 y Fv(\000)22 b Fu( )2107 2637 y Fs(00)2172 2673 y Fv(\000)g Fu( )t( )2405 2637 y Fs(0)2454 2673 y Fu(;)109 2885 y FC(and)28 b Ft(lim)417 2908 y Fr(x)p Fs(!\0061)673 2885 y Fu(\027)721 2908 y Fy(0)761 2885 y FC(\()p Fu(x)p FC(\))35 b Ft(=)f(0)p FC(.)42 b(W)-8 b(e)29 b(will)e(consider)h(\(1.3\))h(either)f(with)g Fu( )38 b Ft(=)d Fu( )2807 2908 y Fr(a)2877 2885 y FC(and)29 b(corresponding)e Fu(\037)3709 2908 y Fr(a)3751 2885 y FC(,)i(or)109 3006 y(with)21 b Fu( )31 b Ft(=)d Fu( )569 3028 y Fr(per)697 3006 y FC(a)23 b(periodic)e(analytic)g(stationary)g (solution)f(for)i(which)g Fu(\037)27 b Ft(=)h Fu(\037)2864 3028 y Fr(per)2998 3006 y Ft(=)f(0)p FC(.)j(Instead)22 b(of)g Fu( )3683 3028 y Fr(a)3725 3006 y FC(,)g(we)109 3126 y(could)28 b(ha)n(v)o(e)h(used)g(the)f(stationary)g(pro\002les)h (\()p Fk(i.e)o(.)p FC(,)g(stationary)f(solutions)f(of)i(\(1.1\)\))g (constructed)g(in)f([10]\),)109 3246 y(or)d(e)n(v)o(en)f(the)g(e)o (xplicit)g(one)736 3513 y Fu(\026)p FC(\()p Fu(x;)17 b(t)p FC(\))28 b Ft(=)1161 3445 y(15)p 1137 3490 147 4 v 1137 3581 a(361)1293 3426 y Fv(p)p 1376 3426 V 87 x Ft(209)1539 3372 y Fq(\022)1612 3513 y Fv(\000)p Ft(9)17 b(tanh)1967 3402 y Fq(\020)2036 3419 y Fs(p)p 2095 3419 106 3 v 55 x Fy(209)p 2036 3490 165 4 v 2084 3547 a(38)2211 3513 y Fu(x)2266 3402 y Fq(\021)2348 3513 y Ft(+)22 b(11)17 b(tanh)2772 3402 y Fq(\020)2842 3419 y Fs(p)p 2900 3419 106 3 v 2900 3474 a Fy(209)p 2842 3490 165 4 v 2889 3547 a(38)3016 3513 y Fu(x)3071 3402 y Fq(\021)3131 3425 y Fy(3)3171 3372 y Fq(\023)109 3776 y FC(found)35 b(by)h(K)o(uramoto)f ([8].)65 b(Note)36 b(that)g(these)g(pro\002les)g(are)h(uniformly)e (bounded.)64 b(The)36 b(adv)n(antage)g(of)109 3896 y(these)24 b(choices)g(w)o(ould)g(ha)n(v)o(e)g(been)g(that)g Fu(\037)1590 3919 y Fr(a)1660 3896 y Ft(=)j(0)p FC(,)e(the)f(disadv)n(antage)f(is)h (the)g(lack)h(of)f(e)o(xplicit)f(formulas,)h(in)109 4017 y(particular)f(for)g(the)g(F)o(ourier)g(transform)f(of)h(the)g (pro\002les.)30 b(While)23 b(adding)f(an)h(inhomogeneous)e(term)i(to)f (the)109 4137 y(equation,)k(the)g(choice)h(of)g Fu( )1112 4159 y Fr(a)1181 4137 y FC(retains)f(the)h(main)f(properties)g(of)h (these)f(stationary)g(pro\002les,)h Fk(e)o(.g)o(.)36 b FC(in)26 b(terms)109 4257 y(of)20 b(analyticity)-6 b(.)28 b(As)20 b(is)g(easily)g(seen,)h(high)e(frequenc)o(y)i(modes)e (are)i(strongly)e(damped)h(by)g(\(1.1\))h(at)f(the)g(linear)109 4378 y(le)n(v)o(el.)45 b(It)30 b(is)f(kno)n(wn)g(\(see)i Fk(e)o(.g)o(.)p FC(,)d([4]\))j(that)e(solutions)f(corresponding)h(to)h (periodic)g(antisymmetric)e(initial)109 4498 y(condition)21 b(in)i(L)670 4462 y Fy(2)709 4498 y FC(\([)p Fv(\000)p Fu(L=)p Ft(2)p Fu(;)17 b(L=)p Ft(2)p FC(]\))24 b(become)f(analytic)f (in)h(\002nite)g(time)f(in)h(a)g(strip)g(of)g(\002nite)g(width)f (around)g(the)109 4619 y(real)g(axis.)29 b(The)22 b(error)g(term)f Fu(\037)1152 4641 y Fr(a)1216 4619 y FC(of)g(the)h(equation)f(\(see)h (\(1.3\)\))f(and)h Fu( )2485 4641 y Fr(a)2549 4619 y FC(are)g(analytic)f(in)g(the)h(strip)f Fv(j)p FC(Im)g Fu(z)t Fv(j)28 b Fu(<)f Ft(1)109 4739 y FC(and)j(uniformly)e(bounded)i (in)f(an)o(y)h(smaller)g(strip\227these)e(tw)o(o)i(f)o(acts)g(are)h (better)f(seen)g(in)g(F)o(ourier)g(space,)109 4859 y(since)24 b(the)h(F)o(ourier)g(transform)1234 4833 y Ft(~)1214 4859 y Fu( )1277 4882 y Fr(a)1344 4859 y FC(of)g Fu( )1515 4882 y Fr(a)1582 4859 y FC(e)o(xists)e(as)i(a)g(distrib)n(ution)d(and)j (is)g(gi)n(v)o(en)e(by)1687 5109 y Ft(~)1667 5135 y Fu( )1730 5157 y Fr(a)1773 5135 y FC(\()p Fu(k)s FC(\))28 b Ft(=)f Fu(a)2085 5068 y(e)2130 5032 y Fs(\000j)p Fr(k)r Fs(j)p 2085 5112 182 4 v 2149 5203 a Fu(k)2302 5135 y(:)1368 b FC(\(1.4\))p 109 5269 1512 4 v 225 5330 a Fo(3)258 5366 y Fn(Note)23 b(that)g(we)g(do)f(not)h(claim)f(\(and)g(it)i(quite)e (probably)f(is)i(not)g(true\))f(that)h Fz(\026)p Fn(\()p Fz(x;)14 b(t)p Fn(\))20 b Fj(\000)g Fz( )2818 5389 y Fi(a)2858 5366 y Fn(\()p Fz(x;)14 b(t)p Fn(\))23 b(stays)g(bounded)d (in)j(L)3688 5330 y Fm(2)3725 5366 y Fn(.)34 b(W)-7 b(e)109 5466 y(will)20 b(rather)f(see)h(that)f(it)i(gro)n(ws)d(\(quickly\))g (in)i(L)1490 5430 y Fm(2)1527 5466 y Fn(.)25 b(But)20 b(the)f(only)g(thing)g(which)g(matters)g(is)i(that)e(it)i(remains)e(in) g(L)3385 5430 y Fm(2)3442 5466 y Fn(and)g(decays)g(at)109 5565 y(in\002nity)-5 b(.)p eop %%Page: 4 4 4 3 bop -128 43 a Fw(4)2277 b FC(F)t Fl(U)t(N)t(C)t(T)t(I)t(O)t(N)q(A)5 b(L)35 b FC(S)t Fl(P)m(A)q(C)t(E)t(S)t FC(,)g(E)t Fl(S)t(T)t(I)t(M)t(A) -5 b(T)t(E)t(S)-128 350 y Fw(De\002nition)26 b(1.1)48 b Fk(Thr)l(oughout,)23 b(we)j(denote)e(by)h Fu(A)g Fk(the)g(oper)o (ator)e Fu(A)28 b Ft(=)2373 265 y Fq(p)p 2473 265 174 4 v 85 x Fv(\000)p Fu(@)2606 321 y Fy(2)2601 373 y Fr(x)2647 350 y Fk(.)-128 584 y Fw(Remark)e(1.2)48 b Fk(W)-9 b(e)25 b(\002x)h Fu(Y)49 b(>)28 b Ft(0)p Fk(,)d(and)f(we)i(tacitly)e(admit)g (that)g(all)g(constants)g(occurring)g(in)h(the)f(sequel)h(may)-128 704 y(depend)g(on)g Fu(Y)c Fk(.)-128 937 y Fw(Theor)n(em)27 b(1.3)48 b Fk(Ther)l(e)31 b(ar)l(e)g(constants)e Fu(c)h Fk(and)g Fu(\014)44 b(>)37 b Ft(0)31 b Fk(suc)o(h)e(that)h(the)g (following)f(holds.)46 b(F)-10 b(or)30 b(any)g(initial)-128 1058 y(condition)24 b Fu(\027)6 b FC(\()p Fv(\001)p Fu(;)17 b Ft(0)p FC(\))25 b Fk(with)f Fu(\027)784 1080 y Fy(0)851 1058 y Fv(\021)29 b(k)p Fu(\027)6 b FC(\()p Fv(\001)p Fu(;)17 b Ft(0)p FC(\))p Fv(k)1298 1080 y Fy(2)1364 1058 y Fu(<)28 b Fv(1)p Fk(,)c(the)h(solution)e(of)h(\(1.3\))h(e)n(xists)f (for)h(all)f Fu(t)k(>)f Ft(0)e Fk(and)1189 1280 y Ft(sup)1205 1360 y Fr(t)p Fs(\025)p Fy(0)1361 1280 y Fu(e)1406 1244 y Fs(\000)p Fr(\014)s(t)1534 1280 y Fv(k)p Fu(\027)6 b FC(\()p Fv(\001)p Fu(;)17 b(t)p FC(\))p Fv(k)1861 1302 y Fy(2)1928 1280 y Fv(\024)28 b Fu(\027)2081 1302 y Fy(0)2143 1280 y Ft(+)22 b Fu(c)j(:)-128 1560 y Fk(Furthermor)l(e)o(,)g(the)h (\003ow)f(is)h(r)l(e)l(gularizing)d(in)i(the)h(sense)f(that)g(ther)l(e) h(e)n(xist)f(constants)f Fu(\016)2926 1583 y Fs(\003)2995 1560 y Fu(>)29 b Ft(0)p Fk(,)c Fu(\015)35 b Fv(\025)29 b Fu(\014)6 b Fk(,)26 b(and)-128 1681 y Fu(C)35 b(<)28 b Fv(1)c Fk(suc)o(h)g(that)973 1903 y Ft(sup)989 1983 y Fr(t)p Fs(\025)p Fy(0)1162 1903 y Fu(e)1207 1867 y Fs(\000)p Fr(\015)t(t)1332 1903 y Fv(k)p Fu(e)1427 1867 y Fy(min)n Fp(\()p Fr(\016)1597 1875 y Fh(\003)1629 1867 y Fr(;)11 b(t)p Fp(\))p Fr(A)1765 1903 y Fu(\027)6 b FC(\()p Fv(\001)p Fu(;)17 b(t)p FC(\))p Fv(k)2042 1925 y Fy(2)2109 1903 y Fv(\024)28 b Fu(\027)2262 1925 y Fy(0)2324 1903 y Ft(+)22 b Fu(C)32 b(:)-128 2194 y Fw(Cor)n(ollary)25 b(1.4)48 b Fk(F)-10 b(or)24 b(e)o(very)h Fu(m)j Ft(=)f(0)p Fu(;)17 b Ft(1)p Fu(;)g(:)g(:)g(:)40 b Fk(ther)l(e)25 b(e)n(xists)g(a)f(constant)g Fu(C)2422 2216 y Fr(m)2513 2194 y Fk(suc)o(h)g(that)639 2462 y Fv(j)p Fu(@)723 2426 y Fr(m)718 2484 y(x)790 2462 y Fu(\027)6 b FC(\()p Fu(x;)17 b(t)p FC(\))p Fv(j)28 b(\024)1382 2394 y Fu(C)1452 2417 y Fr(m)p 1215 2439 469 4 v 1215 2530 a Ft(min)o FC(\()p Fu(\016)1454 2552 y Fs(\003)1489 2530 y Fu(;)33 b(t)p FC(\))1617 2501 y Fr(m)1694 2462 y Fu(e)1739 2426 y Fr(\015)t(t)1859 2462 y Fk(and)116 b Ft(lim)2075 2521 y Fr(x)p Fs(!\0061)2327 2462 y Fu(@)2383 2426 y Fr(m)2378 2484 y(x)2451 2462 y Fu(\027)6 b FC(\()p Fu(x;)17 b(t)p FC(\))28 b Ft(=)g(0)-128 2730 y Fk(for)d(all)f Fu(t)k(>)f Ft(0)p Fk(.)-128 2963 y(Pr)l(oof)o(.)i FC(By)c(Theorem)f(1.3)g(there)g(e)o(xists)f(a)h Fu(C)1441 2928 y Fs(0)1489 2963 y FC(such)g(that)g Fv(k)p Fu(e)1965 2928 y Fy(min)n Fp(\()p Fr(\016)2135 2936 y Fh(\003)2167 2928 y Fr(;)11 b(t)p Fp(\))p Fr(A)2303 2963 y Fu(\027)6 b FC(\()p Fv(\001)p Fu(;)17 b(t)p FC(\))p Fv(k)2580 2986 y Fy(2)2647 2963 y Fv(\024)28 b Fu(C)2829 2928 y Fs(0)2853 2963 y Fu(e)2898 2928 y Fr(\015)t(t)2968 2963 y FC(.)i(By)25 b(the)f(Schw)o(arz)-128 3084 y(inequality)-6 b(,)422 3383 y Fv(k)p Fu(A)545 3348 y Fr(m)617 3383 y Ft(~)-54 b Fu(\027)6 b FC(\()p Fv(\001)p Fu(;)17 b(t)p FC(\))p Fv(k)889 3406 y Fy(1)956 3383 y Fv(\024)1061 3243 y Fq(\022)1134 3248 y(Z)1251 3383 y FC(d)p Fu(k)27 b Fv(j)p Fu(k)s Fv(j)1489 3348 y Fy(2)p Fr(m)1591 3383 y Fu(e)1636 3348 y Fs(\000)p Fy(2)12 b(min)o Fp(\()p Fr(\016)1909 3356 y Fh(\003)1940 3348 y Fr(;)f(t)p Fp(\))p Fs(j)p Fr(k)r Fs(j)2101 3243 y Fq(\023)2174 3265 y Fy(1)p Fr(=)p Fy(2)2301 3383 y Fv(k)p Fu(e)2396 3348 y Fy(min)n Fp(\()p Fr(\016)2566 3356 y Fh(\003)2598 3348 y Fr(;)g(t)p Fp(\))p Fr(A)2734 3383 y Fu(\027)6 b FC(\()p Fv(\001)p Fu(;)17 b(t)p FC(\))p Fv(k)3011 3406 y Fy(2)3075 3383 y Fu(:)-128 3663 y FC(This)25 b(immediately)f(implies)h(the)g(\002rst)h (assertion)f(since)h Ft(sup)2004 3686 y 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FC(the)g(\(Banach\))h(space)g (obtained)e(by)h(completing)f Fv(C)2846 4443 y Fy(0)2885 4421 y FC(\([)p Fu(\034)6 b(;)17 b(\034)37 b Ft(+)26 b Fu(\016)t FC(])p Fu(;)17 b Fv(C)3406 4385 y Fs(1)3400 4444 y Fy(0)3481 4421 y FC(\()p Fw(R)p FC(\)\))-128 4541 y(in)28 b(the)f(norm)g Ft(sup)512 4565 y Fr(t)p Fs(2)p Fp([)p Fr(\034)t(;\034)8 b Fy(+)p Fr(\016)r Fp(])831 4541 y Fv(k)p Fu(e)926 4506 y Fp(\()p Fr(t)p Fs(\000)p Fr(\034)g Fp(\))p Fr(A)1171 4541 y Fv(\001)24 b(k)1273 4564 y Fy(2)1312 4541 y FC(,)k(where,)h(throughout,)d Fv(k)e(\001)g(k)2328 4564 y Fr(p)2395 4541 y FC(is)j(the)g(L)2699 4506 y Fr(p)2766 4541 y FC(norm.)38 b(W)-8 b(e)28 b(also)f(denote)-128 4662 y(by)d Fv(B)61 4684 y Fr(d)130 4662 y Fv(\032)k(W)333 4684 y Fr(\034)t(;\016)450 4662 y FC(the)c(open)g(ball)g(of)g(radius)g Fu(d)p FC(,)g(centered)h(on)f Ft(0)g FC(in)f Fv(W)2221 4684 y Fr(\034)t(;\016)2314 4662 y FC(.)31 b(The)24 b(bounds)f(of)i (this)e(section)g(serv)o(e)-128 4782 y(to)i(control)f(the)h(non-linear) f(and)h(mix)o(ed)e(terms)i(in)f(Eq.\(1.3\).)-128 4989 y Fw(Lemma)i(2.1)48 b Fk(Ther)l(e)26 b(is)e(a)h Fu(\016)860 5012 y Fs(\003)927 4989 y Fu(>)j Ft(0)c Fk(suc)o(h)g(that)g(for)h(all)f Fu(t)k Fv(2)g FC([)p Fu(\034)6 b(;)17 b(\034)33 b Ft(+)22 b Fu(\016)2260 5012 y Fs(\003)2300 4989 y FC(])j Fk(one)g(has)622 5211 y Fv(k)p Fu(e)717 5176 y Fp(\()p Fr(t)p Fs(\000)p Fr(\034)8 b Fp(\))p Fr(A)963 5211 y Fu( )1026 5234 y Fr(a)1068 5211 y FC(\()p Fv(\001)p FC(\))p Fu(f)j FC(\()p Fv(\001)p Fu(;)17 b(t)p FC(\))p Fv(k)1444 5234 y Fy(2)1510 5211 y Fv(\024)29 b Ft(2)p Fu(\031)f(a)d Fv(k)1874 5124 y(p)p 1957 5124 282 4 v 87 x Ft(1)d(+)g Fu(A)2199 5183 y Fy(2)2263 5211 y Fu(e)2308 5176 y Fp(\()p Fr(t)p Fs(\000)p Fr(\034)8 b Fp(\))p Fr(A)2529 5211 y Fu(f)j FC(\()p Fv(\001)p Fu(;)17 b(t)p FC(\))p Fv(k)2811 5234 y Fy(2)2875 5211 y Fu(:)-128 5445 y Fw(Remark)26 b(2.2)48 b Fk(One)23 b(can)f(c)o(hoose)g Fu(\016)1115 5467 y Fs(\003)1183 5445 y Ft(=)27 b(1)c Fk(as)f(will)g(be)h(seen)g(fr)l(om)e(the)i(pr)l (oof)o(.)28 b(\(This)22 b(value)h(is)f(r)l(elated)g(to)g(the)-128 5565 y(domain)i(of)h(analyticity)e(of)i Fu( )910 5588 y Fr(a)952 5565 y Fk(.\))p eop %%Page: 5 5 5 4 bop 111 43 a FC(F)t Fl(U)t(N)t(C)t(T)t(I)t(O)t(N)q(A)t(L)35 b FC(S)t Fl(P)m(A)q(C)t(E)t(S)t FC(,)h(E)t Fl(S)t(T)t(I)t(M)t(A)-5 b(T)t(E)t(S)2283 b Fw(5)109 350 y Fk(Pr)l(oof)o(.)28 b FC(De\002ne)e Fu(F)14 b FC(\()p Fu(x;)j(t)p FC(\))28 b Ft(=)g Fu(e)1130 314 y Fp(\()p Fr(t)p Fs(\000)p Fr(\034)8 b Fp(\))p Fr(A)1375 350 y Fu( )1438 373 y Fr(a)1480 350 y FC(\()p Fu(x)p FC(\))p Fu(f)j FC(\()p Fu(x;)17 b(t)p FC(\).)32 b(W)-8 b(e)26 b(write)e(the)h(F)o(ourier)g(transform)f(of)h Fu(F)38 b FC(as)1209 554 y Ft(~)1187 579 y Fu(F)13 b FC(\()p Fu(k)s(;)k(t)p FC(\))28 b Ft(=)g Fu(e)1639 544 y Fp(\()p Fr(t)p Fs(\000)p Fr(\034)8 b Fp(\))p Fs(j)p Fr(k)r Fs(j)1901 444 y Fq(Z)2018 579 y FC(d)p Fu(`)2145 553 y Ft(~)2126 579 y Fu( )2189 602 y Fr(a)2230 579 y FC(\()p Fu(k)25 b Fv(\000)e Fu(`)p FC(\))2534 553 y Ft(~)2513 579 y Fu(f)11 b FC(\()p Fu(`;)17 b(t)p FC(\))25 b Fu(:)109 804 y FC(Using)f(\(1.4\),)g(we)h(\002nd)g(\(using)f(principal)g(v)n (alues\))1226 1025 y Ft(~)1204 1050 y Fu(F)14 b FC(\()p Fu(k)s(;)j(t)p FC(\))28 b Ft(=)f Fu(a)17 b(e)1724 1014 y Fp(\()p Fr(t)p Fs(\000)p Fr(\034)8 b Fp(\))p Fs(j)p Fr(k)r Fs(j)1987 914 y Fq(Z)2103 1050 y FC(d)p Fu(`)2220 983 y(e)2265 947 y Fs(\000j)p Fr(k)r Fs(\000)p Fr(`)p Fs(j)p 2220 1027 266 4 v 2244 1118 a Fu(k)26 b Fv(\000)c Fu(`)2517 1024 y Ft(~)2496 1050 y Fu(f)10 b FC(\()p Fu(`;)17 b(t)p FC(\))26 b Fu(:)109 1299 y FC(Denote)g Fu(g)t FC(\()p Fu(x;)17 b(t)p FC(\))31 b Ft(=)g Fu(e)857 1264 y Fp(\()p Fr(t)p Fs(\000)p Fr(\034)8 b Fp(\))p Fr(A)1078 1299 y Fu(f)j FC(\()p Fu(x;)17 b(t)p FC(\),)28 b(so)e(that)1703 1273 y Ft(~)1681 1299 y Fu(f)11 b FC(\()p Fu(`;)17 b(t)p FC(\))32 b Ft(=)f Fu(e)2110 1264 y Fs(\000)p Fp(\()p Fr(t)p Fs(\000)p Fr(\034)8 b Fp(\))p Fs(j)p Fr(`)p Fs(j)2404 1299 y Ft(~)-52 b Fu(g)s FC(\()p Fu(`;)17 b(t)p FC(\).)36 b(Rearranging)28 b(e)o(xponentials,)d(we)109 1420 y(get)874 1640 y Ft(~)852 1665 y Fu(F)13 b FC(\()p Fu(k)s(;)k(t)p FC(\))29 b Ft(=)e Fu(a)1343 1530 y Fq(Z)1460 1665 y FC(d)p Fu(`)1577 1598 y(e)1622 1562 y Fs(\000j)p Fr(k)r Fs(\000)p Fr(`)p Fs(j)p 1577 1642 V 1601 1734 a Fu(k)f Fv(\000)c Fu(`)1853 1665 y(e)1898 1630 y Fp(\()p Fr(t)p Fs(\000)p Fr(\034)8 b Fp(\))p Fs(j)p Fr(k)r Fs(j)2143 1665 y Fu(e)2188 1630 y Fs(\000)p Fp(\()p Fr(t)p Fs(\000)p Fr(\034)g Fp(\))p Fs(j)p Fr(`)p Fs(j)2483 1665 y Ft(~)-53 b Fu(g)t FC(\()p Fu(`;)17 b(t)p FC(\))1156 1937 y Ft(=)27 b Fu(a)1343 1802 y Fq(Z)1460 1937 y FC(d)p Fu(`)1577 1870 y(e)1622 1834 y Fs(\000j)p Fr(k)r Fs(\000)p Fr(`)p Fs(j)p Fp(\()p Fy(1)p Fs(\000)p Fp(\()p Fr(t)p Fs(\000)p Fr(\034)8 b Fp(\)\))p 1577 1914 564 4 v 1750 2006 a Fu(k)26 b Fv(\000)c Fu(`)2151 1937 y(e)2196 1900 y Fp(\()p Fr(t)p Fs(\000)p Fr(\034)8 b Fp(\))p Ft(\()p Fs(j)p Fr(k)r Fs(j\000j)p Fr(`)p Fs(j\000j)p Fr(k)r Fs(\000)p Fr(`)p Fs(j)p Ft(\))2860 1937 y(~)-53 b Fu(g)t FC(\()p Fu(`;)17 b(t)p FC(\))25 b Fu(:)109 2266 y FC(W)-8 b(e)25 b(decompose)f(this)g(as)313 2486 y Ft(~)291 2511 y Fu(F)14 b FC(\()p Fu(k)s(;)j(t)p FC(\))28 b Ft(=)f Fu(a)766 2376 y Fq(Z)882 2511 y FC(d)p Fu(`)1008 2444 y(e)1053 2408 y Fs(\000j)p Fr(k)r Fs(\000)p Fr(`)p Fs(j)p Fp(\()p Fy(1)p Fs(\000)p Fp(\()p Fr(t)p Fs(\000)p Fr(\034)8 b Fp(\)\))p 1008 2489 V 1181 2580 a Fu(k)26 b Fv(\000)c Fu(`)1610 2511 y Ft(~)-53 b Fu(g)t FC(\()p Fu(`;)17 b(t)p FC(\))22 b Ft(+)g Fu(a)2031 2376 y Fq(Z)2147 2511 y FC(d)p Fu(`)50 b(G)p FC(\()p Fu(k)s(;)17 b(`;)g(t)22 b Fv(\000)g Fu(\034)6 b(;)17 b FC(\()p Ft(1)22 b Fv(\000)h FC(\()p Fu(t)f Fv(\000)h Fu(\034)11 b FC(\)\)\))29 b Ft(~)-52 b Fu(g)s FC(\()p Fu(`;)17 b(t)p FC(\))25 b Fu(;)109 2736 y FC(where)1079 2962 y Fu(G)p FC(\()p Fu(k)s(;)17 b(`;)g(\030)5 b(;)17 b(\021)t FC(\))26 b Ft(=)1688 2894 y Fu(e)1733 2859 y Fs(\000)p Fr(\021)r Fs(j)p Fr(k)r Fs(\000)p Fr(`)p Fs(j)p 1688 2939 304 4 v 1732 3030 a Fu(k)f Fv(\000)e Fu(`)2043 2881 y Fq(\000)2089 2962 y Fu(e)2134 2926 y Fr(\030)s Fp(\()p Fs(j)p 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Fu(;)17 b(t)p FC(\))p Fv(k)2112 4602 y Fy(2)2176 4580 y Fu(;)109 4763 y FC(pro)o(vided)23 b Fu(\016)529 4785 y Fs(\003)597 4763 y Fv(\024)28 b Ft(1)p FC(.)i(The)25 b(proof)g(of)g(Lemma)f(2.1)g(is)h(complete.)p 2351 4763 50 50 v 109 4949 a Fw(Lemma)g(2.3)48 b Fk(Let)27 b Fu( )850 4972 y Fr(per)983 4949 y Fk(be)f(periodic)f(of)h(period)f Fu(L)p Fk(,)i(let)f Fu(q)33 b Ft(=)2287 4910 y Fy(2)p Fr(\031)p 2287 4926 79 4 v 2302 4984 a(L)2402 4949 y Fk(and)25 b(assume)h(that)f(ther)l(e)h(e)n(xist)g(constants)109 5079 y Fu(c)151 5101 y Fr(per)257 5079 y Fu(;)17 b(\016)31 b Fk(suc)o(h)26 b(that)765 5004 y Fq(P)871 5108 y Fr(m)p Fs(2)p Fg(Z)1045 5079 y Fv(j)p Fu( )1136 5101 y Fr(per)n(;m)1321 5079 y Fv(j)p Fu(e)1394 5043 y Fr(\016)r(q)r Fs(j)p Fr(m)p Fs(j)1599 5079 y Fu(<)32 b(c)1749 5101 y Fr(per)1855 5079 y Fk(,)c(wher)l(e)g Fu( )2238 5101 y Fr(per)n(;m)2450 5079 y Fk(denotes)e(the)h Fu(m)p Fk(-th)h(F)-10 b(ourier)26 b(coef)n(\002cient)109 5199 y(of)e Fu( )274 5221 y Fr(per)381 5199 y Fk(,)h(then)1077 5382 y Fv(k)p Fu(e)1172 5346 y Fp(\()p Fr(t)p Fs(\000)p Fr(\034)8 b Fp(\))p Fr(A)1393 5382 y Fu( )1456 5405 y Fr(per)1562 5382 y FC(\()p Fv(\001)p FC(\))p Fu(f)j FC(\()p Fv(\001)p Fu(;)17 b(t)p FC(\))p Fv(k)1938 5405 y Fy(2)2005 5382 y Fv(\024)28 b Fu(c)2152 5405 y Fr(per)2283 5382 y Fv(k)p Fu(e)2378 5346 y Fp(\()p Fr(t)p Fs(\000)p Fr(\034)8 b Fp(\))p Fr(A)2599 5382 y Fu(f)j FC(\()p Fv(\001)p Fu(;)17 b(t)p FC(\))p Fv(k)2881 5405 y Fy(2)109 5565 y Fk(for)24 b(all)g Fu(t)k Fv(2)g FC([)p Fu(\034)6 b(;)17 b(\034)34 b Ft(+)22 b Fu(\016)t FC(])p Fk(.)p eop %%Page: 6 6 6 5 bop -128 43 a Fw(6)2277 b FC(F)t Fl(U)t(N)t(C)t(T)t(I)t(O)t(N)q(A)5 b(L)35 b FC(S)t Fl(P)m(A)q(C)t(E)t(S)t FC(,)g(E)t Fl(S)t(T)t(I)t(M)t(A) -5 b(T)t(E)t(S)-128 350 y Fk(Pr)l(oof)o(.)37 b FC(As)28 b(in)f(Lemma)g(2.1,)g(we)h(de\002ne)g Fu(F)14 b FC(\()p Fu(x;)j(t)p FC(\))33 b Ft(=)g Fu(e)1789 314 y Fp(\()p Fr(t)p Fs(\000)p Fr(\034)8 b Fp(\))p Fr(A)2037 350 y Fu( )2100 373 y Fr(per)2207 350 y FC(\()p Fu(x)p FC(\))p Fu(f)j FC(\()p Fu(x;)17 b(t)p FC(\).)40 b(The)28 b(F)o(ourier)f (transform)g(of)-128 470 y Fu(F)39 b FC(satis\002es)967 657 y Ft(~)945 682 y Fu(F)14 b FC(\()p Fu(k)s(;)j(t)p FC(\))28 b Ft(=)f Fu(e)1397 647 y Fp(\()p Fr(t)p Fs(\000)p Fr(\034)8 b Fp(\))p Fs(j)p Fr(k)r Fs(j)1664 588 y Fq(X)1660 798 y Fr(m)p Fs(2)p Fg(Z)1830 682 y Fu( )1893 705 y Fr(per)n(;m)2078 682 y Fu(f)j FC(\()p Fu(k)25 b Fv(\000)d Fu(q)t(m)p FC(\))k Fu(;)-128 980 y FC(so)f(that)609 1249 y Fv(k)p Fu(F)14 b FC(\()p Fv(\001)p Fu(;)j(t)p FC(\))p Fv(k)959 1272 y Fy(2)1026 1249 y Fv(\024)1136 1154 y Fq(X)1131 1365 y Fr(m)p Fs(2)p Fg(Z)1301 1249 y Fv(j)p Fu( )1392 1272 y Fr(per)n(;m)1577 1249 y Fv(j)1622 1109 y Fq(\022)1694 1114 y(Z)1810 1249 y FC(d)p Fu(k)28 b(e)1984 1213 y Fy(2)p Fp(\()p Fr(t)p Fs(\000)p Fr(\034)8 b Fp(\))p Fs(j)p Fr(k)r Fy(+)p Fr(q)r(m)p Fs(j)2441 1249 y Fv(j)p Fu(f)j FC(\()p Fu(k)s FC(\))p Fv(j)2676 1213 y Fy(2)2715 1109 y Fq(\023)2789 1131 y Fy(1)p Fr(=)p Fy(2)1026 1517 y Fv(\024)28 b(k)p Fu(e)1226 1481 y Fp(\()p Fr(t)p Fs(\000)p Fr(\034)8 b Fp(\))p Fr(A)1446 1517 y Fu(f)j FC(\()p Fv(\001)p Fu(;)17 b(t)p FC(\))p Fv(k)1728 1540 y Fy(2)1814 1423 y Fq(X)1809 1633 y Fr(m)p Fs(2)p Fg(Z)1979 1517 y Fv(j)p Fu( )2070 1540 y Fr(per)n(;m)2255 1517 y Fv(j)24 b Fu(e)2352 1481 y Fp(\()p Fr(t)p Fs(\000)p Fr(\034)8 b Fp(\))p Fr(q)r Fs(j)p Fr(m)p Fs(j)2681 1517 y Fu(;)-128 1816 y FC(which)25 b(completes)f(the)g(proof)h(of)g(the)g(lemma)f(since)g Ft(0)k Fv(\024)g Fu(t)22 b Fv(\000)h Fu(\034)39 b Fv(\024)28 b Fu(\016)t FC(.)p 2390 1816 50 50 v -128 2035 a Fw(Lemma)e(2.4)48 b Fk(Let)25 b Fu(\015)33 b(>)28 b Ft(0)c Fk(and)h Fv(k)p FC(\()p Ft(1)d(+)g Fu(A)1310 2000 y Fy(2)1349 2035 y FC(\))p Fu(e)1427 2000 y Fr(\015)t(A)1550 2035 y Fu(f)11 b Fv(k)1659 2058 y Fy(2)1720 2035 y Ft(+)22 b Fv(k)p Fu(e)1913 2000 y Fr(\015)t(A)2035 2035 y Fu(g)t Fv(k)2136 2058 y Fy(2)2197 2035 y Ft(+)g Fv(k)p Fu(e)2390 2000 y Fr(\015)t(A)2512 2035 y Fu(h)p Fv(k)2618 2058 y Fy(2)2685 2035 y Fu(<)28 b Fv(1)p Fk(.)i(Then)371 2160 y Fq(\014)371 2220 y(\014)371 2280 y(\014)371 2339 y(\014)404 2169 y(Z)520 2304 y FC(d)p Fu(x)17 b(e)687 2269 y Fr(\015)t(A)809 2304 y Fu(f)33 b Fv(\001)22 b Fu(e)985 2269 y Fr(\015)t(A)1082 2304 y FC(\()p Fu(g)t(h)p FC(\))1255 2269 y Fs(0)1278 2160 y Fq(\014)1278 2220 y(\014)1278 2280 y(\014)1278 2339 y(\014)1356 2304 y Fv(\024)1478 2228 y(p)p 1561 2228 59 4 v 76 x Fu(\031)t Fv(k)p FC(\()p Ft(1)g(+)g Fu(A)1945 2269 y Fy(2)1984 2304 y FC(\))p Fu(e)2062 2269 y Fr(\015)t(A)2185 2304 y Fu(f)11 b Fv(k)2294 2327 y Fy(2)2358 2304 y Fv(k)p Fu(e)2453 2269 y Fr(\015)t(A)2575 2304 y Fu(g)t Fv(k)2676 2327 y Fy(2)2739 2304 y Fv(k)p Fu(e)2834 2269 y Fr(\015)t(A)2956 2304 y Fu(h)p Fv(k)3062 2327 y Fy(2)3127 2304 y Fu(:)-128 2587 y Fk(Pr)l(oof)o(.)29 b FC(Set)c Fu(F)42 b Ft(=)27 b Fu(e)554 2551 y Fr(\015)t(A)677 2587 y Fu(f)11 b FC(,)24 b Fu(G)k Ft(=)f Fu(e)1038 2551 y Fr(\015)t(A)1161 2587 y Fu(g)h FC(and)d Fu(H)35 b Ft(=)27 b Fu(e)1669 2551 y Fr(\015)t(A)1792 2587 y Fu(h)p FC(.)k(Since)25 b Fv(j)p Fu(k)s Fv(j)c(\000)i(j)p Fu(k)h Fv(\000)f Fu(`)p Fv(j)e(\000)i(j)p Fu(`)p Fv(j)k(\024)h Ft(0)d FC(by)f(the)h(triangle) -128 2707 y(inequality)-6 b(,)23 b(we)i(ha)n(v)o(e)205 2829 y Fq(\014)205 2889 y(\014)205 2948 y(\014)205 3008 y(\014)238 2838 y(Z)354 2973 y FC(d)p Fu(x)17 b(e)521 2938 y Fr(\015)t(A)643 2973 y Fu(f)33 b Fv(\001)22 b Fu(e)819 2938 y Fr(\015)t(A)917 2973 y FC(\()p Fu(g)t(h)p FC(\))1090 2938 y Fs(0)1113 2829 y Fq(\014)1113 2889 y(\014)1113 2948 y(\014)1113 3008 y(\014)1174 2973 y Ft(=)1277 2829 y Fq(\014)1277 2889 y(\014)1277 2948 y(\014)1277 3008 y(\014)1310 2838 y(Z)1427 2973 y FC(d)p Fu(k)27 b(ik)1665 2948 y Ft(~)1642 2973 y Fu(F)14 b FC(\()p Fu(k)s FC(\))1856 2838 y Fq(Z)1973 2973 y FC(d)p Fu(`)24 b(e)2133 2938 y Fr(\015)t Fp(\()p Fs(j)p Fr(k)r Fs(j\000j)p Fr(k)r Fs(\000)p Fr(`)p Fs(j\000j)p Fr(`)p Fs(j)p Fp(\))2686 2948 y Ft(~)2664 2973 y Fu(G)p FC(\()p Fu(k)h Fv(\000)e Fu(`)p FC(\))3074 2948 y Ft(~)3049 2973 y Fu(H)8 b FC(\()p Fu(`)p FC(\))3245 2829 y Fq(\014)3245 2889 y(\014)3245 2948 y(\014)3245 3008 y(\014)1174 3259 y Fv(\024)1279 3144 y Fq(\015)1279 3204 y(\015)1279 3264 y(\015)1344 3109 y Fv(p)p 1427 3109 282 4 v 82 x Ft(1)22 b(+)g Fu(A)1669 3163 y Fy(2)p 1344 3236 365 4 v 1344 3256 a Fv(p)p 1427 3256 282 4 v 82 x Ft(1)g(+)g Fu(A)1669 3309 y Fy(2)1719 3259 y Fu(A)1814 3234 y Ft(~)1792 3259 y Fu(F)1868 3144 y Fq(\015)1868 3204 y(\015)1868 3264 y(\015)1924 3328 y Fy(1)1980 3259 y Fv(k)p Fu(G)p Fv(k)2157 3281 y Fy(2)2212 3259 y Fv(k)p Fu(H)8 b Fv(k)2401 3281 y Fy(2)1174 3469 y Fv(\024)1279 3393 y(p)p 1362 3393 59 4 v 76 x Fu(\031)t Fv(k)p FC(\()p Ft(1)22 b(+)g Fu(A)1746 3433 y Fy(2)1785 3469 y FC(\))p Fu(F)14 b Fv(k)1945 3492 y Fy(2)2001 3469 y Fv(k)p Fu(G)p Fv(k)2178 3492 y Fy(2)2233 3469 y Fv(k)p Fu(H)8 b Fv(k)2422 3492 y Fy(2)2486 3469 y Fu(;)-128 3681 y FC(where)26 b(we)f(used)f(again)730 3601 y Fq(R)813 3681 y FC(d)p Fu(k)19 b FC(\()p Ft(1)j(+)g Fu(k)1189 3645 y Fy(2)1229 3681 y FC(\))1262 3645 y Fs(\000)p Fy(1)1384 3681 y Ft(=)27 b Fu(\031)t FC(.)p 1627 3681 50 50 v -128 3801 a(The)e(follo)n(wing)e(proposition)g(estimates)h(ho)n(w)g(close)g Fu( )1823 3824 y Fr(a)1890 3801 y FC(is)h(to)f(a)h(solution)e(of)i(KS) 2734 3824 y Fs(1)2809 3801 y FC(.)-128 3998 y Fw(Pr)n(oposition)g(2.5) 49 b Fk(De\002ne)36 b Fu( )921 4020 y Fr(a)963 3998 y FC(\()p Fu(x)p FC(\))51 b Ft(=)e Fu(a)17 b Ft(arctan)p FC(\()p Fu(x)p FC(\))37 b Fk(and)f(let)h Fu(\037)2141 4020 y Fr(a)2232 3998 y Ft(=)49 b Fv(\000)p Fu( )2501 3962 y Fs(0)q(000)2497 4020 y Fr(a)2614 3998 y Fv(\000)31 b Fu( )2789 3962 y Fs(00)2785 4020 y Fr(a)2862 3998 y Fv(\000)h Fu( )3034 4020 y Fr(a)3076 3998 y Fu( )3143 3962 y Fs(0)3139 4020 y Fr(a)3181 3998 y Fk(.)66 b(Then,)40 b(for)-128 4118 y Ft(0)28 b Fv(\024)g Fu(\016)k Fv(\024)c Fu(\016)277 4141 y Fs(\003)344 4118 y Fv(\021)459 4079 y Fy(1)p 459 4095 36 4 v 459 4153 a(2)505 4118 y Fk(,)c(one)h(has)1355 4330 y Ft(sup)1326 4412 y Fr(t)p Fs(2)p Fp([)p Fy(0)p Fr(;\016)r Fp(])1548 4330 y Fv(k)p Fu(e)1643 4294 y Fr(tA)1726 4330 y Fu(\037)1787 4352 y Fr(a)1828 4330 y Fv(k)1878 4352 y Fy(2)1945 4330 y Fv(\024)j Fu(B)i(;)-128 4603 y Fk(for)25 b(some)f Fu(B)30 b Fk(depending)24 b(only)h(on)f Fu(a)p Fk(.)-128 4799 y(Pr)l(oof)o(.)29 b FC(The)c(F)o(ourier)g (transform)f(of)h Fu(e)1211 4764 y Fr(tA)1294 4799 y Fu( )1361 4764 y Fs(0000)1357 4822 y Fr(a)1466 4799 y FC(is)g(of)f(the)h(form)g Fu(i)2064 4764 y Fy(3)2103 4799 y Fu(ae)2199 4764 y Fr(t)p Fs(j)p Fr(k)r Fs(j)2307 4799 y Fu(k)2361 4764 y Fy(3)2401 4799 y Fu(e)2446 4764 y Fs(\000j)p Fr(k)r Fs(j)2608 4799 y FC(so)f(that)g(for)h Fu(t)j Fv(\024)h Fu(\016)f FC(we)d(get)749 5057 y Fv(k)p Fu(e)844 5021 y Fr(tA)926 5057 y Fu( )993 5021 y Fs(00)q(00)989 5080 y Fr(a)1074 5057 y Fv(k)1124 5021 y Fy(2)1124 5080 y(2)1191 5057 y Fv(\024)j Fu(a)1347 5021 y Fy(2)1403 4922 y Fq(Z)1520 5057 y FC(d)p Fu(k)19 b(e)1685 5021 y Fy(2)p Fp(\()p Fr(\016)r Fs(\000)p Fy(1)p Fp(\))p Fs(j)p Fr(k)r Fs(j)1971 5057 y Fu(k)2025 5021 y Fy(6)2092 5057 y Fv(\024)28 b(O)s FC(\(\()p Ft(1)22 b Fv(\000)h Fu(\016)t FC(\))2596 5021 y Fs(\000)p Fy(7)2690 5057 y FC(\))i Fu(:)-128 5325 y FC(A)33 b(similar)e(bound)h(holds)g(for)g Fu( )1032 5289 y Fs(0)q(0)1028 5347 y Fr(a)1075 5325 y FC(.)54 b(The)32 b(term)h Fu(e)1602 5289 y Fr(tA)1684 5325 y Fu( )1747 5347 y Fr(a)1789 5325 y Fu( )1856 5289 y Fs(0)1852 5347 y Fr(a)1927 5325 y FC(is)f(bounded)g(using)f(Lemma)h (2.1)h(with)e Fu(\016)3397 5347 y Fs(\003)3479 5325 y Ft(=)3607 5285 y Fy(1)p 3607 5302 V 3607 5359 a(2)-128 5445 y FC(instead)24 b(of)g(1)g(and)f Fu(f)39 b Ft(=)27 b Fu( )784 5409 y Fs(0)780 5467 y Fr(a)822 5445 y FC(.)k(This)23 b(yields)g(a)h(bound)g(on)f(the)h(square)g(of)g(the)g(L)2550 5409 y Fy(2)2613 5445 y FC(norm)g(which)f(is)g(of)i(the)e(form)-128 5565 y Fu(a)-77 5530 y Fy(2)-37 5565 y Fv(O)s FC(\()p Ft(1)p FC(\))177 5485 y Fq(R)260 5565 y FC(d)p Fu(k)c FC(\()p Ft(1)k(+)f Fu(k)637 5530 y Fy(2)676 5565 y FC(\))p Fu(e)754 5530 y Fy(2)p Fr(\016)r Fs(j)p Fr(k)r Fs(j)905 5565 y Fu(e)950 5530 y Fs(\000)p Fy(2)p Fs(j)p Fr(k)r Fs(j)1123 5565 y FC(,)i(and)h(combining)e(the)i(bounds)f(completes)f (the)i(proof.)p 3124 5565 50 50 v eop %%Page: 7 7 7 6 bop 111 43 a FC(T)t Fl(H)t(E)30 b FC(L)t Fl(O)t(C)t(A)t(L)j FC(C)t Fl(AU)t(C)t(H)t(Y)g FC(P)t Fl(R)q(O)t(B)t(L)t(E)t(M)h(I)t(N)28 b FC(L)1666 7 y Fy(2)3838 43 y Fw(7)109 350 y FB(3)119 b(The)30 b(Local)g(Cauch)n(y)h(Pr)n(oblem)f(in)g Ff(L)1921 307 y FA(2)109 566 y FC(In)25 b(this)e(section,)h(we)i(consider)e(the)h (local)f(\(in)h(time\))f(Cauchy)h(problem)791 795 y Ft(_)-43 b Fu(\027)34 b Ft(=)27 b Fv(\000)p Fu(\027)1091 760 y Fs(0)r(000)1195 795 y Fv(\000)c Fu(\027)1349 760 y Fs(0)q(0)1414 795 y Fv(\000)1524 756 y Fy(1)p 1524 772 36 4 v 1524 830 a(2)1569 795 y FC(\()p Fu(\027)1656 760 y Fy(2)1696 795 y FC(\))1729 760 y Fs(0)1774 795 y Fv(\000)g FC(\()p Fu(\027)6 b( )2024 818 y Fr(a)2067 795 y FC(\))2100 760 y Fs(0)2145 795 y Ft(+)22 b Fu(\037)2304 818 y Fr(a)2371 795 y Fu(;)141 b(\027)6 b FC(\()p Fu(x;)17 b(\034)11 b FC(\))29 b Ft(=)f Fu(\027)2992 818 y Fy(0)3032 795 y FC(\()p Fu(x)p FC(\))d Fu(;)492 b FC(\(3.1\))109 1024 y(for)24 b(\(1.3\))h(with)f Fu(\027)715 1047 y Fy(0)782 1024 y Fv(2)k FC(L)937 989 y Fy(2)976 1024 y FC(.)j(W)-8 b(e)25 b(will)e(sho)n(w)-6 b(,)23 b(using)h(a)h(contraction)e(ar)n (gument,)h(that)g(it)g(is)g(well)h(posed)f(on)g(an)o(y)109 1145 y(time)30 b(interv)n(al)f Fu(t)39 b Fv(2)g FC([)p Fu(\034)6 b(;)17 b(\034)38 b Ft(+)26 b Fu(\016)t FC(])31 b(with)f Fu(\016)42 b Fv(\024)d Ft(min)o FC(\()p Fu(\016)1894 1167 y Fs(\003)1934 1145 y Fu(;)17 b(C)2048 1167 y Fs(\003)2087 1145 y Fv(k)p Fu(\027)2185 1167 y Fy(0)2224 1145 y Fv(k)2274 1104 y Fs(\000)p Fy(2)2274 1169 y(2)2368 1145 y FC(\).)2426 1109 y Fp(4)2512 1145 y FC(T)-8 b(o)31 b(this)e(end,)j(we)f(construct)f (the)h(map)109 1265 y Fu(\032)d Fv(7!)f(F)10 b FC(\()p Fu(\032)p FC(\))25 b(de\002ned)g(by)g Fv(F)10 b FC(\()p Fu(\032)p FC(\))28 b Ft(=)f Fu(\030)5 b FC(,)24 b(where)i Fu(\030)j FC(is)24 b(the)h(solution)e(of)811 1468 y Ft(_)790 1494 y Fu(\030)33 b Ft(=)27 b Fv(\000)p Fu(\030)1094 1458 y Fs(0000)1197 1494 y Fv(\000)c Fu(\030)1345 1458 y Fs(0)o(0)1409 1494 y Fv(\000)1518 1455 y Fy(1)p 1518 1471 V 1518 1529 a(2)1563 1494 y FC(\()p Fu(\030)5 b(\032)p FC(\))1727 1458 y Fs(0)1773 1494 y Fv(\000)22 b FC(\()p Fu(\030)5 b( )2016 1517 y Fr(a)2058 1494 y FC(\))2091 1458 y Fs(0)2137 1494 y Ft(+)22 b Fu(\037)2296 1517 y Fr(a)2362 1494 y Fu(;)142 b(\030)5 b FC(\()p Fu(x;)17 b(\034)11 b FC(\))28 b Ft(=)f Fu(\027)2976 1517 y Fy(0)3016 1494 y FC(\()p Fu(x)p FC(\))f Fu(;)507 b FC(\(3.2\))109 1723 y(and)27 b(sho)n(w)f(that)h(if)h Fu(\016)j FC(is)c(suf)n (\002ciently)g(small)f(\()p Fu(\016)37 b Fv(\030)c(k)p Fu(\027)1987 1746 y Fy(0)2026 1723 y Fv(k)2076 1682 y Fs(\000)p Fy(2)2076 1748 y(2)2170 1723 y FC(\))28 b(then)f Fv(F)37 b FC(is)27 b(a)h(contraction)e(in)h(a)h(ball)f(of)h(radius)109 1844 y Fu(>)33 b Fv(k)p Fu(\027)316 1866 y Fy(0)355 1844 y Fv(k)405 1866 y Fy(2)472 1844 y FC(in)28 b Fv(W)676 1866 y Fr(\034)t(;\016)769 1844 y FC(.)39 b(Namely)-6 b(,)28 b(let)f Fu(f)44 b Ft(=)34 b Fu(e)1570 1808 y Fp(\()p Fr(t)p Fs(\000)p Fr(\034)8 b Fp(\))p Fr(A)1790 1844 y Fu(\030)33 b FC(and)27 b Fu(g)37 b Ft(=)c Fu(e)2275 1808 y Fp(\()p Fr(t)p Fs(\000)p Fr(\034)8 b Fp(\))p Fr(A)2496 1844 y Fu(\032)p FC(.)40 b(Multiplying)25 b(\(3.1\))j(with)f Fu(f)11 b(e)3643 1808 y Fp(\()p Fr(t)p Fs(\000)p Fr(\034)d Fp(\))p Fr(A)3863 1844 y FC(,)109 1964 y(inte)o(grating)23 b(o)o(v)o(er)h(the)g(space)i(v)n(ariable)e(and)h(using)e(the)i(results) f(of)h(the)g(preceding)f(section,)g(we)h(ha)n(v)o(e)586 2163 y Fy(1)p 586 2180 V 586 2237 a(2)631 2203 y Fu(@)682 2225 y Fr(t)712 2203 y Fv(k)p Fu(f)11 b Fv(k)871 2167 y Fy(2)871 2226 y(2)937 2203 y Fv(\024)29 b(k)p Fu(A)1166 2167 y Fy(1)p Fr(=)p Fy(2)1275 2203 y Fu(f)11 b Fv(k)1384 2167 y Fy(2)1384 2226 y(2)1446 2203 y Fv(\000)22 b(k)p Fu(A)1668 2167 y Fy(2)1708 2203 y Fu(f)11 b Fv(k)1817 2167 y Fy(2)1817 2226 y(2)1878 2203 y Ft(+)22 b Fv(k)p Fu(Af)11 b Fv(k)2208 2167 y Fy(2)2208 2226 y(2)2269 2203 y Ft(+)2377 2108 y Fs(p)p 2436 2108 43 3 v 48 x Fr(\031)p 2377 2180 102 4 v 2410 2237 a Fy(2)2488 2203 y Fv(k)p FC(\()p Ft(1)22 b(+)g Fu(A)2813 2167 y Fy(2)2853 2203 y FC(\))p Fu(f)11 b Fv(k)2995 2225 y Fy(2)3034 2203 y Fv(k)p Fu(f)g Fv(k)3193 2225 y Fy(2)3232 2203 y Fv(k)p Fu(g)t Fv(k)3383 2225 y Fy(2)1009 2365 y Ft(+)22 b(2)p Fu(\031)t(a)p Fv(k)p Fu(Af)11 b Fv(k)1498 2388 y Fy(2)1537 2365 y Fv(k)1587 2278 y(p)p 1670 2278 282 4 v 87 x Ft(1)22 b(+)g Fu(A)1912 2337 y Fy(2)1951 2365 y Fu(f)11 b Fv(k)2060 2388 y Fy(2)2122 2365 y Ft(+)22 b Fu(B)5 b Fv(k)p Fu(f)11 b Fv(k)2458 2388 y Fy(2)2521 2365 y Fu(:)1149 b FC(\(3.3\))109 2594 y(The)23 b(\002rst)h(term)g(on)f(the)h(r)-5 b(.h.s.)22 b(comes)i(from)f(the)h(time)f(deri)n(v)n(ati)n(v)o(e)e(of)j(the)f(e)o (xponential)f Fu(e)3196 2559 y Fp(\()p Fr(t)p Fs(\000)p Fr(\034)8 b Fp(\))p Fr(A)3417 2594 y FC(,)24 b(the)g(second)109 2715 y(and)f(third)g(from)g(the)g(space)h(deri)n(v)n(ati)n(v)o(es)d(of) j Fu(\030)5 b FC(.)29 b(The)24 b(ne)o(xt)f(term)g(uses)g(Lemma)g(2.4,)g (while)g(the)g(last)g(tw)o(o)g(use)109 2835 y(Lemma)h(2.1)g(and)h (Proposition)f(2.5,)g(respecti)n(v)o(ely)-6 b(.)29 b(Then,)24 b(we)h(use)g(the)g(inequalities)712 3064 y Fv(k)p Fu(Af)11 b Fv(k)944 3087 y Fy(2)1011 3064 y Fv(\024)28 b(k)1166 2977 y(p)p 1249 2977 V 87 x Ft(1)21 b(+)h Fu(A)1490 3035 y Fy(2)1530 3064 y Fu(f)11 b Fv(k)1639 3087 y Fy(2)1703 3064 y Fu(;)116 b Fv(k)1896 2977 y(p)p 1979 2977 V 87 x Ft(1)22 b(+)g Fu(A)2221 3035 y Fy(2)2261 3064 y Fu(f)11 b Fv(k)2370 3028 y Fy(2)2370 3087 y(2)2436 3064 y Fv(\024)28 b(k)p FC(\()p Ft(1)22 b(+)g Fu(A)2866 3028 y Fy(2)2906 3064 y FC(\))p Fu(f)11 b Fv(k)3048 3087 y Fy(2)3087 3064 y Fv(k)p Fu(f)g Fv(k)3246 3087 y Fy(2)109 3293 y FC(and)24 b(get)974 3483 y Fy(1)p 974 3499 36 4 v 974 3557 a(2)1020 3522 y Fu(@)1071 3545 y Fr(t)1101 3522 y Fv(k)p Fu(f)11 b Fv(k)1260 3486 y Fy(2)1260 3545 y(2)1326 3522 y Fv(\024)28 b(k)p Fu(A)1554 3486 y Fy(1)p Fr(=)p Fy(2)1664 3522 y Fu(f)11 b Fv(k)1773 3486 y Fy(2)1773 3545 y(2)1834 3522 y Fv(\000)23 b(k)p Fu(A)2057 3486 y Fy(2)2096 3522 y Fu(f)11 b Fv(k)2205 3486 y Fy(2)2205 3545 y(2)2266 3522 y Ft(+)22 b Fv(k)p Fu(Af)11 b Fv(k)2596 3486 y Fy(2)2596 3545 y(2)2658 3522 y Ft(+)22 b Fu(B)5 b Fv(k)p Fu(f)11 b Fv(k)2994 3545 y Fy(2)1398 3704 y Ft(+)1496 3594 y Fq(\020)1566 3610 y Fs(p)p 1624 3610 43 3 v 1624 3658 a Fr(\031)p 1566 3681 102 4 v 1599 3739 a Fy(2)1677 3704 y Fv(k)p Fu(g)t Fv(k)1828 3727 y Fy(2)1889 3704 y Ft(+)22 b(2)p Fu(\031)t(a)2146 3594 y Fq(\021)2222 3704 y Fv(k)p FC(\()p Ft(1)g(+)g Fu(A)2547 3669 y Fy(2)2586 3704 y FC(\))p Fu(f)11 b Fv(k)2728 3727 y Fy(2)2767 3704 y Fv(k)p Fu(f)g Fv(k)2926 3727 y Fy(2)2990 3704 y Fu(:)109 3954 y FC(W)-8 b(e)25 b(also)f(ha)n(v)o(e)964 4204 y Fv(k)p Fu(A)1087 4168 y Fy(1)p Fr(=)p Fy(2)1197 4204 y Fu(f)11 b Fv(k)1306 4168 y Fy(2)1306 4227 y(2)1373 4204 y Fv(\024)28 b(k)p Fu(f)11 b Fv(k)1637 4153 y Fy(3)p Fr(=)p Fy(2)1637 4228 y(2)1746 4204 y Fv(k)p Fu(A)1869 4168 y Fy(2)1909 4204 y Fu(f)g Fv(k)2018 4153 y Fy(1)p Fr(=)p Fy(2)2018 4228 y(2)2155 4204 y Fv(\024)2348 4136 y Ft(3)p 2270 4181 205 4 v 2270 4294 a(4)p Fu(")2365 4243 y Fy(1)p Fr(=)p Fy(3)2365 4318 y(1)2484 4204 y Fv(k)p Fu(f)g Fv(k)2643 4168 y Fy(2)2643 4227 y(2)2704 4204 y Ft(+)2812 4136 y Fu(")2858 4159 y Fy(1)p 2812 4181 86 4 v 2830 4272 a Ft(4)2907 4204 y Fv(k)p Fu(A)3030 4168 y Fy(2)3070 4204 y Fu(f)g Fv(k)3179 4168 y Fy(2)3179 4227 y(2)3242 4204 y Fu(;)1074 4483 y Fv(k)p Fu(Af)g Fv(k)1306 4447 y Fy(2)1306 4506 y(2)1373 4483 y Fv(\024)28 b(k)p Fu(f)11 b Fv(k)1637 4505 y Fy(2)1676 4483 y Fv(k)p Fu(A)1799 4447 y Fy(2)1838 4483 y Fu(f)g Fv(k)1947 4505 y Fy(2)2014 4483 y Fv(\024)2171 4415 y Ft(1)p 2129 4460 134 4 v 2129 4551 a(2)p Fu(")2224 4574 y Fy(2)2273 4483 y Fv(k)p Fu(f)g Fv(k)2432 4447 y Fy(2)2432 4506 y(2)2493 4483 y Ft(+)2600 4415 y Fu(")2646 4438 y Fy(2)p 2600 4460 86 4 v 2619 4551 a Ft(2)2696 4483 y Fv(k)p Fu(A)2819 4447 y Fy(2)2858 4483 y Fu(f)g Fv(k)2967 4447 y Fy(2)2967 4506 y(2)3031 4483 y Fu(;)711 4738 y(X)d Fv(k)p FC(\()p Ft(1)21 b(+)h Fu(A)1124 4703 y Fy(2)1164 4738 y FC(\))p Fu(f)11 b Fv(k)1306 4761 y Fy(2)1373 4738 y Fv(\024)1530 4671 y Ft(1)p 1488 4716 134 4 v 1488 4807 a(2)p Fu(")1583 4829 y Fy(3)1632 4738 y Fu(X)1721 4703 y Fy(2)1782 4738 y Ft(+)1890 4671 y Fu(")1936 4693 y Fy(3)p 1890 4716 86 4 v 1908 4807 a Ft(2)1985 4628 y Fq(\020)2045 4738 y Fv(k)p Fu(A)2168 4703 y Fy(2)2207 4738 y Fu(f)g Fv(k)2316 4703 y Fy(2)2316 4761 y(2)2377 4738 y Ft(+)22 b(2)p Fv(k)p Fu(Af)11 b Fv(k)2756 4703 y Fy(2)2756 4761 y(2)2817 4738 y Ft(+)22 b Fv(k)p Fu(f)11 b Fv(k)3074 4703 y Fy(2)3074 4761 y(2)3113 4628 y Fq(\021)3197 4738 y Fu(;)109 5014 y FC(for)25 b(all)f Fu(X)r(;)17 b(")547 5037 y Fr(i)603 5014 y Fu(>)28 b Ft(0)p Fu(;)17 b(i)27 b Ft(=)h(1)p Fu(;)17 b Ft(2)p Fu(;)g Ft(3)p FC(.)30 b(Using)24 b(these)g(inequalities)g(with)g(suf)n (\002ciently)f(small)h Fu(")3190 5037 y Fr(i)3243 5014 y FC(sho)n(ws)g(that)g(there)109 5135 y(is)g(a)h(positi)n(v)o(e)e (constant)h Fu(c)1005 5157 y Fs(\003)1069 5135 y FC(such)g(that)1311 5364 y Fu(@)1362 5386 y Fr(t)1392 5364 y Fv(k)p Fu(f)11 b Fv(k)1551 5328 y Fy(2)1551 5387 y(2)1618 5364 y Fv(\024)28 b FC(\()p Fu(c)1798 5386 y Fs(\003)1860 5364 y Ft(+)22 b Fv(k)p Fu(g)t Fv(k)2109 5328 y Fy(2)2109 5387 y(2)2147 5364 y FC(\))p Fv(k)p Fu(f)11 b Fv(k)2339 5328 y Fy(2)2339 5387 y(2)2400 5364 y Ft(+)22 b Fu(B)2577 5328 y Fy(2)2642 5364 y Fu(;)p 109 5469 1512 4 v 225 5530 a Fo(4)258 5565 y Fn(Note)e(that)h(by)e(our)h(choice)f(of)h Fz(\016)1185 5588 y Fe(\003)1223 5565 y Fn(,)h(we)f(ha)n(v)o(e)g Fz(t)e Fj(\000)g Fz(\034)33 b Fj(\024)1855 5533 y Fm(1)p 1855 5547 34 4 v 1855 5594 a(2)1898 5565 y Fn(.)p eop %%Page: 8 8 8 7 bop -128 43 a Fw(8)2685 b FC(P)t Fl(R)q(O)t(O)t(F)33 b(O)t(F)e FC(T)t Fl(H)t(E)t(O)t(R)t(E)t(M)i FC(1)t(.)t(3)-128 350 y(from)25 b(which)f(we)h(get)g(that)f(for)i(all)e Fu(\032)h FC(in)g(the)f(ball)h(of)g(radius)f Fu(d)h FC(in)f Fv(W)2238 373 y Fr(\034)t(;\016)2331 350 y FC(,)h Fv(F)10 b FC(\()p Fu(\032)p FC(\))25 b(satis\002es)809 613 y Ft(sup)733 695 y Fr(t)p Fs(2)p Fp([)p Fr(\034)t(;\034)8 b Fy(+)p Fr(\016)r Fp(])1048 613 y Fv(k)p Fu(e)1143 577 y Fp(\()p Fr(t)p Fs(\000)p Fr(\034)g Fp(\))p Fr(A)1364 613 y Fv(F)i FC(\()p Fu(\032)p FC(\))p Fv(k)1612 635 y Fy(2)1679 613 y Fv(\024)28 b Fu(e)1829 577 y Fr(\016)r Fp(\()p Fr(c)1915 585 y Fh(\003)1952 577 y Fy(+)p Fr(d)2043 554 y Fd(2)2078 577 y Fp(\))p Fr(=)p Fy(2)2174 492 y Fq(q)p 2274 492 466 4 v 121 x Fv(k)p Fu(\027)2372 635 y Fy(0)2412 613 y Fv(k)2462 578 y Fy(2)2462 637 y(2)2523 613 y Ft(+)22 b Fu(B)2700 584 y Fy(2)2764 613 y Fu(:)-128 910 y FC(F)o(or)k(all)f Fu(d)k(>)345 824 y Fq(p)p 444 824 515 4 v 444 910 a Fv(k)p Fu(\027)542 932 y Fy(0)582 910 y Fv(k)632 876 y Fy(2)632 934 y(2)693 910 y Ft(+)22 b(2)p Fu(B)919 881 y Fy(2)958 910 y FC(,)k(there)g(e)o(xists)e(a)i Fu(\016)33 b Ft(=)28 b Fv(O)s FC(\()p Fv(k)p Fu(\027)1948 932 y Fy(0)1988 910 y Fv(k)2038 869 y Fs(\000)p Fy(2)2038 934 y(2)2132 910 y FC(\))e(such)f(that)g Fv(F)35 b FC(maps)25 b 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b Fv(\000)h(F)10 b FC(\()p Fu(\032)1615 1804 y Fy(2)1655 1781 y FC(\)\))p Fv(k)1771 1804 y Fy(2)1838 1781 y Fu(<)103 b Ft(sup)1941 1863 y Fr(t)p Fs(2)p Fp([)p Fr(\034)t(;\034)8 b Fy(+)p Fr(\016)r Fp(])2257 1781 y Fv(k)p Fu(e)2352 1746 y Fp(\()p Fr(t)p Fs(\000)p Fr(\034)g Fp(\))p Fr(A)2573 1781 y FC(\()p Fu(\032)2656 1804 y Fy(1)2718 1781 y Fv(\000)22 b Fu(\032)2867 1804 y Fy(2)2907 1781 y FC(\))p Fv(k)2990 1804 y Fy(2)3054 1781 y Fu(;)-128 2052 y FC(so)k(that)f Fv(F)35 b FC(is)25 b(a)h(contraction)f(in)g Fv(B)1073 2075 y Fr(d)1143 2052 y Fv(\032)k(W)1347 2075 y Fr(\034)t(;\016)1440 2052 y FC(.)k(Thus)25 b(the)g(sequence)h(of)g(approximating)d (solutions)h Fu(\027)3410 2075 y Fr(n)p Fy(+1)3576 2052 y Ft(=)-128 2173 y Fv(F)10 b FC(\()p Fu(\027)35 2195 y Fr(n)82 2173 y FC(\))25 b(con)l(v)o(er)n(ges)g(to)f(a)h(unique)g (solution)e(of)h(\(3.1\))h(in)g Fv(B)1867 2195 y Fr(d)1935 2173 y Fv(\032)j(W)2138 2195 y Fr(\034)t(;\016)2231 2173 y FC(.)19 2293 y(Note)c(that)h(the)f(results)g(of)h(this)f(section)g (also)h(hold)f(for)h(the)f(equation)642 2503 y Ft(_)-43 b Fu(\027)34 b Ft(=)28 b Fv(\000)p Fu(\027)943 2468 y Fs(0)q(000)1047 2503 y Fv(\000)22 b Fu(\027)1200 2468 y Fs(0)q(0)1265 2503 y Fv(\000)1375 2464 y Fy(1)p 1375 2481 V 1375 2538 a(2)1420 2503 y FC(\()p Fu(\027)1507 2468 y Fy(2)1547 2503 y FC(\))1580 2468 y Fs(0)1626 2503 y Fv(\000)g FC(\()p Fu(\027)6 b( )1875 2526 y Fr(per)1983 2503 y FC(\))2016 2468 y Fs(0)2064 2503 y Fu(;)141 b(\027)6 b FC(\()p Fu(x;)17 b(\034)11 b FC(\))29 b Ft(=)f Fu(\027)2685 2526 y Fy(0)2725 2503 y FC(\()p Fu(x)p FC(\))d Fu(:)563 b FC(\(3.4\))-128 2714 y(It)19 b(follo)n(ws)f(easily)g(from)h([10])g (\(see)h(also)f([6)o(]\))h(that)f(periodic)f(stationary)g(solutions)f Fu( )2813 2736 y Fr(per)2939 2714 y FC(of)i(\(1.1\))g(satisfy)f(the) -128 2834 y(hypotheses)24 b(of)g(Lemma)h(2.3.)30 b(The)25 b(procedure)g(is)f(then)h(e)o(xactly)f(the)h(same,)f Fk(i.e)o(.)p FC(,)g(to)h(sho)n(w)e(that)i(the)f(analog)-128 2954 y(of)29 b(the)f(map)g Fv(F)37 b FC(is)28 b(a)h(contraction)e(in)h Fv(B)1258 2977 y Fr(d)1333 2954 y Fv(\032)34 b(W)1542 2977 y Fr(\034)t(;\016)1664 2954 y FC(for)28 b(some)g Fu(d;)17 b(\034)39 b FC(and)28 b Fu(\016)t FC(.)41 b(Using)27 b(ob)o(vious)f(notations,)i(we)-128 3075 y(\002nd)d(that)g(\(3.3\))f (is)h(replaced)g(by)-53 3255 y Fy(1)p -53 3272 V -53 3329 a(2)-8 3295 y Fu(@)43 3317 y Fr(t)73 3295 y Fv(k)p Fu(f)11 b Fv(k)232 3259 y Fy(2)232 3318 y(2)298 3295 y Fv(\024)29 b(k)p Fu(A)527 3259 y Fy(1)p Fr(=)p Fy(2)636 3295 y Fu(f)11 b Fv(k)745 3259 y Fy(2)745 3318 y(2)807 3295 y Fv(\000)22 b(k)p Fu(A)1029 3259 y Fy(2)1069 3295 y Fu(f)11 b Fv(k)1178 3259 y Fy(2)1178 3318 y(2)1239 3295 y Ft(+)22 b Fv(k)p Fu(Af)11 b Fv(k)1569 3259 y Fy(2)1569 3318 y(2)1630 3295 y Ft(+)1738 3200 y Fs(p)p 1797 3200 43 3 v 48 x Fr(\031)p 1738 3272 102 4 v 1771 3329 a Fy(2)1849 3295 y Fv(k)p FC(\()p Ft(1)22 b(+)g Fu(A)2174 3259 y Fy(2)2214 3295 y FC(\))p Fu(f)11 b Fv(k)2356 3317 y Fy(2)2395 3295 y Fv(k)p Fu(f)g Fv(k)2554 3317 y Fy(2)2593 3295 y Fv(k)p Fu(g)t Fv(k)2744 3317 y Fy(2)2804 3295 y Ft(+)22 b Fu(c)2944 3317 y Fr(per)3050 3295 y Fv(k)p Fu(Af)11 b Fv(k)3282 3317 y Fy(2)3321 3295 y Fv(k)p Fu(f)g Fv(k)3480 3317 y Fy(2)3544 3295 y Fu(;)-128 3505 y FC(from)25 b(which)f(it)h (follo)n(ws)e(that)1194 3715 y Fu(@)1245 3737 y Fr(t)1276 3715 y Fv(k)p Fu(f)11 b Fv(k)1435 3679 y Fy(2)1435 3738 y(2)1501 3715 y Fv(\024)28 b FC(\()p Fu(c)1681 3737 y Fs(\003)1743 3715 y Ft(+)22 b Fv(k)p Fu(g)t Fv(k)1992 3679 y Fy(2)1992 3738 y(2)2030 3715 y FC(\))p Fv(k)p Fu(f)11 b Fv(k)2222 3679 y Fy(2)2222 3738 y(2)2286 3715 y Fu(;)-128 3925 y FC(for)26 b(some)e(positi)n(v)o(e)e Fu(c)627 3948 y Fs(\003)667 3925 y FC(.)30 b(The)25 b(remainder)g(of)g (the)g(proof)f(is)h(straightforw)o(ard.)-128 4208 y FB(4)120 b(Pr)n(oof)29 b(of)g(Theor)n(em)h(1.3)-128 4418 y FC(Let)g(no)n(w)f Fu(\034)276 4440 y Fy(0)353 4418 y Ft(=)38 b(0)29 b FC(and)h Fu(\027)6 b FC(\()p Fu(x;)17 b(\034)947 4440 y Fy(0)988 4418 y FC(\))37 b Ft(=)h Fu(\027)1220 4440 y Fy(0)1259 4418 y FC(\()p Fu(x)p FC(\))31 b(with)e Fu(\027)1666 4440 y Fy(0)1743 4418 y Fv(2)38 b FC(L)1908 4382 y Fy(2)1947 4418 y FC(,)31 b(and)f(de\002ne)h Fu(\033)2506 4440 y Fy(0)2583 4418 y Ft(=)37 b Fv(k)p Fu(\027)2794 4440 y Fy(0)2834 4418 y Fv(k)2884 4440 y Fy(2)2923 4418 y FC(.)46 b(From)30 b(the)g(results)-128 4538 y(of)f(the)g(preceding)g(section,)g (we)g(kno)n(w)f(that)g(there)i(e)o(xists)d(a)i(unique)f(solution)f(of)i (\(3.1\))g(in)g Fv(B)3175 4560 y Fr(d)3211 4569 y Fd(0)3285 4538 y Fv(\032)36 b(W)3496 4560 y Fr(\034)3527 4569 y Fd(0)3562 4560 y Fr(;\016)3613 4569 y Fd(0)-128 4658 y FC(for)e Fu(d)73 4681 y Fy(0)156 4658 y Ft(=)44 b Fu(D)357 4681 y Fs(\003)396 4658 y Fu(\033)451 4681 y Fy(0)525 4658 y FC(and)33 b Fu(\016)745 4681 y Fy(0)828 4658 y Ft(=)44 b Fu(C)1018 4681 y Fs(\003)1057 4658 y Fu(\033)1116 4617 y Fs(\000)p Fy(2)1112 4683 y(0)1210 4658 y FC(,)36 b(with)d Fu(C)1552 4681 y Fs(\003)1625 4658 y FC(so)g(small)f(that)h Fu(\016)2222 4681 y Fy(0)2306 4658 y Fu(<)43 b(\016)2468 4681 y Fs(\003)2508 4658 y FC(.)57 b(Let)33 b Fu(\034)2798 4681 y Fy(1)2882 4658 y Ft(=)43 b Fu(\034)3043 4681 y Fy(0)3112 4658 y Ft(+)28 b Fu(\016)3259 4681 y Fy(0)3299 4658 y FC(.)56 b(By)34 b(the)-128 4779 y(de\002nition)c(of)g Fv(W)496 4801 y Fr(\034)t(;\016)619 4779 y FC(\(see)g(also)g(Corollary) g(1.4\),)h Fu(\027)37 b FC(and)30 b(all)g(its)f(deri)n(v)n(ati)n(v)o (es)e(tend)j(to)g Ft(0)g FC(as)g Fu(x)38 b Fv(!)f(\0061)30 b FC(for)-128 4899 y(all)i Fu(t)41 b Fv(2)g FC([)p Fu(\034)262 4922 y Fy(0)302 4899 y Fu(;)17 b(\034)388 4922 y Fy(1)428 4899 y FC(].)52 b Fk(In)32 b(particular)-11 b(,)32 b(the)f(trilinear)f (form)1843 4819 y Fq(R)1926 4899 y Fu(\027)6 b FC(\()p Fu(\027)2067 4864 y Fy(2)2107 4899 y FC(\))2140 4864 y Fs(0)2196 4899 y Fk(satis\002es)2543 4819 y Fq(R)2626 4899 y Fu(\027)g FC(\()p Fu(\027)2767 4864 y Fy(2)2808 4899 y FC(\))2841 4864 y Fs(0)2905 4899 y Ft(=)41 b Fv(\000)3109 4860 y Fy(1)p 3109 4876 36 4 v 3109 4934 a(3)3171 4819 y Fq(R)3237 4899 y FC(\()p Fu(\027)3324 4864 y Fy(3)3364 4899 y FC(\))3397 4864 y Fs(0)3462 4899 y Ft(=)f(0)p Fk(.)-128 5020 y FC(Hence,)26 b(multiplying)c(\(1.3\))i(with)g Fu(\027)32 b FC(and)25 b(inte)o(grating)e(o)o(v)o(er)h(the)g(space)i(v) n(ariable,)e(we)h(get:)388 5236 y Fy(1)p 388 5252 V 388 5309 a(2)433 5275 y Fu(@)484 5297 y Fr(t)531 5139 y Fq(Z)647 5275 y Fu(\027)701 5239 y Fy(2)769 5275 y Ft(=)i Fv(\000)966 5139 y Fq(Z)1083 5275 y Fu(\027)6 b(\027)1191 5239 y Fs(0)q(000)1294 5275 y Fv(\000)1394 5139 y Fq(Z)1510 5275 y Fu(\027)g(\027)1618 5239 y Fs(0)q(0)1684 5275 y Fv(\000)1793 5236 y Fy(1)p 1793 5252 V 1793 5309 a(2)1855 5139 y Fq(Z)1971 5275 y Fu(\027)g FC(\()p Fu(\027)2112 5239 y Fy(2)2153 5275 y FC(\))2186 5239 y Fs(0)2232 5275 y Fv(\000)2331 5139 y Fq(Z)2448 5275 y Fu(\027)g FC(\()p Fu(\027)g( )2652 5297 y Fr(a)2695 5275 y FC(\))2728 5239 y Fs(0)2773 5275 y Fv(\000)2873 5139 y Fq(Z)2989 5275 y Fu(\027)g(\037)3104 5297 y Fr(a)769 5529 y Ft(=)27 b Fv(\000)966 5394 y Fq(Z)1066 5529 y FC(\()p Fu(\027)1153 5494 y Fs(0)q(0)1196 5529 y FC(\))1229 5494 y Fy(2)1291 5529 y Ft(+)1389 5394 y Fq(Z)1488 5529 y FC(\()p Fu(\027)1575 5494 y Fs(0)1599 5529 y FC(\))1632 5494 y Fy(2)1694 5529 y Fv(\000)1804 5490 y Fy(1)p 1804 5507 V 1804 5564 a(2)1866 5394 y Fq(Z)1982 5529 y Fu(\027)2036 5494 y Fy(2)2076 5529 y Fu( )2143 5494 y Fs(0)2139 5552 y Fr(a)2203 5529 y Fv(\000)2303 5394 y Fq(Z)2419 5529 y Fu(\027)6 b(\037)2534 5552 y Fr(a)2601 5529 y Fu(:)p eop %%Page: 9 9 9 8 bop 111 43 a FC(R)t Fl(E)t(F)t(E)t(R)t(E)t(N)t(C)5 b(E)g(S)3175 b Fw(9)109 350 y FC(Note)23 b(that)g(because)h(the)g (trilinear)f(form)g(v)n(anishes,)g(we)h(get)f(only)g(quadratic)g(\(and) h(linear\))g(terms)f(in)g Fu(\027)6 b FC(,)24 b(and)109 470 y(therefore)g(it)f(is)h(natural)f(to)h(\002nd)g(an)g(e)o (xponential)e(bound)h(in)g(time)h(for)g(the)f(e)n(v)n(olution)f(of)i (the)g(L)3419 435 y Fy(2)3482 470 y FC(norm;)f(this)109 591 y(is)i(the)h(main)g(e)o(xplanation)f(for)h(the)g(bounds)f(which)h (follo)n(w)f(belo)n(w)-6 b(.)33 b(Using)26 b Fv(\000)p Fu(k)2913 555 y Fy(4)2976 591 y Ft(+)d Fu(k)3129 555 y Fy(2)3199 591 y Fv(\024)3316 552 y Fy(1)p 3316 568 36 4 v 3316 625 a(4)3362 591 y FC(,)j(we)h(ha)n(v)o(e)e(the)109 711 y(inequalities)1160 972 y Fv(\000)1254 836 y Fq(Z)1353 972 y FC(\()p Fu(\027)1440 936 y Fs(0)q(0)1483 972 y FC(\))1516 936 y Fy(2)1578 972 y Ft(+)1676 836 y Fq(Z)1776 972 y FC(\()p Fu(\027)1863 936 y Fs(0)1887 972 y FC(\))1920 936 y Fy(2)1987 972 y Fv(\024)2102 933 y Fy(1)p 2102 949 V 2102 1006 a(4)2164 836 y Fq(Z)2280 972 y Fu(\027)2334 936 y Fy(2)2399 972 y Fu(;)1799 1166 y Fv(j)p Fu( )1894 1130 y Fs(0)1890 1188 y Fr(a)1932 1166 y Fv(j)i(\024)h Fu(a)d(;)1603 1224 y Fq(\014)1603 1283 y(\014)1603 1343 y(\014)1603 1403 y(\014)1653 1233 y(Z)1769 1368 y Fu(\027)6 b(\037)1884 1391 y Fr(a)1926 1224 y Fq(\014)1926 1283 y(\014)1926 1343 y(\014)1926 1403 y(\014)1987 1368 y Fv(\024)2102 1329 y Fy(1)p 2102 1345 V 2102 1403 a(2)2164 1233 y Fq(Z)2280 1368 y Fu(\027)2334 1332 y Fy(2)2396 1368 y Ft(+)2504 1329 y Fy(1)p 2504 1345 V 2504 1403 a(2)2566 1233 y Fq(Z)2682 1368 y Fu(\037)2743 1332 y Fy(2)2743 1391 y Fr(a)2810 1368 y Fu(:)109 1648 y FC(W)-8 b(e)25 b(\002nd)g(for)l(,)g Fu(t)j Fv(2)g FC([)p Fu(\034)838 1671 y Fy(0)878 1648 y Fu(;)17 b(\034)964 1671 y Fy(1)1003 1648 y FC(],)25 b(with)f Fu(\014)34 b Fv(\021)28 b Ft(2)p FC(\()1574 1609 y Fy(3)p 1574 1625 V 1574 1683 a(4)1641 1648 y Ft(+)22 b Fu(a)p FC(\),)979 1865 y Ft(1)p 979 1909 49 4 v 979 2000 a(2)1038 1932 y Fu(@)1089 1954 y Fr(t)1136 1796 y Fq(Z)1252 1932 y Fu(\027)1306 1896 y Fy(2)1390 1932 y Fv(\024)1522 1865 y Fu(\014)p 1522 1909 61 4 v 1528 2000 a Ft(2)1609 1796 y Fq(Z)1725 1932 y Fu(\027)1779 1896 y Fy(2)1841 1932 y Ft(+)1949 1865 y(1)p 1949 1909 49 4 v 1949 2000 a(2)2025 1796 y Fq(Z)2141 1932 y Fu(\037)2202 1896 y Fy(2)2202 1954 y Fr(a)2288 1932 y Fv(\024)2420 1865 y Fu(\014)p 2420 1909 61 4 v 2426 2000 a Ft(2)2507 1796 y Fq(Z)2623 1932 y Fu(\027)2677 1896 y Fy(2)2739 1932 y Ft(+)2847 1865 y Fu(B)2926 1829 y Fy(2)p 2847 1909 119 4 v 2882 2000 a Ft(2)3000 1932 y Fu(:)670 b FC(\(4.1\))109 2189 y(This)24 b(dif)n(ferential)g (inequality)f(is)i(v)n(alid)e(for)i(all)g Fu(t)j Fv(2)g FC([)p Fu(\034)2010 2212 y Fy(0)2050 2189 y Fu(;)17 b(\034)2136 2212 y Fy(1)2175 2189 y FC(],)25 b(and)g(implies)e(that)1253 2444 y Fv(k)p Fu(\027)6 b FC(\()p Fv(\001)p Fu(;)17 b(\034)1504 2467 y Fy(1)1543 2444 y FC(\))p Fv(k)1626 2467 y Fy(2)1693 2444 y Fv(\024)29 b Fu(e)1844 2409 y Fr(\014)s(\034)1918 2418 y Fd(1)1953 2409 y Fr(=)p Fy(2)2028 2323 y Fq(q)p 2127 2323 338 4 v 2127 2444 a Fu(\033)2186 2410 y Fy(2)2182 2469 y(0)2248 2444 y Ft(+)22 b Fu(B)2425 2416 y Fy(2)2492 2444 y Fv(\021)28 b Fu(\033)2652 2467 y Fy(1)2717 2444 y Fu(:)109 2670 y FC(Again,)c(from)h(the)h(results)e(of)i(the)f (preceding)g(section,)g(we)h(no)n(w)e(see)i(that)f(there)h(e)o(xists)e (a)h(unique)g(solution)109 2790 y(of)h(\(3.1\))g(in)g Fv(B)604 2812 y Fr(d)640 2821 y Fd(1)711 2790 y Fv(\032)31 b(W)917 2812 y Fr(\034)948 2821 y Fd(1)983 2812 y Fr(;\016)1034 2821 y Fd(1)1099 2790 y FC(for)c Fu(d)1293 2812 y Fy(1)1362 2790 y Ft(=)k Fu(D)1550 2812 y Fs(\003)1589 2790 y Fu(\033)1644 2812 y Fy(1)1711 2790 y FC(and)26 b Fu(\016)1924 2812 y Fy(1)1994 2790 y Ft(=)31 b Fu(C)2171 2812 y Fs(\003)2210 2790 y Fu(\033)2269 2749 y Fs(\000)p Fy(2)2265 2814 y(1)2363 2790 y FC(.)36 b(Thus)25 b(\(4.1\))i(is)f(v)n(alid)f(for)i(all)f Fu(t)31 b Fv(2)g FC([)p Fu(\034)3690 2812 y Fy(0)3730 2790 y Fu(;)17 b(\034)3816 2812 y Fy(2)3855 2790 y FC(])109 2910 y(with)24 b Fu(\034)353 2933 y Fy(2)420 2910 y Ft(=)k Fu(\034)566 2933 y Fy(1)628 2910 y Ft(+)22 b Fu(\016)769 2933 y Fy(1)808 2910 y FC(,)j(and)g(we)g(get)1253 3166 y Fv(k)p Fu(\027)6 b FC(\()p Fv(\001)p Fu(;)17 b(\034)1504 3188 y Fy(2)1543 3166 y FC(\))p Fv(k)1626 3188 y Fy(2)1693 3166 y Fv(\024)29 b Fu(e)1844 3130 y Fr(\014)s(\034)1918 3139 y Fd(2)1953 3130 y Fr(=)p Fy(2)2028 3044 y Fq(q)p 2127 3044 V 2127 3166 a Fu(\033)2186 3131 y Fy(2)2182 3190 y(0)2248 3166 y Ft(+)22 b Fu(B)2425 3137 y Fy(2)2492 3166 y Fv(\021)28 b Fu(\033)2652 3188 y Fy(2)2717 3166 y Fu(:)109 3391 y FC(Continuing)23 b(by)i(induction,)e(we)i(\002nd)1478 3665 y Fu(\016)1521 3688 y Fr(n)1596 3665 y Ft(=)i Fu(C)1769 3688 y Fs(\003)1809 3665 y Fu(\033)1868 3629 y Fs(\000)p Fy(2)1864 3688 y Fr(n)1990 3665 y Ft(=)2161 3598 y Fu(e)2206 3562 y Fs(\000)p Fr(\014)s(\034)2335 3570 y Fc(n)p 2103 3642 V 2103 3734 a Fu(\033)2162 3699 y Fy(2)2158 3758 y(0)2224 3734 y Ft(+)22 b Fu(B)2401 3705 y Fy(2)2475 3665 y Fu(;)109 3929 y FC(so)i(that)1538 4145 y Fu(\034)1580 4167 y Fr(n)p Fy(+1)1745 4145 y Ft(=)j Fu(\034)1890 4167 y Fr(n)1960 4145 y Ft(+)22 b Fu(E)2130 4167 y Fs(\003)2169 4145 y Fu(e)2214 4109 y Fs(\000)p Fr(\014)s(\034)2343 4117 y Fc(n)2415 4145 y Fu(;)109 4360 y FC(with)k Fu(E)385 4383 y Fs(\003)457 4360 y Ft(=)31 b FC(\()p Fu(\033)656 4325 y Fy(2)652 4383 y(0)720 4360 y Ft(+)24 b Fu(B)899 4325 y Fy(2)938 4360 y FC(\))971 4325 y Fs(\000)p Fy(1)1066 4360 y FC(.)37 b(This)27 b(implies)e(that)i Ft(lim)1965 4383 y Fr(n)p Fs(!1)2170 4360 y Fu(\034)2212 4383 y Fr(n)2291 4360 y Ft(=)32 b Fv(1)p FC(,)c(and)f(therefore)g(\(4.1\))h(is)e(v)n (alid)g(for)i(all)109 4481 y Fu(t)i(>)f Ft(0)p FC(.)34 b(This)25 b(completes)g(the)h(proof)f(of)h(Theorem)g(1.3)g(for)g(the)f (pro\002le)i(case.)34 b(The)26 b(periodic)f(case)i(follo)n(ws)109 4601 y(along)d(the)h(same)f(lines.)109 4810 y Fw(Ackno)o(wledgements) 109 4967 y FC(This)g(research)h(w)o(as)g(partially)f(supported)g(by)h (the)f(F)o(onds)g(National)g(Suisse.)109 5250 y FB(Refer)n(ences)154 5453 y Fx([1])50 b(P)-10 b(.)33 b(Collet)j(and)f(J.-P)-10 b(.)34 b(Eckmann.)70 b(The)35 b(time)f(dependent)k(amplitude)f (equation)g(for)e(the)h(Swift-Hohenber)n(g)309 5565 y(problem.)e Fb(Comm.)22 b(Math.)i(Phys.)f Fa(132)h Fx(\(1990\),)h(139\226153.)p eop %%Page: 10 10 10 9 bop -128 43 a Fw(10)3119 b FC(R)t Fl(E)t(F)t(E)t(R)t(E)t(N)5 b(C)g(E)t(S)-82 350 y Fx([2])50 b(P)-10 b(.)17 b(Collet)j(and)g(J.-P) -10 b(.)18 b(Eckmann.)25 b(Space-time)20 b(beha)n(viour)j(in)c (problems)i(of)e(hydrodynamic)j(type:)28 b(a)19 b(case)h(study)-6 b(.)73 463 y Fb(Nonlinearity)26 b Fa(5)d Fx(\(1992\),)i(1265\2261302.) -82 617 y([3])50 b(P)-10 b(.)32 b(Collet,)38 b(J.-P)-10 b(.)33 b(Eckmann,)k(H.)c(Epstein,)38 b(and)c(J.)g(Stubbe.)69 b(A)40 b(global)35 b(attracting)i(set)e(for)f(the)h(Kuramoto-)73 730 y(Si)n(v)n(ashinsk)o(y)25 b(equation.)35 b Fb(Comm.)22 b(Math.)i(Phys.)f Fa(152)h Fx(\(1993\),)h(203\226214.)-82 885 y([4])50 b(P)-10 b(.)33 b(Collet,)k(J.-P)-10 b(.)33 b(Eckmann,)38 b(H.)33 b(Epstein,)38 b(and)d(J.)e(Stubbe.)69 b(Analyticity)37 b(for)e(the)g(Kuramoto-Si)n(v)n(ashinsk)o(y)73 998 y(equation.)g Fb(Phys.)23 b(D)g Fa(67)g Fx(\(1993\),)i(321\226326.) -82 1152 y([5])50 b(J.)22 b(Goodman.)32 b(Stability)24 b(of)f(the)f(Kuramoto-Si)n(v)n(ashinsk)o(y)27 b(and)c(related)h (systems.)33 b Fb(Comm.)21 b(Pur)m(e)h(Appl.)g(Math.)73 1265 y Fa(47)h Fx(\(1994\),)i(293\226306.)-82 1419 y([6])50 b(Z.)30 b(Gruji)5 b(\264)-35 b(c.)61 b(Spatial)33 b(analyticity)i(on)d (the)g(global)h(attractor)h(for)f(the)f(Kuramoto-Si)n(v)n(ashinsk)o(y)k (equation.)62 b Fb(J)n(.)73 1532 y(Dynam.)23 b(Dif)n(fer)m(ential)j (Equations)f Fa(12)f Fx(\(2000\),)h(217\226228.)-82 1687 y([7])50 b(J.)25 b(S.)f(Il)309 1651 y Fs(0)333 1687 y Fx(yashenk)o(o.)43 b(Global)27 b(analysis)h(of)e(the)g(phase)h (portrait)g(for)g(the)f(Kuramoto-Si)n(v)n(ashinsk)o(y)j(equation.)43 b Fb(J)n(.)73 1800 y(Dynam.)23 b(Dif)n(fer)m(ential)j(Equations)f Fa(4)e Fx(\(1992\),)i(585\226615.)-82 1954 y([8])50 b(Y)-12 b(.)27 b(K)o(uramoto)j(and)f(T)-7 b(.)27 b(Tsuzuki.)52 b(Persistent)30 b(propagation)i(of)d(concentration)k(w)o(a)n(v)o(es)c (in)g(dissipati)n(v)o(e)i(media)73 2067 y(f)o(ar)23 b(for)h(thermal)h (equilibrium.)35 b Fb(Pr)l(o)o(gr)m(ess)25 b(of)e(Theor)-10 b(.)24 b(Phys.)f Fa(55)h Fx(\(1976\),)h(356\226368.)-82 2221 y([9])50 b(P)-10 b(.)29 b(Manne)n(ville.)60 b(The)31 b(Kuramoto-Si)n(v)n(ashinsk)o(y)j(equation:)47 b(a)31 b(progress)i(report.)59 b(In:)45 b Fb(Pr)l(opa)o(gation)34 b(in)d(sys-)73 2334 y(tems)g(far)g(fr)l(om)g(equilibrium)i(\(Les)e (Houc)o(hes,)j(1987\))p Fx(,)g(v)n(olume)e(41)f(of)g Fb(Spring)o(er)j(Ser)-10 b(.)31 b(Syner)m(g)o(etics)j Fx(\(Berlin:)73 2447 y(Springer)l(,)25 b(1988\),)f(pp.)f(265\226280.) -128 2602 y([10])51 b(D.)18 b(Michelson.)27 b(Steady)20 b(solutions)i(of)d(the)h(Kuramoto-Si)n(v)n(ashinsk)o(y)k(equation.)j Fb(Phys.)19 b(D)f Fa(19)i Fx(\(1986\),)i(89\226111.)-128 2756 y([11])51 b(D.)21 b(Michelson.)34 b(Stability)25 b(of)e(the)g(Bunsen)h(\003ame)e(pro\002les)i(in)f(the)g(Kuramoto-Si)n (v)n(ashinsk)o(y)k(equation.)34 b Fb(SIAM)73 2869 y(J)n(.)22 b(Math.)i(Anal.)f Fa(27)g Fx(\(1996\),)i(765\226781.)-128 3023 y([12])51 b(A.)25 b(Mielk)o(e.)43 b(The)26 b(Ginzb)n(ur)n (g-Landau)k(equation)f(in)d(its)h(role)g(as)f(a)g(modulation)j (equation.)45 b(In:)35 b Fb(Handbook)28 b(of)73 3136 y(dynamical)d(systems,)f(V)-10 b(ol.)23 b(2)g Fx(\(Amsterdam:)30 b(North-Holland,)c(2002\),)e(pp.)g(759\226834.)-128 3291 y([13])51 b(A.)40 b(Mielk)o(e)k(and)e(G.)f(Schneider)-5 b(.)95 b(Attractors)44 b(for)f(modulation)h(equations)h(on)e(unbounded) i(domains\227)73 3404 y(e)o(xistence)25 b(and)f(comparison.)36 b Fb(Nonlinearity)26 b Fa(8)d Fx(\(1995\),)i(743\226768.)-128 3558 y([14])51 b(B.)25 b(Nicolaenk)o(o,)30 b(B.)25 b(Scheurer)l(,)30 b(and)d(R.)e(T)-6 b(emam.)43 b(Some)26 b(global)j(dynamical)f (properties)i(of)d(the)g(Kuramoto-)73 3671 y(Si)n(v)n(ashinsk)o(y)e (equations:)32 b(nonlinear)26 b(stability)g(and)e(attractors.)35 b Fb(Phys.)24 b(D)e Fa(16)i Fx(\(1985\),)g(155\226183.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0308070306348--