Content-Type: multipart/mixed; boundary="-------------9811301801365" This is a multi-part message in MIME format. ---------------9811301801365 Content-Type: text/plain; name="98-735.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="98-735.keywords" porous medium, entropy method, Liapunov functional, Fokker-Planck ---------------9811301801365 Content-Type: application/postscript; name="porosos.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="porosos.ps" %!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: porosos.dvi %%Pages: 26 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%DocumentPaperSizes: Letter %%EndComments %DVIPSCommandLine: dvips porosos.dvi -o porosos.ps %DVIPSParameters: dpi=600, compressed, comments removed %DVIPSSource: TeX output 1998.11.30:1418 %%BeginProcSet: texc.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls 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5041 V 1943 5132 a(m)f Fj(\000)h Fl(1)2208 5064 y Fj(r)p Fk(u)2347 5023 y Fp(m)p Fn(\000)p Fr(1)2504 4939 y Fo(\014)2504 4989 y(\014)2504 5039 y(\014)2504 5089 y(\014)2531 4966 y Fr(2)2604 5064 y Fk(dx:)887 b Fl(\(1.7\))1926 5356 y(2)p eop %%Page: 3 3 3 2 bop 75 107 a Fl(T)-8 b(ogether)33 b(with)f(the)h(en)m(trop)m(y)h (equation)1516 299 y Fk(d)p 1498 343 86 4 v 1498 434 a(dt)1594 366 y(H)8 b Fl(\()p Fk(u)p Fj(j)p Fk(u)1861 381 y Fn(1)1934 366 y Fl(\))28 b(=)f Fj(\000)p Fl(2)p Fk(I)8 b Fl(\()p Fk(u)p Fl(\))1212 b(\(1.8\))75 607 y(one)33 b(obtains)f(the)h(equation)f(for)g(the)h(en)m(trop)m(y)h(pro)s(duction) 1460 799 y Fk(d)p 1442 843 V 1442 935 a(dt)1538 866 y(I)8 b Fl(\()p Fk(u)p Fl(\))27 b(=)g Fj(\000)p Fl(2)p Fk(I)8 b Fl(\()p Fk(u)p Fl(\))22 b Fj(\000)h Fk(R)q Fl(\()p Fk(t)p Fl(\))1156 b(\(1.9\))75 1113 y(where)34 b Fk(R)q Fl(\()p Fk(t)p Fl(\))27 b Fj(\025)i Fl(0.)43 b(Com)m(bining)30 b(\(1.8\))i(and)h(\(1.9\))e(one)i(concludes)g(with)f(the)h(exp)s(onen)m (tial)f(deca)m(y)h(of)75 1233 y(the)g(relativ)m(e)f(en)m(trop)m(y)i(to) e(zero)h(at)f(a)h(rate)f(2)p Fk(t)p Fl(.)221 1354 y(As)46 b(far)e(as)h(the)h(p)s(orous)f(medium)e(equation)i(is)f(concerned,)50 b(the)45 b(asymptotic)g(b)s(eha)m(viour)f(of)75 1474 y(the)g(solution)e(has)j(b)s(een)f(describ)s(ed)h(in)e(dimension)f(1)h (b)m(y)i(Kamin)d(in)h([7],)j([8])e(in)f(a)h Fk(L)3403 1438 y Fn(1)3478 1474 y Fl(-setting.)75 1594 y(This)49 b(result)f(has)h(b)s(een)h(subsequen)m(tly)h(generalized)d(to)g(the)h (case)h Fk(N)65 b(>)55 b Fl(1)48 b(b)m(y)i(F)-8 b(riedman)47 b(and)75 1715 y(Kamin)26 b([5].)41 b(The)29 b(approac)m(h)f(used)h(in)d (these)j(pap)s(ers)f(is)f(completely)g(di\013eren)m(t,)i(and)e(uses)i (similarit)m(y)75 1835 y(transformations)35 b(for)h(\(1.1\).)55 b(Recen)m(tly)-8 b(,)38 b(most)e(of)h(the)g(existing)f(results)h(ha)m (v)m(e)h(b)s(een)f(collected)f(b)m(y)75 1956 y(V)-8 b(azquez)39 b(in)e(an)g(excellen)m(t)h(surv)m(ey)i([19)o(].)59 b(Later)37 b(on)g(in)g(this)g(pap)s(er)h(w)m(e)g(will)e(review)i(the)g(existing)75 2076 y(results)33 b(ab)s(out)f(rates)h(of)f(con)m(v)m(ergence)j(of)e (the)g(solution.)221 2196 y(The)f(organization)d(of)h(the)i(pap)s(er)f (is)f(as)h(follo)m(ws.)42 b(Section)31 b(2)f(is)h(dev)m(oted)h(to)f (some)g(preliminary)75 2317 y(material)j(concerning)i(the)h(time-dep)s (enden)m(t)g(scaling)e(of)h(parab)s(olic)e(equations.)56 b(Here,)38 b(the)f(main)75 2437 y(example)28 b(is)g(furnished)h(b)m(y)g (the)g(connection)f(b)s(et)m(w)m(een)j(the)e(heat)f(equation)g(and)h (the)g(linear)e(F)-8 b(okk)m(er-)75 2557 y(Planc)m(k)29 b(equation.)42 b(In)29 b(section)f(3)h(w)m(e)g(study)h(the)f(time)e(ev) m(olution)h(of)g(the)h(en)m(trop)m(y)h(functional)c(\(1.6\))75 2678 y(and)k(of)f(the)i(en)m(trop)m(y)g(pro)s(duction)e(\(1.8\))g(for)h (the)g(nonlinear)e(F)-8 b(okk)m(er-Planc)m(k)31 b(t)m(yp)s(e)g (equation)e(\(1.3\).)75 2798 y(This)40 b(analysis)f(sho)m(ws)i(that)f 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Fp(N)q(=)p Fr(2)2107 369 y Fl(exp)2272 223 y Fo(\()2339 369 y Fj(\000)2426 302 y(j)p Fk(x)p Fj(j)2537 266 y Fr(2)p 2426 346 151 4 v 2460 438 a Fl(4)p Fk(t)2587 223 y Fo(\))75 632 y Fl(and)1413 752 y Fk(G)p Fl(\()p Fk(x)p Fl(;)17 b Fk(t)p Fl(\))28 b(=)f Fk(e)1876 711 y Fp(N)7 b(t)1969 752 y Fj(N)15 b Fl(\()p Fk(e)2149 711 y Fp(t)2178 752 y Fk(x;)i(\014)6 b Fl(\()p Fk(t)p Fl(\)\))75 921 y(cen)m(tered)29 b(at)e(zero,)h(resp)s (ectiv)m(ely)-8 b(,)29 b(where)g Fk(\014)6 b Fl(\()p Fk(t)p Fl(\))27 b(=)1935 881 y Fr(1)p 1935 897 36 4 v 1935 955 a(2)1980 921 y Fl(\()p Fk(e)2063 884 y Fr(2)p Fp(t)2139 921 y Fj(\000)11 b Fl(1\).)41 b(It)27 b(is)g(easy)h(to)f(see) h(that)f(if)f(w)m(e)i(consider)75 1041 y(the)33 b(c)m(hange)g(of)g(v)-5 b(ariables)1417 1161 y Fk(u)p Fl(\()p Fk(x;)17 b(t)p Fl(\))27 b(=)h Fk(e)1859 1120 y Fp(N)7 b(t)1952 1161 y Fk(v)t Fl(\()p Fk(e)2086 1120 y Fp(t)2115 1161 y Fk(x;)17 b(\014)6 b Fl(\()p Fk(t)p Fl(\)\))32 b Fk(;)1141 b Fl(\(2.3\))75 1330 y(then)33 b Fk(u)f Fl(is)g(a)h(solution)e(of)h(\(2.2\))g(for)g(an) m(y)h(solution)e Fk(v)36 b Fl(of)d(\(2.1\).)42 b(Or,)33 b(equiv)-5 b(alen)m(tly)d(,)924 1598 y Fk(v)t Fl(\()p Fk(x;)17 b(t)p Fl(\))28 b(=)g(\(2)p Fk(t)22 b Fl(+)g(1\))1646 1557 y Fn(\000)p Fp(N)q(=)p Fr(2)1832 1598 y Fk(u)1905 1452 y Fo( )2120 1530 y Fk(x)p 1980 1574 336 4 v 1980 1592 a Fj(p)p 2063 1592 253 4 v 79 x Fl(2)p Fk(t)g Fl(+)g(1)2326 1598 y Fk(;)2380 1530 y Fl(1)p 2380 1574 49 4 v 2380 1666 a(2)2455 1598 y(log\(2)p Fk(t)g Fl(+)g(1\))2910 1452 y Fo(!)3624 1598 y Fl(\(2.4\))75 1866 y(is)46 b(a)h(solution)e(of) h(\(2.1\))g(for)g(an)m(y)h(solution)f Fk(u)g Fl(of)g(\(2.2\).)85 b(Therefore,)51 b(the)d(c)m(hange)f(of)f(v)-5 b(ariables)75 1986 y(\(2.3\){\(2.4\))31 b(pro)s(duce)j(an)e(isomorphism)e(b)s(et)m(w) m(een)35 b(the)e(sets)h(of)e(solutions)f(of)h(\(2.1\))g(and)h(\(2.2\).) 221 2107 y(Moreo)m(v)m(er,)28 b(the)e(fundamen)m(tal)e(solution)f(of)h (the)i(heat)f(equation)g(\(2.1\))f(at)g(time)g Fk(t)k Fl(=)3339 2067 y Fr(1)p 3339 2083 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3311 y Fg(I)-10 b(R)1182 3292 y Ff(N)1261 3143 y Fo(\020)1311 3240 y Fl(1)21 b(+)h Fj(j)p Fk(x)p Fj(j)1590 3198 y Fr(2)1652 3240 y Fl(+)g Fj(j)17 b Fl(log)f Fk(v)1984 3255 y Fr(0)2023 3240 y Fl(\()p Fk(x)p Fl(\))p Fj(j)2182 3143 y Fo(\021)2248 3240 y Fk(v)2295 3255 y Fr(0)2335 3240 y Fl(\()p Fk(x)p Fl(\))h Fk(dx)28 b(<)f Fj(1)804 b Fl(\(2.5\))75 3474 y(More)33 b(exactly)-8 b(,)887 3637 y Fj(k)p Fk(v)t Fl(\()p Fk(x;)17 b(t)p Fl(\))22 b Fj(\000)g Fk(M)10 b Fj(N)15 b Fl(\()p Fk(x;)i(t)p Fl(\))p Fj(k)1780 3656 y Fp(L)1828 3637 y Fg(1)1863 3656 y Fr(\()p Fg(I)-10 b(R)1947 3637 y Ff(N)2005 3656 y Fr(\))2064 3637 y Fj(\024)2309 3570 y Fk(C)p 2179 3614 336 4 v 2179 3632 a Fj(p)p 2262 3632 253 4 v 78 x Fl(2)p Fk(t)22 b Fl(+)g(1)2558 3637 y(;)211 b Fk(t)28 b Fj(\025)g Fl(0)75 3853 y(and)33 b(that)f(this)g(b)s(ound)h (is)f(sharp.)44 b(Here,)34 b Fk(M)43 b Fl(is)32 b(the)h(initial)c(mass) j(of)g Fk(v)t Fl(.)221 3973 y(As)41 b(far)e(as)h(the)g(F)-8 b(okk)m(er-Planc)m(k)41 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b(an)m(y)h(result)f(ab)s(out)f(the)h(asymptotic)g(b)s(eha)m (vior)f(of)h Fk(u)p Fl(\()p Fk(x;)17 b(t)p Fl(\))36 b(translates)h(in)m (to)f(a)h(result)f(ab)s(out)75 4866 y(the)d(asymptotic)f(b)s(eha)m (vior)g(of)g Fk(v)t Fl(\()p Fk(x;)17 b(t)p Fl(\))33 b(and)f(vice)h(v)m (ersa.)221 4987 y(W)-8 b(e)26 b(remark)f(that)g(condition)e(\(2.5\))i (is)g(quite)g(natural)f(for)h(the)g(F)-8 b(okk)m(er-Planc)m(k)26 b(equation,)h(where)75 5107 y(the)34 b(usual)g(requiremen)m(ts)g(for)g (the)g(particle)e(densit)m(y)-8 b(,)35 b(in)e(addition)f(to)i(the)g(p)s (ositivit)m(y)-8 b(,)33 b(are)h(exactly)1926 5356 y(4)p eop %%Page: 5 5 5 4 bop 75 107 a Fl(the)34 b(initial)29 b(b)s(ounds)34 b(on)f(mass,)g(energy)h(and)f(en)m(trop)m(y)-8 b(,)35 b(while)d(the)i(same)f(condition)e(is)i(in)f(principle)75 227 y(not)g(so)h(natural)f(for)g(the)h(heat)f(equation.)221 348 y(The)38 b(previous)f(example)g(sho)m(ws)h(that)e(the)i(\\kinetic") d(equation)i(\(1.3\))f(is)g(the)i(natural)d(one)i(to)75 468 y(lo)s(ok)27 b(for)g(the)i(asymptotic)e(deca)m(y)-8 b(.)43 b(This)28 b(is)g(the)g(p)s(oin)m(t)f(of)g(view)i(w)m(e)g(will)c (adopt)j(to)g(treat)g(the)g(p)s(orous)75 588 y(medium)j(equation)h (\(1.1\).)221 709 y(It)44 b(is)f(w)m(ell-kno)m(wn)g(that)g(equation)g (\(1.1\))g(admits)f(a)h(family)e(of)i(self-similar)c(solutions)k(\(in)f (a)75 829 y(w)m(eak)34 b(sense\))g(called)e(Baren)m(blatt-P)m(attle)f (solutions)g(giv)m(en)i(b)m(y)984 1144 y Fk(V)22 b Fl(\()p Fj(j)p Fk(x)p Fj(j)p Fk(;)17 b(t)p Fl(\))27 b(=)h Fk(t)1495 1103 y Fn(\000)p Fp(k)r(N)1672 998 y Fo( )1738 1144 y Fk(C)1808 1159 y Fr(1)1869 1144 y Fj(\000)1979 1077 y Fl(\()p Fk(m)22 b Fj(\000)h Fl(1\))p Fk(k)p 1979 1121 386 4 v 2105 1213 a Fl(2)p Fk(m)2375 1144 y Fj(j)p Fk(x)p Fj(j)2486 1103 y Fr(2)2525 1144 y Fk(t)2560 1103 y Fn(\000)p Fr(2)p Fp(k)2693 998 y Fo(!)2820 994 y Fg(1)p 2769 1006 133 4 v 2769 1047 a Ff(m)p Fe(\000)p Fg(1)2759 1248 y Fr(+)3624 1144 y Fl(\(2.6\))75 1442 y(where)34 b Fk(k)d Fl(=)c(\()p Fk(N)10 b Fl(\()p Fk(m)23 b 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Fl(\()p Fk(u)p Fl(\))21 b(+)h Fk(\025)1978 4952 y Fo(Z)2023 5140 y Fg(I)-10 b(R)2080 5121 y Ff(N)2158 5069 y Fk(u)17 b(dx)1926 5356 y Fl(8)p eop %%Page: 9 9 9 8 bop 75 107 a Fl(has)33 b(v)-5 b(ariation)1137 130 y Fo(Z)1183 318 y Fg(I)-10 b(R)1240 299 y Ff(N)1318 125 y Fo(\024)1362 247 y Fj(j)p Fk(x)p Fj(j)1473 206 y Fr(2)1534 247 y Fl(+)22 b Fk(\025)g Fl(+)1880 179 y(2)p Fk(m)p 1819 223 256 4 v 1819 315 a(m)h Fj(\000)f Fl(1)2085 247 y Fk(u)2141 206 y Fp(m)p Fn(\000)p Fr(1)2297 125 y Fo(\025)2358 247 y Fk(\016)t(u)17 b(dx)27 b Fl(=)g(0)75 451 y(Th)m(us,)34 b(the)f(extremals)f(are)h(solutions)f(of)1138 703 y Fj(j)p Fk(x)p Fj(j)1249 661 y Fr(2)1310 703 y Fl(+)22 b Fk(\025)g Fl(+)1656 635 y(2)p Fk(m)p 1595 679 V 1595 771 a(m)h Fj(\000)g Fl(1)1861 703 y Fk(u)1917 661 y Fp(m)p Fn(\000)p Fr(1)2101 703 y Fl(=)28 b(0)k Fk(;)212 b(u)27 b Fj(\025)h Fl(0)75 954 y(and)e Fk(\025)g Fl(m)m(ust)f(b)s(e)h(suc)m(h)i(that)d (the)h(mass)g(of)f(the)i(extremal)e(is)g(equal)h(to)f(the)h(mass)g(of)f Fk(u)p Fl(\()p Fk(x)p Fl(\).)41 b(Hence,)29 b Fk(u)3751 969 y Fn(1)75 1074 y Fl(is)h(an)g(extremal)g(of)f Fk(H)8 b Fl(\()p Fk(u)p Fl(\).)42 b(This)30 b(is)g(ob)m(viously)g(a)g(minim)m (um)d(since)k(one)g(can)f(construct)i Fk(u)3435 1089 y Fr(1)3474 1074 y Fl(\()p Fk(x)p Fl(\))e(with)75 1195 y(mass)36 b(equal)g(to)g(the)h(mass)f(of)f Fk(u)1287 1210 y Fn(1)1362 1195 y Fl(\()p Fk(x)p Fl(\))h(and)g(with)g(supp)s(ort) h(con)m(tained)f(in)f(a)h(ball)e(of)i(unit)g(measure)75 1315 y(lo)s(cated)k(v)m(ery)j(far)e(from)f(the)i(origin.)67 b(In)42 b(this)f(w)m(a)m(y)h(w)m(e)h(can)e(get)h Fk(H)8 b Fl(\()p Fk(u)2790 1330 y Fr(1)2828 1315 y Fl(\))41 b(larger)f(than)i(an)m(y)g(\014xed)75 1435 y(p)s(ositiv)m(e)32 b(constan)m(t.)221 1556 y(A)38 b(deep)s(er)g(study)h(of)d(the)i(prop)s (erties)f(of)g Fk(H)8 b Fl(\()p Fk(u)p Fj(j)p Fk(u)2056 1571 y Fn(1)2129 1556 y Fl(\))37 b(is)g(p)s(ostp)s(oned)h(to)f(the)h (next)g(section.)58 b(F)-8 b(or)75 1676 y(the)33 b(presen)m(t)h(purp)s (oses,)g(w)m(e)f(only)f(pro)m(v)m(e)i(that)e(the)h(con)m(v)m(ergence)i (to)d(zero)h(of)f(the)h(relativ)m(e)f(en)m(trop)m(y)75 1797 y(is)g(a)g(consequence)k(of)c(\(3.3\).)75 2016 y Fd(Theorem)37 b(3.1)49 b Fc(L)-5 b(et)35 b(the)g(initial)f(c)-5 b(ondition)34 b(for)g(the)h(F)-7 b(okker-Planck)32 b(e)-5 b(quation)34 b(\(1.3\))g(satisfy)h Fl(0)27 b Fk(<)75 2136 y(u)131 2151 y Fr(0)170 2136 y Fl(\()p Fk(x)p Fl(\))43 b Fj(2)g Fk(L)519 2100 y Fr(1)586 2136 y Fj(\\)29 b Fk(L)747 2100 y Fn(1)822 2136 y Fl(\(I)-20 b(R)947 2095 y Fp(N)1014 2136 y Fl(\))43 b Fc(with)g Fj(j)p Fk(x)p Fj(j)1426 2100 y Fr(2+)p Fp(\016)1553 2136 y Fk(u)1609 2151 y Fr(0)1648 2136 y Fl(\()p Fk(x)p Fl(\))g Fj(2)g Fk(L)1997 2100 y Fr(1)2037 2136 y Fl(\(I)-20 b(R)2162 2095 y Fp(N)2229 2136 y Fl(\))43 b Fc(for)f(some)g Fk(\016)47 b(>)42 b Fl(0)p Fc(.)69 b(Then)42 b(the)g(r)-5 b(elative)75 2257 y(entr)g(opy)35 b Fk(H)8 b Fl(\()p Fk(u)p Fl(\()p Fk(t)p Fl(\))p Fj(j)p Fk(u)803 2272 y Fn(1)876 2257 y Fl(\))35 b Fc(is)f(monotonic)-5 b(al)5 b(ly)34 b(de)-5 b(cr)g(e)g(asing,)33 b(and)i(c)-5 b(onver)g(ges)33 b(to)i(zer)-5 b(o)35 b(as)f Fk(t)28 b Fj(!)g(1)p Fc(.)221 2476 y(Pr)-5 b(o)g(of)p Fl(.)40 b(Using)21 b(\(3.3\),)j Fk(H)8 b Fl(\()p Fk(u)p Fl(\()p Fk(t)p Fl(\))p Fj(j)p Fk(u)1408 2491 y Fn(1)1481 2476 y Fl(\))21 b(is)h(decreasing)g(and)g(b)s(ounded)g(from)f(b)s(elo)m (w)g(so)h Fk(H)8 b Fl(\()p Fk(u)p Fl(\()p Fk(t)p Fl(\))p Fj(j)p Fk(u)3627 2491 y Fn(1)3700 2476 y Fl(\))28 b Fj(!)75 2596 y Fk(H)164 2560 y Fn(\003)233 2596 y Fl(as)i Fk(t)d Fj(!)h(1)p Fl(.)42 b(W)-8 b(e)30 b(ha)m(v)m(e)h(to)e(pro)m(v)m(e)i (that)f Fk(H)1770 2560 y Fn(\003)1837 2596 y Fl(=)d(0.)42 b(No)m(w,)31 b(remark)f(that)f(the)i(en)m(trop)m(y)g(pro)s(duction)75 2717 y(for)h(\(1.3\))g(giv)m(en)h(in)f(\(3.3\))1163 2959 y Fk(I)8 b Fl(\()p Fk(u)p Fl(\))27 b(=)1477 2842 y Fo(Z)1523 3030 y Fg(I)-10 b(R)1579 3012 y Ff(N)1658 2959 y Fk(u)1714 2859 y Fo(\014)1714 2909 y(\014)1714 2959 y(\014)o Fk(x)22 b Fl(+)2012 2892 y Fk(m)p 1926 2936 V 1926 3027 a(m)h Fj(\000)f Fl(1)2192 2959 y Fj(r)p Fk(u)2331 2918 y Fp(m)p Fn(\000)p Fr(1)2487 2859 y Fo(\014)2487 2909 y(\014)2487 2959 y(\014)2515 2886 y Fr(2)2571 2959 y Fk(dx)33 b(;)75 3210 y Fl(is)f(summable)f(o)m(v)m(er)j(the)f(time)e(in)m(terv)-5 b(al)31 b([0)p Fk(;)17 b Fl(+)p Fj(1)p Fl(\).)43 b(In)33 b(fact,)f(\(3.3\))g(can)h(b)s(e)g(written)f(as)1405 3405 y Fk(d)p 1387 3449 86 4 v 1387 3540 a(dt)1483 3472 y(H)8 b Fl(\()p Fk(u)p Fl(\()p Fk(t)p Fl(\))p Fj(j)p Fk(u)1861 3487 y Fn(1)1934 3472 y Fl(\))28 b(=)f Fj(\000)p Fl(2)p Fk(I)8 b Fl(\()p Fk(u)p Fl(\()p Fk(t)p Fl(\)\))1101 b(\(3.4\))75 3710 y(whic)m(h)33 b(implies)1317 3752 y Fo(Z)1400 3778 y Fr(+)p Fn(1)1363 3940 y Fr(0)1547 3869 y Fk(I)8 b Fl(\()p Fk(u)p Fl(\()p Fk(s)p Fl(\)\))17 b Fk(ds)26 b Fj(\024)2107 3802 y Fl(1)p 2107 3846 49 4 v 2107 3937 a(2)2165 3869 y Fk(H)8 b Fl(\()p Fk(u)2348 3884 y Fr(0)2387 3869 y Fj(j)p Fk(u)2471 3884 y Fn(1)2545 3869 y Fl(\))75 4074 y(Hence,)51 b(there)46 b(exists)h(a)e(sequence)k(of)c(times)g Fj(f)p Fk(t)1949 4089 y Fp(k)1991 4074 y Fj(g)h Fl(with)f Fk(t)2357 4089 y Fp(k)2450 4074 y Fj(!)50 b(1)c Fl(when)h Fk(k)53 b Fj(!)d(1)45 b Fl(suc)m(h)i(that)75 4194 y Fk(I)8 b Fl(\()p Fk(u)p Fl(\()p Fk(t)293 4209 y Fp(k)335 4194 y Fl(\)\))28 b Fj(!)f Fl(0)32 b(as)g Fk(k)f Fj(!)c(1)p Fl(.)43 b(Let)32 b(us)h(denote)g Fk(u)p Fl(\()p Fk(t)1887 4209 y Fp(k)1929 4194 y Fl(\))f(b)m(y)h Fk(u)2190 4209 y Fp(k)2232 4194 y Fl(.)43 b(Since)33 b Fk(H)8 b Fl(\()p Fk(u)p Fl(\()p Fk(t)2813 4209 y Fp(k)2855 4194 y Fl(\))p Fj(j)p Fk(u)2977 4209 y Fn(1)3050 4194 y Fl(\))32 b(is)g(b)s(ounded,)h (th)m(us)1309 4319 y Fo(Z)1355 4508 y Fg(I)-10 b(R)1412 4489 y Ff(N)1490 4437 y Fj(j)p Fk(x)p Fj(j)1601 4395 y Fr(2)1640 4437 y Fk(u)1696 4452 y Fp(k)1755 4437 y Fk(dx)65 b Fl(and)2165 4319 y Fo(Z)2211 4508 y Fg(I)-10 b(R)2267 4489 y Ff(N)2346 4437 y Fk(u)2402 4395 y Fp(m)2402 4461 y(k)2485 4437 y Fk(dx)75 4677 y Fl(are)33 b(b)s(ounded.)221 4798 y(No)m(w,)h(dev)m(eloping)e(the)h(v)-5 b(alue)32 b(of)g Fk(I)8 b Fl(\()p Fk(u)p Fl(\))32 b(and)g(applying)g(div)m (ergence)i(theorem,)e(w)m(e)i(deduce)513 5064 y Fk(I)8 b Fl(\()p Fk(u)p Fl(\))27 b(=)826 4947 y Fo(Z)872 5135 y Fg(I)-10 b(R)929 5116 y Ff(N)1007 5064 y Fj(j)p Fk(x)p Fj(j)1118 5023 y Fr(2)1157 5064 y Fk(u)17 b(dx)22 b Fj(\000)g Fl(2)p Fk(N)1611 4947 y Fo(Z)1657 5135 y Fg(I)-10 b(R)1714 5116 y Ff(N)1792 5064 y Fk(u)1848 5023 y Fp(m)1931 5064 y Fk(dx)22 b Fl(+)2157 4943 y Fo(\022)2313 4996 y Fk(m)p 2228 5041 256 4 v 2228 5132 a(m)h Fj(\000)f Fl(1)2494 4943 y Fo(\023)2555 4966 y Fr(2)2611 4947 y Fo(Z)2657 5135 y Fg(I)-10 b(R)2713 5116 y Ff(N)2792 5064 y Fk(u)p Fj(jr)p Fk(u)3015 5023 y Fp(m)p Fn(\000)p Fr(1)3170 5064 y Fj(j)3198 5023 y Fr(2)3254 5064 y Fk(dx:)1926 5356 y Fl(9)p eop %%Page: 10 10 10 9 bop 75 107 a Fl(Hence)83 350 y Fk(I)8 b Fl(\()p Fk(u)228 365 y Fp(k)270 350 y Fl(\))22 b(+)g Fk(N)10 b Fl(\()p Fk(m)23 b Fj(\000)f Fl(1\))p Fk(H)8 b Fl(\()p Fk(u)1031 365 y Fp(k)1073 350 y Fl(\))27 b(=)h([)p Fk(N)10 b(m)23 b Fj(\000)g Fk(N)32 b Fl(+)22 b(1])1866 233 y Fo(Z)1912 421 y Fg(I)-10 b(R)1968 402 y Ff(N)2047 350 y Fj(j)p Fk(x)p Fj(j)2158 309 y Fr(2)2197 350 y Fk(u)17 b(dx)k Fl(+)2495 229 y Fo(\022)2652 282 y Fl(2)p Fk(m)p 2566 327 305 4 v 2566 418 a Fl(2)p Fk(m)i Fj(\000)f Fl(1)2881 229 y Fo(\023)2942 245 y Fr(2)2998 233 y Fo(Z)3044 421 y Fg(I)-10 b(R)3101 402 y Ff(N)3179 350 y Fj(jr)p Fk(u)3346 299 y Fp(m)p Fn(\000)p Fr(\(1)p Fp(=)p Fr(2\))3346 375 y Fp(k)3627 350 y Fj(j)3655 309 y Fr(2)3711 350 y Fk(dx)75 596 y Fl(Since)33 b Fk(I)8 b Fl(\()p Fk(u)475 611 y Fp(k)517 596 y Fl(\))27 b Fj(!)h Fl(0)k(and)h Fk(H)8 b Fl(\()p Fk(u)1164 611 y Fp(k)1205 596 y Fl(\))33 b(is)f(monotone)g (nonincreasing,)g(the)h(sequence)1540 716 y Fo(Z)1587 905 y Fg(I)-10 b(R)1643 886 y Ff(N)1721 833 y Fj(jr)p Fk(u)1888 782 y Fp(m)p Fn(\000)p Fr(\(1)p Fp(=)p Fr(2\))1888 858 y Fp(k)2170 833 y Fj(j)2198 792 y Fr(2)2253 833 y Fk(dx)75 1069 y Fl(is)32 b(b)s(ounded.)44 b(On)33 b(the)g(other)g (hand,)g(w)m(e)g(ha)m(v)m(e)h(that)1264 1293 y Fj(r)p Fk(u)1403 1252 y Fp(m)1497 1293 y Fl(=)1782 1226 y Fk(m)p 1610 1270 430 4 v 1610 1361 a(m)23 b Fj(\000)f Fl(\(1)p Fk(=)p Fl(2\))2050 1293 y Fk(u)2106 1252 y Fr(1)p Fp(=)p Fr(2)2215 1293 y Fj(r)p Fk(u)2354 1252 y Fp(m)p Fn(\000)p Fr(\(1)p Fp(=)p Fr(2\))75 1550 y Fl(and)33 b(th)m(us,)g(w)m(e)h(can)f (apply)f(H\177)-49 b(older)32 b(inequalit)m(y)g(to)g(deduce)i(that)520 1696 y Fo(Z)566 1884 y Fg(I)-10 b(R)622 1865 y Ff(N)701 1813 y Fj(jr)p Fk(u)868 1772 y Fp(m)868 1837 y(k)933 1813 y Fj(j)17 b Fk(dx)60 b Fj(\024)1464 1745 y Fk(m)p 1292 1790 V 1292 1881 a(m)22 b Fj(\000)h Fl(\(1)p Fk(=)p Fl(2\))1747 1692 y Fo(\024)1791 1696 y(Z)1837 1884 y Fg(I)-10 b(R)1894 1865 y Ff(N)1972 1813 y Fk(u)2028 1828 y Fp(k)2087 1813 y Fk(dx)2193 1692 y Fo(\025)2237 1715 y Fr(1)p Fp(=)p Fr(2)2364 1692 y Fo(\024)2407 1696 y(Z)2453 1884 y Fg(I)g(R)2510 1865 y Ff(N)2588 1813 y Fj(jr)p Fk(u)2755 1762 y Fp(m)p Fn(\000)p Fr(\(1)p Fp(=)p Fr(2\))2755 1838 y Fp(k)3037 1813 y Fj(j)3065 1772 y Fr(2)3120 1813 y Fk(dx)3226 1692 y Fo(\025)3270 1715 y Fr(1)p Fp(=)p Fr(2)75 2088 y Fl(and)33 b(hence,)h(the)f(sequence)i Fk(u)1191 2052 y Fp(m)1191 2113 y(k)1290 2088 y Fl(is)d(b)s(ounded)h (in)f Fk(W)2006 2052 y Fr(1)p Fp(;)p Fr(1)2100 2088 y Fl(\(I)-20 b(R)2225 2046 y Fp(N)2292 2088 y Fl(\).)221 2208 y(T)-8 b(aking)37 b(in)m(to)f(accoun)m(t)i(that)f(the)g(initial)d (data)i(is)h(uniformly)e(b)s(ounded)i(w)m(e)h(deduce)h(that)e Fk(u)f Fl(is)75 2329 y(uniformly)30 b(b)s(ounded)k(in)e(\(0)p Fk(;)17 b Fj(1)p Fl(\))k Fj(\002)h Fl(I)-20 b(R)1509 2287 y Fp(N)1608 2329 y Fl(and)33 b(th)m(us,)1263 2449 y Fo(Z)1309 2637 y Fg(I)-10 b(R)1366 2618 y Ff(N)1444 2566 y Fj(j)p Fk(x)p Fj(j)1555 2525 y Fp(\016)1593 2566 y Fk(u)1649 2525 y Fp(m)1731 2566 y Fk(dx)61 b Fj(\024)f Fk(C)2129 2449 y Fo(Z)2175 2637 y Fg(I)-10 b(R)2231 2618 y Ff(N)2310 2566 y Fj(j)p Fk(x)p Fj(j)2421 2525 y Fp(\016)2459 2566 y Fk(u)17 b(dx)986 b Fl(\(3.5\))75 2819 y(for)32 b(an)m(y)h Fk(t)28 b Fj(\025)g Fl(0)33 b(and)f(an)m(y)i Fk(\016)d Fj(\024)d Fl(2,)33 b(and)f(hence,)j(the)e(sequence)i Fj(j)p Fk(x)p Fj(j)p Fk(u)2547 2783 y Fp(m)2547 2844 y(k)2645 2819 y Fl(is)d(b)s(ounded)i(in)d Fk(L)3321 2783 y Fr(1)3361 2819 y Fl(\(I)-20 b(R)3487 2778 y Fp(N)3554 2819 y Fl(\).)221 2940 y(Using)43 b(the)h(previous)f(facts)h(w)m(e)g (deduce)h(that)e Fk(u)2093 2904 y Fp(m)2093 2964 y(k)2205 2940 y Fj(!)i Fk(g)2401 2904 y Fp(m)2510 2940 y Fl(in)e Fk(L)2701 2904 y Fr(1)2741 2940 y Fl(\(I)-20 b(R)2866 2898 y Fp(N)2933 2940 y Fl(\))43 b(\(after)g(passing)g(to)g(a)75 3060 y(subsequence\).)k(Since)33 b(0)27 b Fj(\024)h Fk(I)8 b Fl(\()p Fk(u)1283 3075 y Fp(k)1325 3060 y Fl(\))28 b Fj(!)f Fl(0,)33 b(w)m(e)g(ha)m(v)m(e)h Fk(I)8 b Fl(\()p Fk(g)t Fl(\))27 b(=)g(0)33 b(b)m(y)g(F)-8 b(atou)32 b(lemma.)41 b(Let)1338 3280 y Fk(B)5 b Fl(\()p Fk(g)t Fl(\))27 b(=)1675 3183 y Fo(n)1730 3280 y Fk(x)h Fj(2)g Fl(I)-20 b(R)1995 3238 y Fp(N)2090 3280 y Fl(:)28 b Fk(g)t Fl(\()p Fk(x)p Fl(\))f Fk(>)g Fl(0)2506 3183 y Fo(o)75 3506 y Fl(Then,)j Fk(I)8 b Fl(\()p Fk(g)t Fl(\))27 b(=)g(0)h(implies)e Fk(g)t Fl(\()p Fk(x)p Fl(\))h(=)h Fk(c)13 b Fj(\000)1533 3467 y Fp(m)p Fn(\000)p Fr(1)p 1533 3483 153 4 v 1561 3541 a(2)p Fp(m)1696 3506 y Fj(j)p Fk(x)p Fj(j)1807 3470 y Fr(2)1874 3506 y Fl(for)28 b(all)e Fk(x)i Fj(2)g Fk(B)5 b Fl(\()p Fk(g)t Fl(\),)28 b(where)i Fk(c)e Fl(is)g(constan)m(t.)42 b(It)29 b(is)e(no)m(w)75 3627 y(immediate)j(to)i(conclude,)h(b)m(y)h(p) s(ositivit)m(y)d(and)i(mass)f(conserv)-5 b(ation,)33 b(that)f Fk(g)t Fl(\()p Fk(x)p Fl(\))27 b(=)h Fk(u)3267 3642 y Fn(1)3341 3627 y Fl(\()p Fk(x)p Fl(\))65 b Fk(a:e:)221 3747 y Fl(It)33 b(remains)f(to)g(sho)m(w)i(that)e Fk(H)8 b Fl(\()p Fk(u)1446 3762 y Fp(k)1488 3747 y Fj(j)p Fk(u)1572 3762 y Fn(1)1645 3747 y Fl(\))28 b Fj(!)f Fl(0.)44 b(It)32 b(su\016ces)j(to)d(pro)m(v)m(e)i(that)1028 3867 y Fo(Z)1074 4056 y Fg(I)-10 b(R)1130 4037 y Ff(N)1209 3984 y Fj(j)p Fk(x)p Fj(j)1320 3943 y Fr(2)1359 3984 y Fk(u)1415 3999 y Fp(k)1474 3984 y Fk(dx)60 b Fj(!)1800 3867 y Fo(Z)1846 4056 y Fg(I)-10 b(R)1902 4037 y Ff(N)1981 3984 y Fj(j)p Fk(x)p Fj(j)2092 3943 y Fr(2)2131 3984 y Fk(u)2187 3999 y Fn(1)2278 3984 y Fk(dx)33 b Fl(as)f Fk(k)f Fj(!)c(1)p Fk(:)75 4220 y Fl(It)35 b(is)e(clear)h(that)h(w)m(e)g(cannot)g(pro)m(v) m(e)g(this,)g(unless)g(w)m(e)h(ha)m(v)m(e)g(some)e(extra)h(momen)m(t)e (b)s(ounded.)50 b(Ap-)75 4340 y(plying)31 b(div)m(ergence)j(theorem)f (it)e(is)h(easy)i(to)e(see)i(that)385 4523 y Fk(d)p 367 4568 86 4 v 367 4659 a(dt)479 4474 y Fo(Z)526 4662 y Fg(I)-10 b(R)582 4643 y Ff(N)660 4591 y Fj(j)p Fk(x)p Fj(j)771 4550 y Fr(2+)p Fp(\016)899 4591 y Fk(u)17 b(dx)60 b Fl(=)g(\(2)22 b(+)g Fk(\016)t Fl(\)\()p Fk(N)32 b Fl(+)22 b Fk(\016)t Fl(\))1914 4474 y Fo(Z)1960 4662 y Fg(I)-10 b(R)2016 4643 y Ff(N)2094 4591 y Fj(j)p Fk(x)p Fj(j)2205 4550 y Fp(\016)2243 4591 y Fk(u)2299 4550 y Fp(m)2382 4591 y Fk(dx)22 b Fj(\000)h Fl(\(2)e(+)h Fk(\016)t Fl(\))2918 4474 y Fo(Z)2964 4662 y Fg(I)-10 b(R)3020 4643 y Ff(N)3099 4591 y Fj(j)p Fk(x)p Fj(j)3210 4550 y Fr(2+)p Fp(\016)3338 4591 y Fk(u)17 b(dx:)75 4832 y Fl(Using)32 b(\(3.5\))g(and)h(the)g(v)-5 b(ariation)30 b(of)i(constan)m(ts)i(form)m(ula,)d(w)m(e)i(deduce)i (that)1651 4952 y Fo(Z)1697 5140 y Fg(I)-10 b(R)1753 5121 y Ff(N)1832 5069 y Fj(j)p Fk(x)p Fj(j)1943 5028 y Fr(2+)p Fp(\016)2071 5069 y Fk(u)17 b(dx)1901 5356 y Fl(10)p eop %%Page: 11 11 11 10 bop 75 107 a Fl(is)32 b(b)s(ounded)h(uniformly)e(on)h Fk(t)p Fl(.)44 b(As)33 b(a)f(consequence,)1028 234 y Fo(Z)1074 423 y Fg(I)-10 b(R)1130 404 y Ff(N)1209 351 y Fj(j)p Fk(x)p Fj(j)1320 310 y Fr(2)1359 351 y Fk(u)1415 366 y Fp(k)1474 351 y Fk(dx)60 b Fj(!)1800 234 y Fo(Z)1846 423 y Fg(I)-10 b(R)1902 404 y Ff(N)1981 351 y Fj(j)p Fk(x)p Fj(j)2092 310 y Fr(2)2131 351 y Fk(u)2187 366 y Fn(1)2278 351 y Fk(dx)33 b Fl(as)f Fk(k)f Fj(!)c(1)p Fk(:)75 600 y Fl(and)33 b Fk(H)354 564 y Fn(\003)420 600 y Fl(=)28 b(0.)p 676 541 41 4 v 676 597 4 56 v 713 597 V 676 600 41 4 v 221 841 a(Eq.)44 b(3.3)31 b(relates)h(the)g (relativ)m(e)f(en)m(trop)m(y)i(to)e(the)h(en)m(trop)m(y)h(pro)s (duction.)42 b(F)-8 b(rom)30 b(this)h(relation)f(w)m(e)75 961 y(just)35 b(concluded)h(with)f(the)h(con)m(v)m(ergence)h(to)e(zero) h(of)e(the)i(solution)e(in)g(relativ)m(e)g(en)m(trop)m(y)-8 b(,)37 b(without)75 1081 y(an)m(y)42 b(rate.)70 b(As)42 b(discussed)h(in)e(the)h(in)m(tro)s(duction,)g(to)f(\014nd)h(the)g (rate)f(of)g(con)m(v)m(ergence)j(requires)e(a)75 1202 y(further)33 b(step.)44 b(Ha)m(ving)33 b(this)f(in)g(mind,)f(let)h(us)h (compute)g(the)g(time)e(ev)m(olutions)h(of)g Fk(I)8 b Fl(\()p Fk(u)p Fl(\()p Fk(t)p Fl(\)\),)415 1404 y Fk(d)p 397 1448 86 4 v 397 1540 a(dt)493 1471 y(I)g Fl(\()p Fk(u)p Fl(\))82 b(=)917 1354 y Fo(Z)963 1543 y Fg(I)-10 b(R)1019 1524 y Ff(N)1108 1404 y Fk(@)5 b(u)p 1108 1448 113 4 v 1118 1540 a(@)g(t)1230 1372 y Fo(\014)1230 1421 y(\014)1230 1471 y(\014)p Fk(x)22 b Fl(+)1529 1404 y Fk(m)p 1443 1448 256 4 v 1443 1540 a(m)h Fj(\000)g Fl(1)1709 1471 y Fj(r)p Fk(u)1848 1430 y Fp(m)p Fn(\000)p Fr(1)2005 1372 y Fo(\014)2005 1421 y(\014)2005 1471 y(\014)2032 1398 y Fr(2)2088 1471 y Fk(dx)917 1718 y Fl(+2)1059 1600 y Fo(Z)1104 1789 y Fg(I)-10 b(R)1161 1770 y Ff(N)1239 1718 y Fk(u)1295 1621 y Fo(\020)1344 1718 y Fk(x)23 b Fl(+)1615 1650 y Fk(m)p 1530 1694 V 1530 1786 a(m)f Fj(\000)h Fl(1)1796 1718 y Fj(r)p Fk(u)1935 1676 y Fp(m)p Fn(\000)p Fr(1)2091 1621 y Fo(\021)2163 1718 y Fj(\001)2240 1650 y Fk(@)p 2222 1694 93 4 v 2222 1786 a(@)5 b(t)2324 1621 y Fo(h)2459 1650 y Fk(m)p 2374 1694 256 4 v 2374 1786 a(m)22 b Fj(\000)h Fl(1)2639 1718 y Fj(r)p Fk(u)2778 1676 y Fp(m)p Fn(\000)p Fr(1)2935 1621 y Fo(i)2991 1718 y Fk(dx)k Fl(=)h Fk(I)3271 1733 y Fr(1)3333 1718 y Fl(+)22 b Fk(I)3474 1733 y Fr(2)221 1981 y Fl(Let)33 b(us)g(set)h Fk(y)c Fl(=)e Fk(x)22 b Fl(+)1087 1942 y Fp(m)p 1041 1958 153 4 v 1041 2016 a(m)p Fn(\000)p Fr(1)1204 1981 y Fj(r)p Fk(u)1343 1945 y Fp(m)p Fn(\000)p Fr(1)1499 1981 y Fl(.)44 b(The)33 b(second)h(term)e(can)h(b)s(e)g(written)f(as) 1041 2267 y Fk(I)1084 2282 y Fr(2)1124 2267 y Fl(\()p Fk(u)p Fl(\))82 b(=)h Fj(\000)p Fl(2)1718 2200 y Fk(m)p 1633 2244 256 4 v 1633 2335 a(m)23 b Fj(\000)f Fl(1)1916 2150 y Fo(Z)1962 2338 y Fg(I)-10 b(R)2018 2320 y Ff(N)2096 2267 y Fl(div\()p Fk(uy)t Fl(\))2439 2200 y Fk(@)p 2423 2244 93 4 v 2423 2335 a(@)5 b(t)2524 2267 y(u)2580 2226 y Fp(m)p Fn(\000)p Fr(1)2753 2267 y Fk(dx)1338 2513 y Fl(=)83 b Fj(\000)p Fl(2)p Fk(m)1725 2396 y Fo(Z)1771 2585 y Fg(I)-10 b(R)1828 2566 y Ff(N)1906 2513 y Fk(u)1962 2472 y Fp(m)p Fn(\000)p Fr(2)2118 2513 y Fl(div\()p Fk(uy)t Fl(\))2445 2446 y Fk(@)5 b(u)p 2445 2490 113 4 v 2455 2582 a(@)g(t)2583 2513 y(dx)1338 2735 y Fl(=)83 b Fj(\000)p Fl(2)p Fk(m)1725 2618 y Fo(Z)1771 2807 y Fg(I)-10 b(R)1828 2788 y Ff(N)1906 2735 y Fk(u)1962 2694 y Fp(m)p Fn(\000)p Fr(2)2118 2735 y Fl(div\()p Fk(uy)t Fl(\))2435 2694 y Fr(2)2490 2735 y Fk(dx)75 2984 y Fl(The)34 b(\014rst)f(term)f(can)g(b)s (e)h(written)g(as)1106 3204 y Fk(I)1149 3219 y Fr(1)1188 3204 y Fl(\()p Fk(u)p Fl(\))27 b(=)h Fj(\000)p Fl(2)1594 3087 y Fo(Z)1640 3275 y Fg(I)-10 b(R)1696 3256 y Ff(N)1775 3204 y Fk(u)p Fl(\()p Fk(y)24 b Fj(\001)55 b Fl(Jacob\()p Fk(y)t Fl(\))21 b Fj(\001)h Fk(y)2520 3163 y Fp(T)2574 3204 y Fl(\))17 b Fk(dx)32 b(:)75 3462 y Fl(Since)h(Jacob\()p Fk(y)t Fl(\))27 b(=)g Fk(I)876 3477 y Fp(N)966 3462 y Fl(+)1119 3423 y Fp(m)p 1074 3439 153 4 v 1074 3497 a(m)p Fn(\000)p Fr(1)1269 3462 y Fl(Hess)q(\()p Fk(u)1556 3426 y Fp(m)p Fn(\000)p Fr(1)1712 3462 y Fl(\))33 b(w)m(e)g(ha)m(v)m(e)888 3729 y Fk(I)931 3744 y Fr(1)970 3729 y Fl(\()p Fk(u)p Fl(\))83 b(=)f Fj(\000)p Fl(2)1486 3611 y Fo(Z)1532 3800 y Fg(I)-10 b(R)1589 3781 y Ff(N)1667 3729 y Fk(u)p Fj(j)p Fk(y)t Fj(j)1831 3687 y Fr(2)1885 3729 y Fk(dx)1343 3950 y Fj(\000)p Fl(2)1565 3883 y Fk(m)p 1479 3927 256 4 v 1479 4019 a(m)23 b Fj(\000)g Fl(1)1762 3833 y Fo(Z)1808 4022 y Fg(I)-10 b(R)1864 4003 y Ff(N)1943 3950 y Fk(u)p Fl(\()p Fk(y)25 b Fj(\001)54 b Fl(Hess)q(\()p Fk(u)2479 3909 y Fp(m)p Fn(\000)p Fr(1)2636 3950 y Fl(\))22 b Fj(\001)g Fk(y)2798 3909 y Fp(T)2852 3950 y Fl(\))17 b Fk(dx)75 4204 y Fl(No)m(w,)33 b(consider)g(that)g(the)g(last)e(in)m(tegral)g (can)i(b)s(e)g(written)f(in)g(the)h(follo)m(wing)d(w)m(a)m(y)843 4376 y Fo(Z)890 4565 y Fg(I)-10 b(R)946 4546 y Ff(N)1024 4493 y Fk(u)p Fl(\()p Fk(y)25 b Fj(\001)d Fl(Hess)q(\()p Fk(u)1528 4452 y Fp(m)p Fn(\000)p Fr(1)1684 4493 y Fl(\))h Fj(\001)e Fk(y)1846 4452 y Fp(T)1901 4493 y Fl(\))c Fk(dx)27 b Fl(=)2244 4385 y Fp(N)2216 4410 y Fo(X)2192 4592 y Fp(i;j)t Fr(=1)2375 4376 y Fo(Z)2421 4565 y Fg(I)-10 b(R)2478 4546 y Ff(N)2556 4493 y Fk(uy)2660 4508 y Fp(i)2687 4493 y Fk(y)2735 4508 y Fp(j)2781 4426 y Fk(@)2837 4390 y Fr(2)2878 4426 y Fk(u)2934 4390 y Fp(m)p Fn(\000)p Fr(1)p 2781 4470 309 4 v 2791 4561 a Fk(@)5 b(x)2902 4576 y Fp(i)2931 4561 y Fk(@)g(x)3042 4576 y Fp(j)3117 4493 y Fk(dx)1038 4808 y Fl(=)1194 4700 y Fp(N)1165 4725 y Fo(X)1142 4908 y Fp(i;j)t Fr(=1)1325 4691 y Fo(Z)1371 4880 y Fg(I)-10 b(R)1427 4861 y Ff(N)1519 4741 y Fl(1)p 1516 4785 56 4 v 1516 4877 a Fk(u)1581 4808 y Fl(\()p Fk(uy)1723 4823 y Fp(i)1750 4808 y Fl(\)\()p Fk(uy)1930 4823 y Fp(j)1966 4808 y Fl(\))2014 4741 y Fk(@)2070 4705 y Fr(2)2110 4741 y Fk(u)2166 4705 y Fp(m)p Fn(\000)p Fr(1)p 2014 4785 309 4 v 2024 4877 a Fk(@)5 b(x)2135 4892 y Fp(i)2164 4877 y Fk(@)g(x)2275 4892 y Fp(j)2349 4808 y Fk(dx)1901 5356 y Fl(11)p eop %%Page: 12 12 12 11 bop 75 107 a Fl(Using)32 b(div)m(ergence)i(theorem,)f(it)e(is)h (straigh)m(tforw)m(ard)g(to)h(c)m(hec)m(k)h(that)368 263 y Fp(N)340 288 y Fo(X)317 470 y Fp(i;j)t Fr(=1)500 254 y Fo(Z)546 443 y Fg(I)-10 b(R)602 424 y Ff(N)694 304 y Fl(1)p 691 348 56 4 v 691 439 a Fk(u)756 371 y Fl(\()p Fk(uy)898 386 y Fp(i)925 371 y Fl(\)\()p Fk(uy)1105 386 y Fp(j)1141 371 y Fl(\))1189 304 y Fk(@)1245 268 y Fr(2)1285 304 y Fk(u)1341 268 y Fp(m)p Fn(\000)p Fr(1)p 1189 348 309 4 v 1199 439 a Fk(@)5 b(x)1310 454 y Fp(i)1339 439 y Fk(@)g(x)1450 454 y Fp(j)1524 371 y Fk(dx)158 686 y Fl(=)83 b Fj(\000)463 578 y Fp(N)434 603 y Fo(X)411 786 y Fp(i;j)t Fr(=1)594 569 y Fo(Z)640 758 y Fg(I)-10 b(R)696 739 y Ff(N)785 619 y Fk(@)5 b(u)897 583 y Fp(m)p Fn(\000)p Fr(1)p 785 663 270 4 v 849 755 a Fk(@)g(x)960 770 y Fp(i)1080 540 y Fo(\()1147 686 y Fj(\000)1258 619 y Fl(1)p 1234 663 95 4 v 1234 755 a Fk(u)1290 726 y Fr(2)1340 686 y Fl(\()p Fk(uy)1482 701 y Fp(i)1509 686 y Fl(\)\()p Fk(uy)1689 701 y Fp(j)1724 686 y Fl(\))1790 619 y Fk(@)g(u)p 1772 663 149 4 v 1772 755 a(@)g(x)1883 770 y Fp(j)1953 686 y Fl(+)2064 619 y(1)p 2061 663 56 4 v 2061 755 a Fk(u)2137 619 y(@)g Fl([\()p Fk(uy)2362 634 y Fp(i)2390 619 y Fl(\)\()p Fk(uy)2570 634 y Fp(j)2605 619 y Fl(\)])p 2137 663 534 4 v 2329 755 a Fk(@)g(x)2440 770 y Fp(j)2680 540 y Fo(\))2780 686 y Fk(dx)158 1014 y Fl(=)83 b(\()p Fk(m)22 b Fj(\000)h Fl(1\))666 840 y Fo(8)665 915 y(<)665 1064 y(:)791 907 y Fp(N)762 931 y Fo(X)739 1114 y Fp(i;j)t Fr(=1)922 897 y Fo(Z)968 1086 y Fg(I)-10 b(R)1024 1067 y Ff(N)1103 1014 y Fk(u)1159 973 y Fp(m)p Fn(\000)p Fr(2)1315 1014 y Fk(y)1363 1029 y Fp(i)1391 1014 y Fk(y)1439 1029 y Fp(j)1499 947 y Fk(@)5 b(u)p 1485 991 141 4 v 1485 1083 a(@)g(x)1596 1098 y Fp(i)1696 947 y Fk(@)g(u)p 1678 991 149 4 v 1678 1083 a(@)g(x)1789 1098 y Fp(j)1853 1014 y Fk(dx)22 b Fj(\000)2133 907 y Fp(N)2104 931 y Fo(X)2081 1114 y Fp(i;j)t Fr(=1)2264 897 y Fo(Z)2310 1086 y Fg(I)-10 b(R)2366 1067 y Ff(N)2445 1014 y Fk(u)2501 973 y Fp(m)p Fn(\000)p Fr(2)2681 947 y Fk(@)5 b(u)p 2667 991 141 4 v 2667 1083 a(@)g(x)2778 1098 y Fp(i)2834 868 y Fo(")2882 1014 y Fk(y)2930 1029 y Fp(i)2968 947 y Fk(@)g Fl(\()p Fk(uy)3166 962 y Fp(j)3203 947 y Fl(\))p 2968 991 273 4 v 3030 1083 a Fk(@)g(x)3141 1098 y Fp(j)3273 1014 y Fl(+)22 b Fk(y)3419 1029 y Fp(j)3465 947 y Fk(@)5 b Fl(\()p Fk(uy)3663 962 y Fp(i)3691 947 y Fl(\))p 3465 991 265 4 v 3522 1083 a Fk(@)g(x)3633 1098 y Fp(j)3739 868 y Fo(#)3820 1014 y Fk(dx)3926 840 y Fo(9)3926 915 y(=)3926 1064 y(;)158 1347 y Fl(=)83 b(\()p Fk(m)22 b Fj(\000)h Fl(1\))666 1172 y Fo(8)665 1247 y(<)665 1396 y(:)791 1239 y Fp(N)762 1264 y Fo(X)739 1446 y 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Fg(I)-10 b(R)950 1718 y Ff(N)1029 1666 y Fk(u)1085 1625 y Fp(m)p Fn(\000)p Fr(2)1265 1599 y Fk(@)5 b(u)p 1251 1643 141 4 v 1251 1734 a(@)g(x)1362 1749 y Fp(i)1418 1520 y Fo(")1467 1666 y Fk(y)1515 1681 y Fp(i)1552 1599 y Fk(@)g Fl(\()p Fk(uy)1750 1614 y Fp(j)1787 1599 y Fl(\))p 1552 1643 273 4 v 1614 1734 a Fk(@)g(x)1725 1749 y Fp(j)1857 1666 y Fj(\000)22 b Fk(y)2004 1681 y Fp(j)2051 1599 y Fk(@)5 b Fl(\()p Fk(uy)2249 1614 y Fp(i)2277 1599 y Fl(\))p 2051 1643 265 4 v 2108 1734 a Fk(@)g(x)2219 1749 y Fp(j)2325 1520 y Fo(#)2406 1666 y Fk(dx)75 1966 y Fl(Owing)32 b(to)g(the)h(iden)m(tit)m(y)1047 1923 y Fp(@)t(y)1123 1933 y Ff(i)p 1041 1942 114 4 v 1041 2000 a Fp(@)t(x)1122 2010 y Ff(j)1192 1966 y Fl(=)1306 1916 y Fp(@)t(y)1382 1926 y Ff(j)p 1306 1942 109 4 v 1307 2000 a Fp(@)t(x)1388 2010 y Ff(i)1424 1966 y Fl(,)g(w)m(e)h(obtain)577 2157 y Fp(N)548 2182 y Fo(X)525 2364 y Fp(i;j)t Fr(=1)732 2197 y Fk(@)5 b(u)p 718 2241 141 4 v 718 2333 a(@)g(x)829 2348 y Fp(i)885 2119 y Fo(")933 2265 y 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Fj(\000)23 b Fl(\()p Fk(y)i Fj(\001)d(r)p Fl(\(div)p Fk(y)t Fl(\)\))59 b(=)2334 323 y Fp(N)2306 347 y Fo(X)2283 530 y Fp(i;j)t Fr(=1)2466 334 y Fo(\020)2533 363 y Fk(@)5 b(y)2637 378 y Fp(i)p 2525 407 149 4 v 2525 499 a Fk(@)g(x)2636 514 y Fp(j)2684 334 y Fo(\021)2733 357 y Fr(2)75 728 y Fl(and)33 b(as)f(a)h (consequence)935 925 y Fk(d)p 918 969 86 4 v 918 1060 a(dt)1013 992 y(I)8 b Fl(\()p Fk(u)p Fl(\))83 b(=)g Fj(\000)p Fl(2)p Fk(I)8 b Fl(\()p Fk(u)p Fl(\))21 b Fj(\000)i Fl(2\()p Fk(m)f Fj(\000)h Fl(1\))2266 875 y Fo(Z)2311 1063 y Fg(I)-10 b(R)2367 1045 y Ff(N)2446 992 y Fk(u)2502 951 y Fp(m)2568 896 y Fo(h)2607 992 y Fl(div)p Fk(y)2792 896 y Fo(i)2830 919 y Fr(2)2886 992 y Fk(dx)1438 1271 y Fj(\000)p Fl(2)1581 1154 y Fo(Z)1627 1343 y Fg(I)g(R)1683 1324 y Ff(N)1761 1271 y Fk(u)1817 1230 y Fp(m)1900 1100 y Fo(2)1900 1249 y(4)2007 1163 y Fp(N)1978 1188 y Fo(X)1955 1370 y Fp(i;j)t Fr(=1)2138 1175 y Fo(\020)2206 1204 y Fk(@)5 b(y)2310 1219 y Fp(i)p 2198 1248 149 4 v 2198 1339 a 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41 4 v 75 4303 a Fd(Remark)37 b(3.5)49 b Fl(The)34 b(classical)d(logarithmic)e(Sob)s(olev)k(inequalit)m(y)f(b)m (y)h(Gross)g([6])g(can)g(b)s(e)g(obtained)75 4424 y(b)m(y)k(\(3.11\))f (simply)f(subtracting)h(to)f(b)s(oth)i(sides)1998 4385 y Fr(1)p 1939 4401 153 4 v 1939 4458 a Fp(m)p Fn(\000)p Fr(1)2102 4424 y Fj(k)p Fk(f)11 b Fj(k)2261 4439 y Fr(1)2300 4424 y Fl(,)37 b(and)f(letting)f Fk(m)f Fj(!)g Fl(1.)54 b(This)36 b(giv)m(es)h(an)75 4544 y(alternativ)m(e)32 b(w)m(a)m(y)i(to)e(\014nd)h(the)g(remainder)f(\014rst)h(obtained)f(in)f ([15].)1901 5356 y(14)p eop %%Page: 15 15 15 14 bop 75 123 a Fm(4)161 b(Relativ)l(e)55 b(en)l(trop)l(y)c(and)j Fb(L)1950 71 y Fl(1)2003 123 y Fm(-estimates)75 342 y Fl(W)-8 b(e)42 b(shall)e(no)m(w)i(b)s(e)g(concerned)h(with)e(pro)m (ving)g(a)h(generalized)f(Csiszar-Kullbac)m(k)g(inequalit)m(y)f([4])75 463 y([10],)c(whic)m(h)f(is)g(an)g(estimate)g(for)f(the)i Fk(L)1592 427 y Fr(1)1632 463 y Fl(-distance)f(of)g(t)m(w)m(o)g 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Fl(\))28 b(=)f Fj(j)p Fk(x)p Fj(j)923 320 y Fr(2)963 361 y Fl(\()p Fk(t)22 b Fj(\000)g Fl(1\))g(+)1364 265 y Fo(\020)1424 294 y Fl(2)p Fk(C)1543 309 y Fr(2)1582 294 y Fl(\()p Fk(M)10 b Fl(\))p 1424 338 339 4 v 1465 429 a Fk(m)23 b Fj(\000)f Fl(1)1795 361 y Fj(\000)1904 294 y(j)p Fk(x)p Fj(j)2015 258 y Fr(2)p 1904 338 151 4 v 1937 429 a Fk(m)2064 265 y Fo(\021)2114 361 y Fl(\()p Fk(t)2187 320 y Fp(m)2276 361 y Fj(\000)h Fl(1\))32 b Fk(;)212 b(t)27 b Fj(\025)i Fl(0)j Fk(;)49 b(x)28 b Fj(2)g Fk(B)5 b Fl(\()p Fk(M)10 b Fl(\))75 620 y(Then,)34 b Fk( )t Fl(\()p Fk(x;)17 b Fl(1\))27 b(=)h(0.)43 b(Expanding)33 b Fk( )t Fl(\()p Fk(x;)17 b(t)p Fl(\))33 b(in)e(T)-8 b(a)m(ylor's)33 b(series)g(of)f Fk(t)h Fl(up)g(to)f(order)h(t)m(w)m(o)g (w)m(e)h(obtain)587 885 y Fk( )t Fl(\()p Fk(x;)17 b(t)p Fl(\))28 b(=)1066 817 y(2)p Fk(m)p 1005 862 256 4 v 1005 953 a(m)23 b Fj(\000)g Fl(1)1271 885 y Fk(C)1341 900 y Fr(2)1381 885 y Fl(\()p Fk(M)10 b Fl(\)\()p Fk(t)22 b Fj(\000)h Fl(1\))f(+)g Fk(mu)2104 844 y Fp(m)p Fn(\000)p Fr(1)2104 909 y Fn(1)2261 885 y Fl(\(1)f(+)h Fk(\022)s Fl(\()p Fk(t)h Fj(\000)f Fl(1\)\))2835 844 y Fp(m)p Fn(\000)p Fr(2)2992 885 y Fl(\()p Fk(t)g Fj(\000)h Fl(1\))3274 844 y Fr(2)75 1144 y Fl(with)32 b Fk(\022)f Fj(2)d Fl(\(0)p Fk(;)17 b Fl(1\).)43 b(Using)32 b(that)g Fk(u)g Fl(and)h Fk(u)1574 1159 y Fn(1)1681 1144 y Fl(ha)m(v)m(e)h(equal)e(mass)h(w)m(e) g(deduce)h(that)637 1394 y Fk(H)8 b Fl(\()p Fk(u)p Fj(j)p Fk(u)904 1409 y Fn(1)977 1394 y Fl(\))83 b(=)1257 1277 y Fo(Z)1303 1466 y Fp(B)s Fr(\()p Fp(M)7 b Fr(\))1510 1394 y Fk( )1577 1298 y Fo(\020)1627 1394 y Fk(x;)1773 1327 y(u)p 1736 1371 131 4 v 1736 1462 a(u)1792 1477 y Fn(1)1876 1298 y Fo(\021)1926 1394 y Fk(u)1982 1409 y Fn(1)2072 1394 y Fk(dx)1098 1633 y Fl(=)83 b Fk(m)1359 1515 y Fo(Z)1405 1704 y Fp(B)s Fr(\()p Fp(M)7 b Fr(\))1612 1633 y Fk(u)1668 1591 y Fp(m)p Fn(\000)p Fr(2)1668 1657 y Fn(1)1824 1536 y Fo(\020)1874 1633 y Fl(1)22 b(+)g Fk(\022)2091 1536 y Fo(\020)2188 1565 y Fk(u)p 2151 1609 V 2151 1701 a(u)2207 1716 y Fn(1)2313 1633 y Fj(\000)h Fl(1)2462 1536 y Fo(\021)o(\021)2560 1559 y Fp(m)p Fn(\000)p Fr(2)2717 1633 y Fl(\()p Fk(u)f Fj(\000)g Fk(u)2988 1648 y Fn(1)3063 1633 y Fl(\))3101 1591 y Fr(2)3157 1633 y Fk(dx)75 1898 y Fl(T)-8 b(aking)32 b Fk(m)c Fj(\025)g Fl(2)33 b(w)m(e)g(ha)m(v)m(e)h(that)1131 2142 y Fk(H)8 b Fl(\()p Fk(u)p Fj(j)p Fk(u)1398 2157 y Fn(1)1470 2142 y Fl(\))28 b Fj(\025)g Fk(m)1743 2025 y Fo(Z)1789 2214 y Fp(u>u)1926 2222 y Fe(1)2012 2142 y Fk(u)2068 2101 y Fp(m)p Fn(\000)p Fr(2)2068 2167 y Fn(1)2224 2142 y Fl(\()p Fk(u)22 b Fj(\000)g Fk(u)2495 2157 y Fn(1)2569 2142 y Fl(\))2607 2101 y Fr(2)2663 2142 y Fk(dx)75 2399 y Fl(If)33 b(1)27 b Fk(<)h(m)g(<)f Fl(2,)32 b(analogously)f(w)m(e)j(ha) m(v)m(e)1117 2644 y Fk(H)8 b Fl(\()p Fk(u)p Fj(j)p Fk(u)1384 2659 y Fn(1)1457 2644 y Fl(\))27 b Fj(\025)i Fk(m)1730 2526 y Fo(Z)1776 2715 y Fp(u)f Fl(2,)g(H\177)-49 b(older)43 b(inequalit)m(y)g(giv)m(es)i(a)e(b)s(ounded)i(righ)m(t-hand) e(side)i(only)e(if)75 3686 y Fk(u)131 3650 y Fr(\(2)p Fn(\000)p Fp(m)p Fr(\)\()p Fp(q)r Fn(\000)p Fr(1\))131 3711 y Fn(1)554 3686 y Fl(is)32 b(in)m(tegrable.)42 b(This)33 b(is)f(clearly)g(true)h(for)f(an)m(y)h Fk(p)27 b(>)2504 3647 y Fr(2)p Fp(m)p Fn(\000)p Fr(3)p 2504 3663 188 4 v 2522 3721 a Fp(m)p Fn(\000)p Fr(1)2702 3686 y Fl(.)p 2805 3627 41 4 v 2805 3683 4 56 v 2842 3683 V 2805 3686 41 4 v 221 3927 a(A)44 b(simple)e(consequence)47 b(of)c(the)h(previous) g(lemma)d(is)i(the)h(follo)m(wing)d(Csiszar-Kullbac)m(k)i(in-)75 4048 y(equalit)m(y)75 4276 y Fd(Lemma)37 b(4.2)49 b Fc(F)-7 b(or)39 b Fk(m)f(>)g Fl(1)p Fc(,)j(let)g Fk(u)1407 4291 y Fn(1)1521 4276 y Fc(b)-5 b(e)40 b(the)g(stationary)h(solution)f(of)g (e)-5 b(q.)60 b(\(1.3\),)41 b(and)f(let)g Fk(u)d Fj(\025)h Fl(0)75 4396 y Fc(and)29 b Fk(u)315 4411 y Fn(1)419 4396 y Fc(have)g(the)g(same)g(mass)g Fk(M)40 b Fc(and)29 b(supp)-5 b(ort)30 b Fk(B)5 b Fl(\()p Fk(M)10 b Fl(\))p Fc(.)44 b(Then,)29 b(ther)-5 b(e)30 b(exists)f(a)g(p)-5 b(ositive)29 b(c)-5 b(onstant)75 4517 y Fk(D)30 b Fl(=)e Fk(D)s Fl(\()p Fk(m;)17 b(M)10 b Fl(\))35 b Fc(such)g(that)1189 4669 y Fo(\022)1250 4673 y(Z)1296 4862 y Fp(B)s Fr(\()p Fp(M)7 b Fr(\))1503 4790 y Fj(j)p Fk(u)22 b Fj(\000)g Fk(u)1764 4805 y Fn(1)1839 4790 y Fj(j)17 b Fk(dx)1990 4669 y Fo(\023)2050 4692 y Fr(2)2117 4790 y Fj(\024)28 b Fk(D)s(H)8 b Fl(\()p Fk(u)p Fj(j)p Fk(u)2573 4805 y Fn(1)2646 4790 y Fl(\))p Fk(:)913 b Fl(\(4.1\))1901 5356 y(16)p eop %%Page: 17 17 17 16 bop 221 107 a Fc(Pr)-5 b(o)g(of)p Fl(.)44 b(Using)32 b(that)g Fk(u)g Fl(and)h Fk(u)1338 122 y Fn(1)1445 107 y Fl(ha)m(v)m(e)g(equal)g(mass)f(w)m(e)i(obtain)772 294 y(1)p 772 338 49 4 v 772 429 a(2)847 244 y Fo(Z)893 433 y Fp(B)s Fr(\()p Fp(M)7 b Fr(\))1100 361 y Fj(j)p Fk(u)22 b Fj(\000)g Fk(u)1361 376 y Fn(1)1435 361 y Fj(j)17 b Fk(dx)28 b Fl(=)1717 244 y Fo(Z)1763 433 y Fp(u>u)1900 441 y Fe(1)1986 361 y Fj(j)p Fk(u)21 b Fj(\000)i Fk(u)2247 376 y Fn(1)2321 361 y Fj(j)k Fl(=)2480 244 y Fo(Z)2526 433 y Fp(uC)1722 1879 y Fo(\024)1766 2000 y Fl(\()p Fj(j)p Fk(x)p Fj(j)1915 1959 y Fr(2)1976 2000 y Fj(\000)23 b Fk(C)7 b Fl(\))p Fk(u)21 b Fl(+)2480 1932 y(2)p 2376 1977 256 4 v 2376 2068 a Fk(m)i Fj(\000)f Fl(1)2642 2000 y Fk(u)2698 1959 y Fp(m)2764 1879 y Fo(\025)2841 2000 y Fk(dx)35 b(;)75 2223 y Fc(and)f Fk(C)h Fl(=)27 b(2)576 2184 y Fp(m)p 531 2200 153 4 v 531 2257 a(m)p Fn(\000)p Fr(1)694 2223 y Fk(C)764 2238 y Fr(2)803 2223 y Fl(\()p Fk(M)10 b Fl(\))p Fc(.)45 b(Both)35 b Fk(H)1375 2238 y Fr(1)1414 2223 y Fl(\()p Fk(u)p Fj(j)p Fk(u)1592 2238 y Fn(1)1665 2223 y Fl(\))g Fc(and)g Fk(H)2009 2238 y Fr(2)2048 2223 y Fl(\()p Fk(u)p Fj(j)p Fk(u)2226 2238 y Fn(1)2299 2223 y Fl(\))g Fc(ar)-5 b(e)35 b(nonne)-5 b(gative.)221 2446 y(Pr)g(o)g(of)p Fl(.)44 b(Let)32 b(us)h(consider)g (the)g(sets)1085 2661 y Fk(A)1158 2676 y Fr(1)1280 2661 y Fl(=)83 b Fj(f)p Fk(x)28 b Fj(2)g Fl(I)-20 b(R)1754 2620 y Fp(N)1849 2661 y Fl(:)27 b Fk(u)h Fj(\025)g Fk(u)2148 2676 y Fn(1)2255 2661 y Fk(;)49 b Fj(j)p Fk(x)p Fj(j)2442 2620 y Fr(2)2509 2661 y Fk(<)27 b(C)7 b Fj(g)32 b Fk(;)1085 2807 y(A)1158 2822 y Fr(2)1280 2807 y Fl(=)83 b Fj(f)p Fk(x)28 b Fj(2)g Fl(I)-20 b(R)1754 2765 y Fp(N)1849 2807 y Fl(:)27 b Fk(u)h(<)f(u)2146 2822 y Fn(1)2253 2807 y Fk(;)49 b Fj(j)p Fk(x)p Fj(j)2440 2765 y Fr(2)2507 2807 y Fk(<)28 b(C)7 b Fj(g)221 2952 y Fl(and)1085 3122 y Fk(A)1158 3137 y Fr(3)1280 3122 y Fl(=)83 b Fj(f)p Fk(x)28 b Fj(2)g Fl(I)-20 b(R)1754 3080 y Fp(N)1849 3122 y Fl(:)27 b Fj(j)p Fk(x)p Fj(j)2014 3081 y Fr(2)2081 3122 y Fk(>)h(C)7 b Fj(g)32 b Fk(:)75 3338 y Fl(W)-8 b(e)33 b(write)586 3572 y Fk(H)8 b Fl(\()p Fk(u)p Fj(j)p Fk(u)853 3587 y Fn(1)926 3572 y Fl(\))83 b(=)1206 3455 y Fo(Z)1252 3644 y Fp(A)1305 3653 y Fg(3)1360 3451 y Fo(\022)1421 3572 y Fj(j)p Fk(x)p Fj(j)1532 3531 y Fr(2)1572 3572 y Fk(u)21 b Fl(+)1861 3505 y(2)p 1757 3549 256 4 v 1757 3641 a Fk(m)i Fj(\000)f Fl(1)2023 3572 y Fk(u)2079 3531 y Fp(m)2145 3451 y Fo(\023)2239 3572 y Fk(dx)h Fl(+)1401 3818 y(+)1494 3701 y Fo(Z)1540 3889 y Fp(A)1593 3898 y Fg(2)1648 3697 y Fo(\024)1692 3818 y Fj(j)p Fk(x)p Fj(j)1803 3777 y Fr(2)1842 3818 y Fl(\()p Fk(u)e Fj(\000)i Fk(u)2113 3833 y Fn(1)2187 3818 y Fl(\))f(+)2459 3751 y(2)p 2355 3795 V 2355 3886 a Fk(m)h Fj(\000)f Fl(1)2621 3818 y(\()p Fk(u)2715 3777 y Fp(m)2803 3818 y Fj(\000)h Fk(u)2959 3777 y Fp(m)2959 3843 y Fn(1)3033 3818 y Fl(\))3071 3697 y Fo(\025)3148 3818 y Fk(dx)1401 4064 y Fl(+)1494 3946 y Fo(Z)1540 4135 y Fp(A)1593 4144 y Fg(1)1648 3942 y Fo(\024)1692 4064 y Fj(j)p Fk(x)p Fj(j)1803 4022 y Fr(2)1842 4064 y Fl(\()p Fk(u)e Fj(\000)i Fk(u)2113 4079 y Fn(1)2187 4064 y Fl(\))f(+)2459 3996 y(2)p 2355 4040 V 2355 4132 a Fk(m)h Fj(\000)f Fl(1)2621 4064 y(\()p Fk(u)2715 4022 y Fp(m)2803 4064 y Fj(\000)h Fk(u)2959 4022 y Fp(m)2959 4088 y Fn(1)3033 4064 y Fl(\))3071 3942 y Fo(\025)3148 4064 y Fk(dx)33 b(:)75 4317 y Fl(Using)f(the)h(same)g(T)-8 b(a)m(ylor)32 b(expansion)h(as)g(in)e(Lemma)h(4.1)g(w)m(e)h(deduce)i (that)571 4454 y Fo(Z)617 4643 y Fp(A)670 4652 y Fg(2)725 4450 y Fo(\024)769 4571 y Fj(j)p Fk(x)p Fj(j)880 4530 y Fr(2)919 4571 y Fl(\()p Fk(u)22 b Fj(\000)g Fk(u)1190 4586 y Fn(1)1265 4571 y Fl(\))g(+)1536 4504 y(2)p 1433 4548 V 1433 4639 a Fk(m)g Fj(\000)h Fl(1)1699 4571 y(\()p Fk(u)1793 4530 y Fp(m)1881 4571 y Fj(\000)f Fk(u)2036 4530 y Fp(m)2036 4596 y Fn(1)2111 4571 y Fl(\))2149 4450 y Fo(\025)2226 4571 y Fk(dx)27 b Fl(=)571 4826 y(=)g Fk(C)768 4709 y Fo(Z)814 4898 y Fp(A)867 4907 y Fg(2)906 4826 y Fl(\()p Fk(u)21 b Fj(\000)i Fk(u)1177 4841 y Fn(1)1251 4826 y Fl(\))17 b Fk(dx)22 b Fl(+)g Fk(m)1634 4709 y Fo(Z)1680 4898 y Fp(A)1733 4907 y Fg(2)1788 4826 y Fk(u)1844 4785 y Fp(m)p Fn(\000)p Fr(2)1844 4851 y Fn(1)2017 4705 y Fo(\022)2078 4826 y Fl(1)g(+)g Fk(\022)2295 4730 y Fo(\020)2392 4759 y Fk(u)p 2355 4803 131 4 v 2355 4894 a(u)2411 4909 y Fn(1)2517 4826 y Fj(\000)h Fl(1)2666 4730 y Fo(\021)2715 4705 y(\023)2776 4728 y Fp(m)p Fn(\000)p Fr(2)2950 4826 y Fl(\()p Fk(u)e Fj(\000)i Fk(u)3221 4841 y Fn(1)3295 4826 y Fl(\))3333 4785 y Fr(2)3389 4826 y Fk(dx)571 5060 y Fl(=)k Fk(C)768 4943 y Fo(Z)814 5131 y Fp(A)867 5140 y Fg(2)906 5060 y Fl(\()p Fk(u)21 b Fj(\000)i Fk(u)1177 5075 y Fn(1)1251 5060 y Fl(\))f(+)g Fk(L)p Fl(\()p Fk(u)p Fl(\))1901 5356 y(17)p eop %%Page: 18 18 18 17 bop 75 107 a Fl(where)29 b Fk(L)p Fl(\()p Fk(u)p Fl(\))f Fj(\025)g Fl(0.)41 b(Since)28 b Fk(u)g Fl(and)g Fk(u)1375 122 y Fn(1)1477 107 y Fl(ha)m(v)m(e)h(equal)f(mass)g(and)g (the)g(supp)s(ort)g(of)g Fk(u)3052 122 y Fn(1)3154 107 y Fl(is)f Fj(j)p Fk(x)p Fj(j)3358 71 y Fr(2)3425 107 y Fj(\024)i Fk(C)34 b Fl(then)1136 236 y Fo(Z)1183 425 y Fp(A)1236 434 y Fg(2)1274 353 y Fl(\()p Fk(u)22 b Fj(\000)g Fk(u)1545 368 y Fn(1)1620 353 y Fl(\))17 b Fk(dx)27 b Fl(=)g Fj(\000)2005 236 y Fo(Z)2052 425 y Fp(u>u)2189 433 y Fe(1)2257 353 y Fl(\()p Fk(u)22 b Fj(\000)h Fk(u)2529 368 y Fn(1)2603 353 y Fl(\))17 b Fk(dx)75 608 y Fl(and)457 825 y Fk(H)8 b Fl(\()p Fk(u)p Fj(j)p Fk(u)724 840 y Fn(1)797 825 y Fl(\))83 b(=)g Fk(H)1158 840 y Fr(1)1197 825 y Fl(\()p Fk(u)p Fj(j)p Fk(u)1375 840 y Fn(1)1448 825 y Fl(\))22 b Fj(\000)h Fk(C)1702 708 y Fo(Z)1748 896 y Fp(A)1801 905 y Fg(1)1839 825 y Fl(\()p Fk(u)f Fj(\000)h Fk(u)2111 840 y Fn(1)2185 825 y Fl(\))17 b Fk(dx)1272 1070 y Fl(+)1365 953 y Fo(Z)1410 1142 y Fp(A)1463 1151 y Fg(1)1519 949 y Fo(\024)1562 1070 y Fj(j)p Fk(x)p Fj(j)1673 1029 y Fr(2)1713 1070 y Fl(\()p Fk(u)k Fj(\000)i Fk(u)1984 1085 y Fn(1)2058 1070 y Fl(\))f(+)2330 1003 y(2)p 2226 1047 256 4 v 2226 1139 a Fk(m)h Fj(\000)f Fl(1)2492 1070 y(\()p Fk(u)2586 1029 y Fp(m)2674 1070 y Fj(\000)h Fk(u)2830 1029 y Fp(m)2830 1095 y Fn(1)2904 1070 y Fl(\))2942 949 y Fo(\025)3019 1070 y Fk(dx)f Fl(+)g Fk(L)p Fl(\()p Fk(u)p Fl(\))75 1330 y(It)34 b(is)g(clear)f(that)h Fk(H)810 1345 y Fr(1)849 1330 y Fl(\()p Fk(u)p Fj(j)p Fk(u)1027 1345 y Fn(1)1101 1330 y Fl(\))g(and)g Fk(L)p Fl(\()p Fk(u)p Fl(\))g(are)g(p)s(ositiv)m(e.)47 b(Let)34 b(us)h(pro)m(v)m(e)g (that)f(the)h(sum)f(of)f(the)i(other)75 1451 y(t)m(w)m(o)e(terms)g(is)f (also)g(p)s(ositiv)m(e.)42 b(This)33 b(sum)g(can)f(b)s(e)h(written)g (as)767 1706 y Fj(\000)916 1639 y Fl(2)p Fk(m)p 854 1683 V 854 1775 a(m)23 b Fj(\000)g Fl(1)1137 1589 y Fo(Z)1183 1778 y Fp(A)1236 1787 y Fg(1)1291 1706 y Fk(u)1347 1665 y Fp(m)p Fn(\000)p Fr(1)1347 1731 y Fn(1)1504 1706 y Fl(\()p Fk(u)e Fj(\000)i Fk(u)1775 1721 y Fn(1)1849 1706 y Fl(\))17 b Fk(dx)22 b Fl(+)2243 1639 y(2)p 2140 1683 V 2140 1775 a Fk(m)g Fj(\000)h Fl(1)2422 1589 y Fo(Z)2468 1778 y Fp(A)2521 1787 y Fg(1)2560 1706 y Fl(\()p Fk(u)2654 1665 y Fp(m)2742 1706 y Fj(\000)g Fk(u)2898 1665 y Fp(m)2898 1731 y Fn(1)2972 1706 y Fl(\))17 b Fk(dx)75 1961 y Fl(Using)32 b(b)m(y)i(T)-8 b(a)m(ylor)32 b(expansion)h(that)970 2177 y Fk(u)1026 2136 y Fp(m)1115 2177 y Fj(\000)22 b Fk(u)1270 2136 y Fp(m)1270 2202 y Fn(1)1372 2177 y Fl(=)28 b Fk(m)1561 2081 y Fo(\020)1611 2177 y Fk(\030)5 b(u)1715 2136 y Fp(m)p Fn(\000)p Fr(1)1892 2177 y Fl(+)22 b(\(1)g Fj(\000)h Fk(\030)5 b Fl(\))p Fk(u)2341 2136 y Fp(m)p Fn(\000)p Fr(1)2341 2202 y Fn(1)2497 2081 y Fo(\021)2546 2177 y Fl(\()p Fk(u)22 b Fj(\000)g Fk(u)2817 2192 y Fn(1)2892 2177 y Fl(\))75 2407 y(where)34 b Fk(\030)e Fj(2)c Fl(\(0)p Fk(;)17 b Fl(1\))32 b(w)m(e)h(ha)m(v)m(e)1356 2527 y Fk(u)1412 2486 y Fp(m)1500 2527 y Fj(\000)23 b Fk(u)1656 2486 y Fp(m)1656 2552 y Fn(1)1758 2527 y Fj(\025)28 b Fk(mu)2004 2486 y Fp(m)p Fn(\000)p Fr(1)2004 2552 y Fn(1)2161 2527 y Fl(\()p Fk(u)21 b Fj(\000)i Fk(u)2432 2542 y Fn(1)2506 2527 y Fl(\))75 2700 y(in)32 b Fk(A)262 2715 y Fr(1)301 2700 y Fl(.)44 b(Finally)-8 b(,)30 b Fk(H)804 2715 y Fr(2)843 2700 y Fl(\()p Fk(u)p Fj(j)p Fk(u)1021 2715 y Fn(1)1094 2700 y Fl(\))j(is)f(non)h(negativ)m(e)f(and)h(the)g(lemma)d (is)i(pro)m(v)m(ed.)p 2995 2641 41 4 v 2995 2696 4 56 v 3033 2696 V 2995 2699 41 4 v 221 2941 a(W)-8 b(e)33 b(no)m(w)h(pro)m(v)m(e)f(our)g(main)e(result)75 3165 y Fd(Theorem)37 b(4.4)49 b Fc(L)-5 b(et)35 b(the)f(initial)g(c)-5 b(ondition)34 b(for)g(the)h(F)-7 b(okker-Planck)32 b(e)-5 b(quation)34 b(\(1.3\))g(satisfy)g Fl(0)28 b Fj(\024)75 3285 y Fk(u)131 3300 y Fr(0)170 3285 y Fl(\()p Fk(x)p Fl(\))g Fj(2)g Fk(L)489 3249 y Fr(1)529 3285 y Fl(\(I)-20 b(R)654 3244 y Fp(N)721 3285 y Fl(\))32 b Fc(and)g Fk(H)8 b Fl(\()p Fk(u)1161 3300 y Fr(0)1200 3285 y Fl(\))27 b Fk(<)h Fj(1)p Fc(.)43 b(Then)32 b(ther)-5 b(e)32 b(exist)h(a)f(p)-5 b(ositive)32 b(c)-5 b(onstant)32 b Fk(C)i Fl(=)28 b Fk(C)7 b Fl(\()p Fk(m;)17 b(H)8 b Fl(\()p Fk(u)3711 3300 y Fr(0)3749 3285 y Fl(\)\))75 3406 y Fc(such)35 b(that)535 3660 y Fj(k)p Fk(u)p Fl(\()p Fk(t;)17 b(x)p Fl(\))22 b Fj(\000)g Fk(u)1028 3675 y Fn(1)1103 3660 y Fl(\()p Fk(x)p Fl(\))p Fj(k)1284 3679 y Fp(L)1332 3660 y Fg(1)1366 3679 y Fr(\()p Fg(I)-10 b(R)1450 3660 y Ff(N)1508 3679 y Fr(\))1567 3660 y Fj(\024)28 b Fk(C)c Fl(exp)1931 3514 y Fo(\()1998 3660 y Fj(\000)2105 3592 y Fk(N)10 b Fl(\()p Fk(m)23 b Fj(\000)g Fl(1\))f(+)g(2)p 2085 3637 629 4 v 2085 3728 a(\()p Fk(N)33 b Fl(+)22 b(2\))p Fk(m)g Fj(\000)h Fk(N)2724 3660 y(t)2759 3514 y Fo(\))2878 3660 y Fk(;)216 b(t)28 b Fj(\025)g Fl(0)p Fk(:)221 3941 y Fc(Pr)-5 b(o)g(of)p Fl(.)43 b(Firstly)-8 b(,)29 b(assume)h(that)g(the)g(initial)c(data)k (satis\014es)h(the)f(additional)d(h)m(yp)s(otheses)32 b(needed)75 4062 y(in)c(Theorems)i(3.1)f(and)g(3.2.)42 b(Let)30 b Fk(B)i Fl(=)c Fk(H)8 b Fl(\()p Fk(u)1731 4077 y Fr(0)1769 4062 y Fj(j)p Fk(u)1853 4077 y Fn(1)1927 4062 y Fl(\).)43 b(W)-8 b(e)29 b(kno)m(w)h(that)f Fk(H)8 b Fl(\()p Fk(u)p Fl(\()p Fk(t)p Fl(\))p Fj(j)p Fk(u)3036 4077 y Fn(1)3109 4062 y Fl(\))28 b Fj(\024)g Fk(B)5 b(e)3404 4025 y Fn(\000)p Fr(2)p Fp(t)3524 4062 y Fl(,)30 b Fk(t)e Fj(\025)g Fl(0.)75 4182 y(Therefore,)459 4437 y Fk(H)540 4452 y Fr(1)580 4437 y Fl(\()p Fk(u)p Fj(j)p Fk(u)758 4452 y Fn(1)831 4437 y Fl(\))g(=)1000 4320 y Fo(Z)1046 4509 y Fn(j)p Fp(x)p Fn(j)1126 4490 y Fg(2)1160 4509 y Fp(>C)1290 4316 y Fo(\024)1334 4437 y Fl(\()p Fj(j)p Fk(x)p Fj(j)1483 4396 y Fr(2)1545 4437 y Fj(\000)22 b Fk(C)7 b Fl(\))p Fk(u)22 b Fl(+)2048 4370 y(2)p 1945 4414 256 4 v 1945 4506 a Fk(m)g Fj(\000)h Fl(1)2211 4437 y Fk(u)2267 4396 y Fp(m)2333 4316 y Fo(\025)2410 4437 y Fk(dx)k Fj(\024)i Fk(B)5 b(e)2773 4396 y Fn(\000)p Fr(2)p Fp(t)2925 4437 y Fk(;)212 b(t)28 b Fj(\025)g Fl(0)k Fk(:)75 4699 y Fl(First,)g(w)m(e)h(pro)m(v)m(e)h(that)1143 4727 y Fo(Z)1190 4916 y Fn(j)p Fp(x)p Fn(j)1270 4897 y Fg(2)1303 4916 y Fp(>C)1434 4844 y Fk(u)p Fl(\()p Fk(x;)17 b(t)p Fl(\))g Fk(dx)27 b Fj(\024)h Fk(D)s(e)2084 4803 y Fn(\000)p Fp(\015)t(t)2241 4844 y Fk(;)212 b(t)28 b Fj(\025)g Fl(0)k Fk(:)868 b Fl(\(4.2\))75 5089 y(where)34 b Fk(D)s Fl(\()p Fk(m)p Fl(\))27 b Fk(>)h Fl(1)k(is)g(a)h(b)s(ounded)g (constan)m(t,)g(and)g Fk(\015)5 b Fl(\()p Fk(m)p Fl(\))28 b(=)2361 5043 y Fr(2)p Fp(N)7 b Fr(\()p Fp(m)p Fn(\000)p Fr(1\)+4)p 2361 5066 397 4 v 2365 5124 a(\()p Fp(N)g Fr(+2\))p Fp(m)p Fn(\000)p Fp(N)2767 5089 y Fl(.)1901 5356 y(18)p eop %%Page: 19 19 19 18 bop 221 107 a Fl(F)-8 b(or)32 b(an)m(y)h(giv)m(en)g Fk(\032)28 b(>)g Fl(0)k(w)m(e)h(write)173 217 y Fo(Z)219 406 y Fn(j)p Fp(x)p Fn(j)299 387 y Fg(2)333 406 y Fp(>C)464 334 y Fk(u)p Fl(\()p Fk(x;)17 b(t)p Fl(\))g Fk(dx)27 b Fj(\024)985 217 y Fo(Z)1031 406 y Fp(C)5 b(<)p Fn(j)p Fp(x)p Fn(j)1221 387 y Fg(2)1254 406 y Fp(<)p Fr(\()1336 349 y Fn(p)p 1396 349 55 3 v 1396 406 a Fp(C)g Fr(+)p Fp(\032=)p Fr(2\))1639 387 y Fg(2)1695 334 y Fk(u)p Fl(\()p Fk(x;)17 b(t)p Fl(\))g Fk(dx)22 b Fl(+)2234 267 y(4)p 2214 311 90 4 v 2214 403 a Fk(\032)2264 374 y Fr(2)2330 217 y Fo(Z)2376 406 y Fn(j)p Fp(x)p Fn(j)2456 387 y Fg(2)2489 406 y Fp(>)p Fr(\()2571 349 y Fn(p)p 2631 349 55 3 v 2631 406 a Fp(C)5 b Fr(+)p Fp(\032=)p Fr(2\))2874 387 y Fg(2)2913 334 y Fl(\()p Fj(j)p Fk(x)p Fj(j)3062 293 y Fr(2)3124 334 y Fj(\000)22 b Fk(C)7 b Fl(\))p Fk(u)p Fl(\()p Fk(x;)17 b(t)p Fl(\))g Fk(dx)75 568 y Fl(th)m(us,)34 b(b)m(y)f(H\177)-49 b(older)32 b(inequalit)m(y)75 695 y Fo(Z)121 884 y Fn(j)p Fp(x)p Fn(j)201 865 y Fg(2)235 884 y Fp(>C)365 812 y Fk(u)p Fl(\()p Fk(x;)17 b(t)p Fl(\))g Fk(dx)83 b Fj(\024)g Fk(A)1070 827 y Fp(N)1138 691 y Fo(\022)1199 695 y(Z)1245 884 y Fp(C)5 b(<)p Fn(j)p Fp(x)p Fn(j)1435 865 y Fg(2)1468 884 y Fp(<)p Fr(\()1550 827 y Fn(p)p 1609 827 V 57 x Fp(C)g Fr(+)p Fp(\032=)p Fr(2\))1852 865 y Fg(2)1909 812 y Fk(u)1965 771 y Fp(m)2031 691 y Fo(\023)2092 714 y Fr(1)p Fp(=m)2229 812 y Fk(\032)2279 771 y Fp(N)i Fr(\()p Fp(m)p Fn(\000)p Fr(1\))p Fp(=m)2674 812 y Fl(+)2802 745 y(4)p 2782 789 90 4 v 2782 881 a Fk(\032)2832 852 y Fr(2)2898 695 y Fo(Z)2944 884 y Fn(j)p Fp(x)p Fn(j)3024 865 y Fg(2)3058 884 y Fp(>C)3172 812 y Fl(\()p Fj(j)p Fk(x)p Fj(j)3321 771 y Fr(2)3382 812 y Fj(\000)23 b Fk(C)7 b Fl(\))p Fk(u)p Fl(\()p Fk(x;)17 b(t)p Fl(\))g Fk(dx)837 1089 y Fj(\024)83 b Fk(A)1070 1104 y Fp(N)1154 968 y Fo(\022)1225 1021 y Fk(m)23 b Fj(\000)f Fl(1)p 1225 1066 256 4 v 1329 1157 a(2)1491 1089 y Fk(H)1572 1104 y Fr(1)1611 1089 y Fl(\()p Fk(u)p Fj(j)p Fk(u)1789 1104 y Fn(1)1863 1089 y Fl(\))1901 968 y Fo(\023)1962 983 y Fr(1)p Fp(=m)2116 1089 y Fk(\032)2166 1048 y Fp(N)7 b Fr(\()p Fp(m)p Fn(\000)p Fr(1\))p Fp(=m)2560 1089 y Fl(+)2689 1021 y(4)p 2668 1066 90 4 v 2668 1157 a Fk(\032)2718 1128 y Fr(2)2768 1089 y Fk(H)2849 1104 y Fr(1)2888 1089 y Fl(\()p Fk(u)p Fj(j)p Fk(u)3066 1104 y Fn(1)3140 1089 y Fl(\))75 1335 y(Optimizing)29 b(o)m(v)m(er)34 b Fk(\032)f Fl(w)m(e)h(obtain)d(\(4.2\).)43 b(No)m(w,)33 b(let)f(us)h(c)m(ho)s(ose)h Fk(\013)q Fl(\()p Fk(t)p Fl(\))27 b Fj(2)h Fl(I)-20 b(R)2802 1293 y Fr(+)2893 1335 y Fl(suc)m(h)34 b(that)1037 1554 y Fk(\013)q Fl(\()p Fk(t)p Fl(\))1228 1437 y Fo(Z)1274 1625 y Fn(j)p Fp(x)p Fn(j)1354 1606 y Fg(2)1387 1625 y Fp(C)1498 3833 y Fk(u)17 b(dx)22 b Fl(+)g(\()p Fk(\013)q Fl(\()p Fk(t)p Fl(\))g Fj(\000)g Fl(1\))2234 3716 y Fo(Z)2280 3905 y Fn(j)p Fp(x)p Fn(j)2360 3886 y Fg(2)2393 3905 y Fp()g Fl(0,)j(where)g Fk(!)1058 1479 y Fp(\017)1127 1464 y Fl(is)f(a)g(regularizing)d(sequence.)58 b(No)m(w,)38 b Fk(u)2612 1428 y Fp(\017)2612 1489 y Fr(0)2687 1464 y Fl(satis\014es)f(all)d(the)j(h)m(yp)s(otheses)75 1584 y(in)32 b(Theorems)h(3.1)f(and)h(3.2.)43 b(Moreo)m(v)m(er)34 b(w)m(e)g(can)f(assume)g(that)1517 1773 y Fj(k)p Fk(u)1623 1732 y Fp(\017)1623 1798 y Fr(0)1684 1773 y Fj(\000)22 b Fk(u)1839 1788 y Fr(0)1878 1773 y Fj(k)1928 1792 y Fp(L)1976 1773 y Fg(1)2011 1792 y Fr(\()p Fg(I)-10 b(R)2095 1773 y Ff(N)2152 1792 y Fr(\))2211 1773 y Fj(\024)29 b Fk(\017:)75 1962 y Fl(Using)j(no)m(w)h(the)g Fk(L)786 1925 y Fr(1)859 1962 y Fl(con)m(traction)f(prop)s(ert)m(y)h(of)f(the)h (solutions)f(of)g(\(1.3\))g(w)m(e)i(deduce)g(that)1115 2150 y Fj(k)p Fk(u)1221 2109 y Fp(\017)1253 2150 y Fl(\()p Fk(t)p Fl(\))22 b Fj(\000)h Fk(u)p Fl(\()p Fk(t)p Fl(\))p Fj(k)1703 2169 y Fp(L)1751 2150 y Fg(1)1785 2169 y Fr(\()p Fg(I)-10 b(R)1869 2150 y Ff(N)1926 2169 y Fr(\))1986 2150 y Fj(\024)28 b(k)p Fk(u)2197 2109 y Fp(\017)2197 2175 y Fr(0)2258 2150 y Fj(\000)22 b Fk(u)2413 2165 y Fr(0)2452 2150 y Fj(k)2502 2169 y Fp(L)2550 2150 y Fg(1)2585 2169 y Fr(\()p Fg(I)-10 b(R)2668 2150 y Ff(N)2726 2169 y Fr(\))2758 2150 y Fk(:)221 2339 y Fl(As)33 b(a)g(consequence,)i(w)m (e)f(ha)m(v)m(e)g(that)257 2527 y Fj(k)p Fk(u)p Fl(\()p Fk(t)p Fl(\))22 b Fj(\000)h Fk(u)652 2542 y Fn(1)726 2527 y Fj(k)776 2546 y Fp(L)824 2527 y Fg(1)858 2546 y Fr(\()p Fg(I)-10 b(R)942 2527 y Ff(N)1000 2546 y Fr(\))1059 2527 y Fj(\024)28 b(k)p Fk(u)1270 2486 y Fp(\017)1302 2527 y Fl(\()p Fk(t)p Fl(\))22 b Fj(\000)h Fk(u)1591 2486 y Fp(\017)1591 2552 y Fn(1)1665 2527 y Fj(k)1715 2546 y Fp(L)1763 2527 y Fg(1)1798 2546 y Fr(\()p Fg(I)-10 b(R)1881 2527 y Ff(N)1939 2546 y Fr(\))1993 2527 y Fl(+)22 b Fj(k)p Fk(u)2197 2486 y Fp(\017)2197 2552 y Fr(0)2258 2527 y Fj(\000)g Fk(u)2413 2542 y Fr(0)2452 2527 y Fj(k)2502 2546 y Fp(L)2550 2527 y Fg(1)2585 2546 y Fr(\()p Fg(I)-10 b(R)2668 2527 y Ff(N)2726 2546 y Fr(\))2780 2527 y Fl(+)22 b Fj(k)p Fk(u)2984 2486 y Fp(\017)2984 2552 y Fn(1)3080 2527 y Fj(\000)h Fk(u)3236 2542 y Fn(1)3310 2527 y Fj(k)3360 2546 y Fp(L)3408 2527 y Fg(1)3442 2546 y Fr(\()p Fg(I)-10 b(R)3526 2527 y Ff(N)3584 2546 y Fr(\))3615 2527 y Fk(:)75 2716 y Fl(T)i(aking)32 b(in)m(to)g(accoun)m(t)h(the)g(\014rst)g(part)g (of)f(the)h(theorem,)f(then)i(it)d(is)h(easy)i(to)e(conclude.)p 3448 2657 41 4 v 3448 2713 4 56 v 3485 2713 V 3448 2716 41 4 v 75 3029 a Fd(Remark)37 b(4.5)49 b Fl(F)-8 b(or)40 b Fk(m)i(>)f Fl(1)g(the)g(deca)m(y)h(to)m(w)m(ards)g(the)g(steady)f (state)h(of)e(the)h(nonlinear)f(F)-8 b(okk)m(er-)75 3149 y(Planc)m(k)39 b(t)m(yp)s(e)h(equation)e(\(1.3\))g(is)g(exp)s(onen)m (tial)g(at)g(a)g(rate)h(that)f(for)g Fk(m)g Fj(!)g Fl(1)g(coincides)g (with)h(the)75 3269 y(rate)33 b(of)g(con)m(v)m(ergence)k(of)c(the)h (classical)e(F)-8 b(okk)m(er-Planc)m(k)34 b(equation)f([3],)h(and)g (tends)g(to)f Fk(N)r(=)p Fl(\()p Fk(N)h Fl(+)22 b(2\))75 3390 y(as)33 b Fk(m)c Fj(!)g(1)p Fl(.)45 b(Our)33 b(analysis)f(can)i(b) s(e)f(applied)f(to)h(the)h(fast)f(di\013usion)f(case)i(in)f(the)g(in)m (terv)-5 b(al)3602 3351 y Fp(N)p 3557 3367 154 4 v 3557 3424 a(N)7 b Fr(+2)3749 3390 y Fk(<)75 3510 y(m)28 b(<)g Fl(1)k(with)g(minor)f(mo)s(di\014cations)p Fc(.)75 3838 y Fm(5)161 b(En)l(trop)l(y)52 b(pro)t(duction)g(and)i Fb(L)2173 3786 y Fj(1)2276 3838 y Fa(\(I)-32 b(R)o(\))p Fm(-estimates)75 4057 y Fl(This)34 b(last)f(section)i(is)e(dev)m(oted)j (to)d(\014nd)i(the)f(rate)g(of)g(exp)s(onen)m(tial)f(con)m(v)m(ergence) k(to)m(w)m(ards)e(equilib-)75 4178 y(rium)c(in)h Fk(L)488 4142 y Fn(1)563 4178 y Fl(\(I)-20 b(R\).)44 b(The)33 b(main)e(result)i(is)f(giv)m(en)g(b)m(y)i(the)f(follo)m(wing)75 4370 y Fd(Theorem)k(5.1)49 b Fc(L)-5 b(et)35 b(the)f(initial)g(c)-5 b(ondition)34 b(for)g(the)h(F)-7 b(okker-Planck)32 b(e)-5 b(quation)34 b(\(1.3\))g(satisfy)g Fl(0)28 b Fj(\024)75 4491 y Fk(u)131 4506 y Fr(0)170 4491 y Fl(\()p Fk(x)p Fl(\))i Fj(2)f Fk(L)492 4454 y Fr(1)532 4491 y Fl(\(I)-20 b(R\))36 b Fc(and)f Fk(H)8 b Fl(\()p Fk(u)1104 4506 y Fr(0)1142 4491 y Fl(\))29 b Fk(<)h Fj(1)p Fc(.)46 b(Then,)35 b(for)h(al)5 b(l)35 b Fk(t)30 b Fj(\025)f Fk(t)2279 4506 y Fr(0)2348 4491 y Fk(>)h Fl(0)p Fc(,)35 b Fk(u)p Fl(\()p Fk(t)p Fl(\))29 b Fj(2)h Fk(L)2926 4454 y Fn(1)3001 4491 y Fl(\(I)-20 b(R\))35 b Fc(and)h(ther)-5 b(e)35 b(exist)75 4611 y(a)g(p)-5 b(ositive)34 b(c)-5 b(onstant)34 b Fk(C)h Fl(=)28 b Fk(C)23 b Fl(\()p Fk(m;)17 b(H)8 b Fl(\()p Fk(u)1551 4626 y Fr(0)1590 4611 y Fl(\))p Fk(;)17 b(I)8 b Fl(\()p Fk(u)p Fl(\()p Fk(t)1890 4626 y Fr(0)1928 4611 y Fl(\)\)\))35 b Fc(such)f(that)997 4862 y Fj(k)p Fk(u)p Fl(\()p Fk(t)p Fl(\))22 b Fj(\000)g Fk(u)1391 4877 y Fn(1)1466 4862 y Fj(k)1516 4877 y Fp(L)1564 4858 y Fe(1)1629 4877 y Fr(\()p Fg(I)-10 b(R)p Fr(\))1771 4862 y Fj(\024)28 b Fk(C)c Fl(exp)2136 4716 y Fo(\()2203 4862 y Fj(\000)2396 4794 y Fk(m)e Fl(+)g(1)p 2290 4839 466 4 v 2290 4930 a Fk(m)p Fl(\(3)p Fk(m)g Fj(\000)h Fl(1\))2766 4862 y Fk(t)2801 4716 y Fo(\))75 5107 y Fc(for)35 b(any)f Fk(t)28 b Fj(\025)h Fk(t)621 5122 y Fr(0)660 5107 y Fc(.)1901 5356 y Fl(20)p eop %%Page: 21 21 21 20 bop 221 107 a Fc(Pr)-5 b(o)g(of)p Fl(.)43 b(Firstly)-8 b(,)29 b(assume)h(that)g(the)g(initial)c(data)k(satis\014es)h(the)f (additional)d(h)m(yp)s(otheses)32 b(needed)75 227 y(in)g(Theorems)h (3.1)f(and)h(3.2.)43 b(W)-8 b(e)33 b(can)g(write)f(the)h(en)m(trop)m(y) h(pro)s(duction)e(as)868 517 y Fk(I)8 b Fl(\()p Fk(u)p Fl(\))26 b(=)1181 400 y Fo(Z)1227 588 y Fg(I)-10 b(R)1304 517 y Fk(x)1359 476 y Fr(2)1399 517 y Fk(u)17 b(dx)k Fj(\000)i Fl(2)1765 400 y Fo(Z)1811 588 y Fg(I)-10 b(R)1888 517 y Fk(u)1944 476 y Fp(m)2026 517 y Fk(dx)23 b Fl(+)2253 400 y Fo(Z)2299 588 y Fg(I)-10 b(R)2389 449 y Fl(1)p 2386 493 56 4 v 2386 585 a Fk(u)2468 371 y Fo( )2571 449 y Fk(@)p 2544 493 113 4 v 2544 585 a(@)5 b(x)2666 517 y(u)2722 476 y Fp(m)2788 371 y Fo(!)2854 394 y Fr(2)2926 517 y Fk(dx)75 799 y Fl(Let)33 b(as)g(usual)f Fk(B)5 b Fl(\()p Fk(M)10 b Fl(\))28 b(=)g Fj(fj)p Fk(x)p Fj(j)1177 763 y Fr(2)1243 799 y Fj(\024)h Fk(C)7 b Fj(g)32 b Fl(b)s(e)h(the)g (supp)s(ort)g(of)f Fk(u)2337 814 y Fn(1)2411 799 y Fl(,)h(and)f(let)g Fk(A)2874 763 y Fr(2)2941 799 y Fl(=)c Fk(C)7 b Fl(.)43 b(If)33 b Fk(x)28 b Fj(\024)g(\000)p Fk(A)809 1065 y(u)865 1024 y Fp(m)931 1065 y Fl(\()p Fk(x)p Fl(\))84 b(=)1306 948 y Fo(Z)1389 974 y Fp(x)1352 1137 y Fn(\0001)1508 998 y Fk(@)5 b(u)1620 961 y Fp(m)p 1508 1042 180 4 v 1544 1133 a Fk(@)g(y)1714 1065 y(dy)30 b Fj(\024)1948 948 y Fo(Z)2031 974 y Fn(\000)p Fp(A)1994 1137 y Fn(\0001)2160 965 y Fo(\014)2160 1015 y(\014)2160 1065 y(\014)2197 998 y Fk(@)5 b(u)2309 961 y Fp(m)p 2197 1042 V 2233 1133 a Fk(@)g(y)2386 965 y Fo(\014)2386 1015 y(\014)2386 1065 y(\014)17 b Fk(dy)30 b Fj(\024)1145 1352 y(\024)1306 1231 y Fo(\022)1367 1235 y(Z)1450 1261 y Fn(\000)p Fp(A)1413 1423 y Fn(\0001)1592 1284 y Fl(1)p 1588 1329 56 4 v 1588 1420 a Fk(u)1670 1206 y Fo( )1746 1284 y Fk(@)5 b(u)1858 1248 y Fp(m)p 1746 1329 180 4 v 1782 1420 a Fk(@)g(y)1935 1206 y Fo(!)2001 1229 y Fr(2)2073 1352 y Fk(dy)2176 1231 y Fo(\023)2236 1254 y Fr(1)p Fp(=)p Fr(2)2346 1231 y Fo(\022)2408 1235 y(Z)2491 1261 y Fn(\000)p Fp(A)2454 1423 y Fn(\0001)2619 1352 y Fk(u)p Fl(\()p Fk(y)t Fl(\))17 b Fk(dy)2923 1231 y Fo(\023)2981 1254 y Fr(1)p Fp(=)p Fr(2)75 1630 y Fl(Using)32 b(the)h(de\014nition)f(of)g Fk(I)8 b Fl(\()p Fk(u)p Fl(\),)32 b(w)m(e)h(obtain)467 1808 y Fo(Z)551 1835 y Fn(\000)p Fp(A)514 1997 y Fn(\0001)692 1858 y Fl(1)p 689 1902 56 4 v 689 1994 a Fk(u)771 1779 y Fo( )847 1858 y Fk(@)5 b(u)959 1822 y Fp(m)p 847 1902 180 4 v 880 1994 a Fk(@)g(x)1036 1779 y Fo(!)1101 1802 y Fr(2)1174 1925 y Fk(dx)83 b Fj(\024)h Fk(I)8 b Fl(\()p Fk(u)p Fl(\))21 b(+)h(2)1892 1808 y Fo(Z)1938 1997 y Fg(I)-10 b(R)2015 1925 y Fk(u)2071 1884 y Fp(m)2153 1925 y Fk(dx)1363 2190 y Fj(\024)84 b Fk(I)8 b Fl(\()p Fk(u)p Fl(\))21 b(+)h(\()p Fk(m)h Fj(\000)f Fl(1\))2158 2069 y Fo(\024)2315 2122 y Fl(2)p 2212 2167 256 4 v 2212 2258 a Fk(m)g Fj(\000)h Fl(1)2494 2073 y Fo(Z)2540 2261 y Fg(I)-10 b(R)2617 2190 y Fk(u)2673 2149 y Fp(m)2756 2190 y Fk(dx)22 b Fl(+)2982 2073 y Fo(Z)3028 2261 y Fg(I)-10 b(R)3105 2190 y Fk(x)3160 2149 y Fr(2)3200 2190 y Fk(u)17 b(dx)3379 2069 y Fo(\025)1363 2384 y Fj(\024)84 b Fk(I)8 b Fl(\()p Fk(u)p Fl(\))21 b(+)h(\()p Fk(m)h Fj(\000)f Fl(1\))p Fk(H)8 b Fl(\()p Fk(u)p Fj(j)p Fk(u)2425 2399 y Fn(1)2498 2384 y Fl(\))27 b Fj(\024)i Fk(D)s(e)2798 2342 y Fn(\000)p Fr(2)p Fp(t)2950 2384 y Fk(;)212 b(t)27 b Fj(\025)i Fl(0)p Fk(:)75 2600 y Fl(In)k(addition,)e(if)g Fk(x)d Fj(\024)g(\000)p Fk(A)1112 2742 y Fo(Z)1195 2769 y Fn(\000)p Fp(A)1158 2931 y Fn(\0001)1323 2860 y Fk(u)17 b(dx)27 b Fj(\024)1676 2792 y Fl(1)p 1644 2836 113 4 v 1644 2928 a Fk(A)1717 2899 y Fr(2)1783 2742 y Fo(Z)1866 2769 y Fn(\000)p Fp(A)1829 2931 y Fn(\0001)1995 2860 y Fk(x)2050 2818 y Fr(2)2090 2860 y Fk(u)g Fj(\024)2320 2792 y Fl(1)p 2288 2836 V 2288 2928 a Fk(A)2361 2899 y Fr(2)2411 2860 y Fk(H)8 b Fl(\()p Fk(u)p Fj(j)p Fk(u)2678 2875 y Fn(1)2750 2860 y Fl(\))75 3117 y(Finally)-8 b(,)37 b(if)g Fk(t)h Fj(\025)g Fk(t)751 3132 y Fr(0)828 3117 y Fk(>)g Fl(0)g(b)m(y)h(Theorem)g(3.2,)g(eq.)62 b(\(3.7\))38 b(w)m(e)h(kno)m(w)h(that)e Fk(I)8 b Fl(\()p Fk(u)p Fl(\()p Fk(t)p Fl(\)\))38 b(is)g(b)s(ounded)h(and)75 3237 y(exp)s(onen)m (tially)44 b(decreasing.)81 b(Collecting)43 b(the)j(previous)f (inequalities)e(w)m(e)j(conclude)f(that)g(there)75 3358 y(exists)33 b(a)g(p)s(ositiv)m(e)f(constan)m(t)h(suc)m(h)h(that)1603 3574 y Fk(u)1659 3533 y Fp(m)1725 3574 y Fl(\()p Fk(x)p Fl(\))28 b Fj(\024)g Fk(D)s(e)2118 3533 y Fn(\000)p Fr(2)p Fp(t)2270 3574 y Fk(:)75 3790 y Fl(Analogously)j(for)h Fk(x)c(>)g(A)p Fl(.)44 b(Th)m(us,)34 b(outside)e(the)h(supp)s(ort)g(of) f Fk(u)2414 3805 y Fn(1)2521 3790 y Fl(w)m(e)i(ha)m(v)m(e)1328 4006 y Fj(j)p Fk(u)p Fl(\()p Fk(x;)17 b(t)p Fl(\))22 b Fj(\000)h Fk(u)1800 4021 y Fn(1)1874 4006 y Fl(\()p Fk(x)p Fl(\))p Fj(j)k(\024)i Fk(D)s(e)2295 3965 y Fn(\000)p Fr(2)p Fp(t=m)2545 4006 y Fk(:)1052 b Fl(\(5.1\))75 4222 y(Supp)s(ose)34 b(no)m(w)f(that)f Fj(j)p Fk(x)p Fj(j)983 4186 y Fr(2)1050 4222 y Fj(\024)c Fk(C)7 b Fl(.)44 b(W)-8 b(e)33 b(uses)h(the)f(iden)m(tit)m(y)95 4488 y Fk(u)151 4447 y Fp(m)217 4488 y Fl(\()p Fk(x)p Fl(\))23 b Fj(\000)f Fk(u)526 4447 y Fp(m)526 4513 y Fn(1)601 4488 y Fl(\()p Fk(x)p Fl(\))83 b(=)g Fk(u)1030 4447 y Fp(m)1096 4488 y Fl(\()p Fj(\000)p Fk(A)p Fl(\))22 b(+)1442 4371 y Fo(Z)1526 4397 y Fp(x)1489 4560 y Fn(\000)p Fp(A)1653 4421 y Fk(@)p 1627 4465 109 4 v 1627 4556 a(@)5 b(y)1745 4488 y Fl([)p Fk(u)1828 4447 y Fp(m)1916 4488 y Fj(\000)23 b Fk(u)2072 4447 y Fp(m)2072 4513 y Fn(1)2146 4488 y Fl(])17 b Fk(dy)815 4758 y Fl(=)83 b Fk(u)1030 4717 y Fp(m)1096 4758 y Fl(\()p Fj(\000)p Fk(A)p Fl(\))22 b(+)1442 4641 y Fo(Z)1526 4668 y Fp(x)1489 4830 y Fn(\000)p Fp(A)1617 4758 y Fk(m)1719 4612 y Fo(\()1786 4758 y Fk(u)1842 4717 y Fp(m)p Fn(\000)p Fr(1)2008 4691 y Fk(@)5 b(u)p 2008 4735 113 4 v 2010 4827 a(@)g(y)2153 4758 y Fj(\000)22 b Fk(u)2308 4717 y Fp(m)p Fn(\000)p Fr(1)2308 4783 y Fn(1)2475 4691 y Fk(@)5 b(u)2587 4706 y Fn(1)p 2475 4735 188 4 v 2514 4827 a Fk(@)g(y)2672 4612 y Fo(\))2772 4758 y Fk(dy)815 5041 y Fl(=)83 b Fk(u)1030 5000 y Fp(m)1096 5041 y Fl(\()p Fj(\000)p Fk(A)p Fl(\))22 b(+)1538 4973 y Fk(m)p 1452 5018 256 4 v 1452 5109 a(m)h Fj(\000)g Fl(1)1735 4924 y Fo(Z)1818 4950 y Fp(x)1781 5112 y Fn(\000)p Fp(A)1910 4895 y Fo(\()1976 5041 y Fk(u)2032 4944 y Fo(h)2081 4973 y Fk(@)5 b(u)2193 4937 y Fp(m)p Fn(\000)p Fr(1)p 2081 5018 270 4 v 2162 5109 a Fk(@)g(y)2382 5041 y Fj(\000)2492 4973 y Fk(@)g(u)2604 4937 y Fp(m)p Fn(\000)p Fr(1)2604 4998 y Fn(1)p 2492 5018 V 2573 5109 a Fk(@)g(y)2771 4944 y Fo(i)2832 5041 y Fl(+)22 b(\()p Fk(u)g Fj(\000)h Fk(u)3202 5056 y Fn(1)3276 5041 y Fl(\))3324 4973 y Fk(@)5 b(u)3436 4937 y Fp(m)p Fn(\000)p Fr(1)3436 4998 y Fn(1)p 3324 5018 V 3404 5109 a Fk(@)g(y)3603 4895 y Fo(\))3703 5041 y Fk(dy)1901 5356 y Fl(21)p eop %%Page: 22 22 22 21 bop 75 107 a Fl(Therefore)427 361 y Fj(j)p Fk(u)511 320 y Fp(m)576 361 y Fl(\()p Fk(x)p Fl(\))23 b Fj(\000)f Fk(u)885 376 y Fn(1)960 361 y Fl(\()p Fk(x)p Fl(\))1091 320 y Fp(m)1157 361 y Fj(j)83 b(\024)h(j)p Fk(u)1513 320 y Fp(m)1578 361 y Fl(\()p Fj(\000)p Fk(A)p Fl(\))p Fj(j)23 b Fl(+)2048 294 y Fk(m)p 1963 338 256 4 v 1963 429 a(m)f Fj(\000)h Fl(1)2245 244 y Fo(Z)2328 270 y Fp(A)2291 433 y Fn(\000)p Fp(A)2420 361 y Fk(u)2476 261 y Fo(\014)2476 311 y(\014)2476 361 y(\014)2513 294 y Fk(@)5 b(u)2625 258 y Fp(m)p Fn(\000)p Fr(1)p 2513 338 270 4 v 2591 429 a Fk(@)g(x)2814 361 y Fj(\000)2924 294 y Fk(@)g(u)3036 258 y Fp(m)p Fn(\000)p Fr(1)3036 318 y Fn(1)p 2924 338 V 3002 429 a Fk(@)g(x)3203 261 y Fo(\014)3203 311 y(\014)3203 361 y(\014)16 b Fk(dx)22 b Fl(+)1624 619 y(+)1795 552 y Fk(m)p 1710 596 256 4 v 1710 687 a(m)g Fj(\000)h Fl(1)1992 502 y Fo(Z)2075 528 y Fp(A)2038 691 y Fn(\000)p Fp(A)2167 619 y Fj(j)p Fk(u)e Fj(\000)i Fk(u)2428 634 y Fn(1)2502 619 y Fj(j)2562 519 y Fo(\014)2562 569 y(\014)2562 619 y(\014)2600 552 y Fk(@)5 b(u)2712 516 y Fp(m)p Fn(\000)p Fr(1)2712 576 y Fn(1)p 2600 596 270 4 v 2678 687 a Fk(@)g(x)2879 519 y Fo(\014)2879 569 y(\014)2879 619 y(\014)16 b Fk(dx)75 875 y Fl(No)m(w,)33 b(consider)g(that)1532 981 y Fk(@)5 b(u)1644 945 y Fp(m)p Fn(\000)p Fr(1)1644 1006 y Fn(1)p 1532 1025 V 1610 1117 a Fk(@)g(x)1839 1049 y Fl(=)27 b Fj(\000)2029 981 y Fk(m)c Fj(\000)g Fl(1)p 2029 1025 256 4 v 2115 1117 a Fk(m)2296 1049 y(x:)75 1248 y Fl(Th)m(us,)34 b(applying)e(Cauc)m(h)m(y-Sc)m(h)m(w)m(artz)k(inequalit)m(y)31 b(w)m(e)j(obtain)197 1524 y Fj(j)p Fk(u)281 1483 y Fp(m)346 1524 y Fl(\()p Fk(x)p Fl(\))23 b Fj(\000)f Fk(u)655 1483 y Fp(m)655 1548 y Fn(1)730 1524 y Fl(\()p Fk(x)p Fl(\))p Fj(j)83 b(\024)g(j)p Fk(u)1216 1483 y Fp(m)1282 1524 y Fl(\()p Fj(\000)p Fk(A)p Fl(\))p Fj(j)22 b Fl(+)1656 1403 y Fo(\024)1700 1407 y(Z)1783 1433 y Fp(A)1746 1595 y Fn(\000)p Fp(A)1875 1524 y Fk(u)1931 1427 y Fo(\020)1980 1524 y Fk(x)g Fl(+)2251 1456 y Fk(m)p 2165 1501 V 2165 1592 a(m)h Fj(\000)f Fl(1)2474 1456 y Fk(@)5 b(u)2586 1420 y Fp(m)p Fn(\000)p Fr(1)p 2474 1501 270 4 v 2552 1592 a Fk(@)g(x)2753 1427 y Fo(\021)2802 1450 y Fr(2)2858 1524 y Fk(dx)2964 1403 y Fo(\025)3008 1426 y Fr(1)p Fp(=)p Fr(2)3118 1403 y Fo(\022)3179 1407 y(Z)3262 1433 y Fp(A)3226 1595 y Fn(\000)p Fp(A)3354 1524 y Fk(u)17 b(dx)3533 1403 y Fo(\023)3593 1426 y Fr(1)p Fp(=)p Fr(2)1327 1783 y Fl(+)1403 1662 y Fo(\022)1464 1666 y(Z)1547 1693 y Fp(A)1510 1855 y Fn(\000)p Fp(A)1639 1783 y Fj(j)p Fk(u)k Fj(\000)i Fk(u)1900 1798 y Fn(1)1974 1783 y Fj(j)17 b Fk(dx)2125 1662 y Fo(\023)2186 1685 y Fr(1)p Fp(=)p Fr(2)2296 1662 y Fo(\022)2357 1666 y(Z)2440 1693 y Fp(A)2403 1855 y Fn(\000)p Fp(A)2531 1783 y Fk(x)2586 1742 y Fr(2)2626 1783 y Fj(j)p Fk(u)22 b Fj(\000)g Fk(u)2887 1798 y Fn(1)2961 1783 y Fj(j)17 b Fk(dx)3112 1662 y Fo(\023)3173 1685 y Fr(1)p Fp(=)p Fr(2)75 2045 y Fl(Using)32 b(\(5.1\))g(w)m(e)i(ha)m(v)m (e)691 2265 y Fj(j)p Fk(u)775 2223 y Fp(m)840 2265 y Fl(\()p Fk(x;)17 b(t)p Fl(\))23 b Fj(\000)f Fk(u)1228 2223 y Fp(m)1228 2289 y Fn(1)1303 2265 y Fl(\()p Fk(x)p Fl(\))p Fj(j)27 b(\024)h Fk(D)s(e)1723 2223 y Fn(\000)p Fr(2)p Fp(t)1865 2265 y Fl(+)22 b Fk(I)8 b Fl(\()p Fk(u)p Fl(\))2146 2223 y Fr(1)p Fp(=)p Fr(2)2256 2265 y Fk(M)2360 2223 y Fr(1)p Fp(=)p Fr(2)2493 2265 y Fl(+)22 b Fk(A)2664 2223 y Fr(2)2703 2265 y Fj(k)p Fk(u)g Fj(\000)g Fk(u)2986 2280 y Fn(1)3060 2265 y Fj(k)3110 2280 y Fr(1)3182 2265 y Fk(:)415 b Fl(\(5.2\))75 2485 y(Since)44 b(\()p Fk(x)31 b Fj(\000)f Fk(y)t Fl(\))662 2448 y Fp(m)758 2485 y Fl(+)g Fk(y)916 2448 y Fp(m)1026 2485 y Fl(is)43 b(con)m(v)m(ex)k(for)d Fk(m)j(>)h Fl(1,)f Fk(x;)17 b(y)50 b(>)e Fl(0,)f(\()p Fk(x)30 b Fj(\000)g Fk(y)t Fl(\))2771 2448 y Fp(m)2867 2485 y Fl(+)g Fk(y)3025 2448 y Fp(m)3138 2485 y Fj(\024)48 b Fk(x)3318 2448 y Fp(m)3429 2485 y Fl(whenev)m(er)75 2605 y Fk(x)28 b Fj(\025)g Fk(y)t Fl(.)42 b(Hence,)33 b(if)d Fk(u)d Fj(\025)h Fk(u)1032 2620 y Fn(1)1107 2605 y Fl(,)j(\()p Fk(u)19 b Fj(\000)g Fk(u)1430 2620 y Fn(1)1504 2605 y Fl(\))1542 2569 y Fp(m)1637 2605 y Fj(\024)28 b Fk(u)1798 2569 y Fp(m)1883 2605 y Fj(\000)19 b Fk(u)2035 2569 y Fp(m)2035 2630 y Fn(1)2110 2605 y Fl(.)43 b(The)32 b(same)f(argumen)m(t)f(w)m(orks)j(for)d Fk(u)e Fj(\024)g Fk(u)3724 2620 y Fn(1)3798 2605 y Fl(.)75 2725 y(Therefore)1058 2846 y Fj(j)p Fk(u)22 b Fj(\000)g Fk(u)1319 2861 y Fn(1)1393 2846 y Fj(j)28 b(\024)g(j)p Fk(u)1638 2805 y Fp(m)1726 2846 y Fj(\000)23 b Fk(u)1882 2805 y Fp(m)1882 2870 y Fn(1)1956 2846 y Fj(j)1984 2805 y Fr(1)p Fp(=m)2121 2846 y Fk(;)211 b(t)28 b Fj(\025)g Fk(t)2562 2861 y Fr(0)2630 2846 y Fk(>)f Fl(0)33 b Fk(:)75 3020 y Fl(By)39 b(theorem)e(3.2)h(w)m (e)h(reco)m(v)m(er)h(the)e(sharp)h(deca)m(y)g(of)f Fk(I)8 b Fl(\()p Fk(u)p Fl(\()p Fk(t)p Fl(\)\),)38 b(while)f(theorem)h(4.4)g (furnishes)h(the)75 3140 y(deca)m(y)i(of)d(the)i Fk(L)711 3104 y Fr(1)751 3140 y Fl(-distance.)63 b(Hence,)42 b(substituting)d (in)m(to)f(\(5.2\))h(w)m(e)h(conclude)g(the)g(pro)s(of)e(for)h(the)75 3261 y(initial)29 b(data)j(as)h(in)f(Theorems)h(3.1)f(and)h(3.2.)221 3381 y(T)-8 b(o)38 b(\014nish)f(the)h(pro)s(of,)f(w)m(e)h(use)h(again)c (a)i(densit)m(y)i(argumen)m(t)e(\(see)h([19)o(]\).)58 b(Consider)38 b(an)f(initial)75 3502 y(data)32 b(0)c Fj(\024)g Fk(u)535 3517 y Fr(0)574 3502 y Fl(\()p Fk(x)p Fl(\))g Fj(2)g Fk(L)893 3465 y Fr(1)933 3502 y Fl(\(I)-20 b(R)1058 3460 y Fp(N)1125 3502 y Fl(\))33 b(with)f Fk(H)8 b Fl(\()p Fk(u)1601 3517 y Fr(0)1639 3502 y Fl(\))28 b Fk(<)g Fj(1)k Fl(and)g Fk(I)8 b Fl(\()p Fk(u)2275 3517 y Fr(0)2314 3502 y Fl(\))28 b Fk(<)f Fj(1)p Fl(.)43 b(Let)1469 3731 y Fk(u)1525 3690 y Fp(\017)1525 3756 y Fr(0)1592 3731 y Fl(=)27 b(\()p Fk(!)1794 3746 y Fp(\017)1849 3731 y Fj(\003)22 b Fk(u)1977 3746 y Fr(0)2016 3731 y Fl(\))g(+)g Fk(\017e)2258 3690 y Fn(\000j)p Fp(x)p Fn(j)2393 3666 y Fg(2)75 3951 y Fl(for)29 b(an)m(y)h Fk(\017)f(>)e Fl(0,)j(where)h Fk(!)1019 3966 y Fp(\017)1081 3951 y Fl(is)e(a)h(regularizing)d (sequence.)45 b(No)m(w,)31 b Fk(u)2527 3915 y Fp(\017)2527 3976 y Fr(0)2596 3951 y Fl(satis\014es)f(all)e(the)i(h)m(yp)s(otheses)i (in)75 4071 y(Theorems)27 b(3.1)f(and)g(3.2.)41 b(Therefore,)29 b(there)e(exist)f(a)g(p)s(ositiv)m(e)g(constan)m(t)h Fk(C)34 b Fl(=)28 b Fk(C)23 b Fl(\()p Fk(m;)17 b(H)8 b Fl(\()p Fk(u)3445 4086 y Fr(0)3484 4071 y Fl(\))p Fk(;)17 b(I)8 b Fl(\()p Fk(u)3711 4086 y Fr(0)3749 4071 y Fl(\)\))75 4192 y(suc)m(h)34 b(that)960 4350 y Fj(k)p Fk(u)1066 4308 y Fp(\017)1098 4350 y Fl(\()p Fk(t)p Fl(\))22 b Fj(\000)h Fk(u)1387 4308 y Fp(\017)1387 4374 y Fn(1)1461 4350 y Fj(k)1511 4365 y Fp(L)1559 4346 y Fe(1)1624 4365 y Fr(\()p Fg(I)-10 b(R)q Fr(\))1767 4350 y Fj(\024)28 b Fk(C)c Fl(exp)2131 4203 y Fo(\()2198 4350 y Fj(\000)2392 4282 y Fk(m)e Fl(+)g(1)p 2285 4326 466 4 v 2285 4418 a Fk(m)p Fl(\(3)p Fk(m)h Fj(\000)g Fl(1\))2762 4350 y Fk(t)2797 4203 y Fo(\))2913 4350 y Fk(:)684 b Fl(\(5.3\))75 4586 y(Using)30 b(the)h(H\177)-49 b(older)30 b(estimate)f(\(3.2\))h(in) g(com)m(bination)e(with)i(the)h(uniform)d(b)s(ound)j(\(3.1\))f(w)m(e)h (deduce)75 4707 y(that)37 b(the)h(sequence)i Fk(u)929 4670 y Fp(\017)962 4707 y Fl(\()p Fk(t)p Fl(\))d(is)g(relativ)m(ely)f (compact)h(in)g Fk(C)7 b Fl(\([)p Fk(t)2334 4722 y Fr(1)2374 4707 y Fk(;)17 b(t)2453 4722 y Fr(2)2492 4707 y Fl(])26 b Fj(\002)g Fl(\012\))37 b(for)g(an)m(y)i(0)c Fk(<)h(t)3368 4722 y Fr(1)3444 4707 y Fk(<)g(t)3591 4722 y Fr(2)3668 4707 y Fl(and)75 4827 y(an)m(y)30 b(compact)f(in)m(terv)-5 b(al)27 b(\012.)43 b(It)29 b(is)g(easy)h(to)f(see)h(using)f(the)g (uniqueness)i(of)e(strong)g(solution)f(that)h(the)75 4947 y(limit)k(m)m(ust)k(b)s(e)g(the)g(solution)e(of)h(the)h(equation)g (\(1.3\))f(with)g(initial)d(data)j Fk(u)2961 4962 y Fr(0)3000 4947 y Fl(.)56 b(Therefore,)39 b(taking)75 5068 y(a)32 b(subsequence)k(w)m(e)e(deduce)f(that)g Fk(u)1443 5032 y Fp(\017)1475 5068 y Fl(\()p Fk(t)p Fl(\))22 b Fj(\000)g Fk(u)1763 5032 y Fp(\017)1763 5092 y Fn(1)1870 5068 y Fl(con)m(v)m(erges)34 b(to)m(w)m(ards)g Fk(u)p Fl(\()p Fk(t)p Fl(\))21 b Fj(\000)h Fk(u)3013 5083 y Fn(1)3120 5068 y Fl(in)31 b Fk(C)7 b Fl(\([)p Fk(t)3410 5083 y Fr(1)3450 5068 y Fk(;)17 b(t)3529 5083 y Fr(2)3568 5068 y Fl(])22 b Fj(\002)h Fl(\012\))1901 5356 y(22)p eop %%Page: 23 23 23 22 bop 75 107 a Fl(Using)37 b(that)h(the)g(b)s(ound)h(\(5.3\))e(is)g (uniform)f(in)h Fk(\017)i Fl(w)m(e)g(ha)m(v)m(e)g(that)e Fk(u)2611 71 y Fp(\017)2644 107 y Fl(\()p Fk(t)p Fl(\))25 b Fj(\000)i Fk(u)2940 71 y Fp(\017)2940 131 y Fn(1)3052 107 y Fl(con)m(v)m(erges)40 b(to)m(w)m(ards)75 227 y Fk(u)p Fl(\()p Fk(t)p Fl(\))10 b Fj(\000)g Fk(u)395 242 y Fn(1)495 227 y Fl(in)26 b Fk(C)7 b Fl(\([)p Fk(t)780 242 y Fr(0)820 227 y Fk(;)17 b Fj(1)p Fl(\))10 b Fj(\002)g Fl(\012\))26 b(for)g(an)m(y)h(0)h Fk(<)f(t)1769 242 y Fr(0)1835 227 y Fl(and)g(an)m(y)g(compact)f(in)m(terv)-5 b(al)26 b(\012.)41 b(As)28 b(a)e(consequence,)1017 507 y Fj(k)p Fk(u)p Fl(\()p Fk(t)p Fl(\))c Fj(\000)g Fk(u)1411 522 y Fn(1)1486 507 y Fj(k)1536 523 y Fp(L)1584 504 y Fe(1)1649 523 y Fr(\(\012\))1786 507 y Fj(\024)28 b Fk(C)c Fl(exp)2151 361 y Fo(\()2217 507 y Fj(\000)2411 440 y Fk(m)e Fl(+)g(1)p 2304 484 466 4 v 2304 576 a Fk(m)p Fl(\(3)p Fk(m)h Fj(\000)g Fl(1\))2781 507 y Fk(t)2816 361 y Fo(\))75 782 y Fl(for)39 b(an)m(y)i(compact)f(in)m(terv)-5 b(al)38 b(\012)j(with)e Fk(C)47 b Fl(indep)s(enden)m(t)41 b(of)e(\012.)66 b(Therefore,)43 b(the)e(pro)s(of)e(is)g(\014nished)75 902 y(pro)m(vided)30 b Fk(I)8 b Fl(\()p Fk(u)618 917 y Fr(0)657 902 y Fl(\))27 b Fk(<)h Fj(1)p Fl(.)42 b(T)-8 b(o)29 b(eliminate)e(this)i(h)m(yp)s(othesis)i Fk(I)8 b Fl(\()p Fk(u)2365 917 y Fr(0)2403 902 y Fl(\))28 b Fk(<)f Fj(1)i Fl(it)g(su\016ces)i(to)f(c)m(hec)m(k)h(that)e(for)75 1023 y(an)34 b(initial)c(data)k(suc)m(h)i(that)e(0)c Fj(\024)g Fk(u)1396 1038 y Fr(0)1435 1023 y Fl(\()p Fk(x)p Fl(\))h Fj(2)g Fk(L)1760 986 y Fr(1)1800 1023 y Fl(\(I)-20 b(R\))34 b(and)g Fk(H)8 b Fl(\()p Fk(u)2371 1038 y Fr(0)2410 1023 y Fl(\))30 b Fk(<)g Fj(1)p Fl(,)k(the)h(solution)d(satis\014es)j (that)75 1143 y Fk(I)8 b Fl(\()p Fk(u)p Fl(\()p Fk(t)p Fl(\)\))27 b Fk(<)h Fj(1)k Fl(for)g(an)m(y)h Fk(t)28 b Fj(\025)g Fk(t)1168 1158 y Fr(0)1235 1143 y Fk(>)g Fl(0.)43 b(Therefore,)34 b(y)m(ou)f(apply)f(the)h(previous)h(result)e (to)g Fk(u)p Fl(\()p Fk(t)3455 1158 y Fr(0)3494 1143 y Fl(\).)p 3635 1084 41 4 v 3635 1140 4 56 v 3673 1140 V 3635 1143 41 4 v 75 1475 a Fm(6)161 b(Conclusion)75 1695 y Fl(The)28 b(relations)d(\(2.7\))i(and)g(\(2.8\))f(and)h(the)h (results)f(of)g(the)g(previous)g(section)h(on)e(the)i(con)m(v)m (ergence)h(of)75 1815 y(the)34 b(solution)d(to)i(the)g(F)-8 b(okk)m(er-Planc)m(k)34 b(equation)f(\(1.3\))g(to)m(w)m(ards)h(the)f (unique)h(stationary)e(solution)75 1935 y(giv)m(e)43 b(a)h(description)f(of)f(the)i(asymptotic)f(b)s(eha)m(viour)g(of)g(the) h(solution)e(to)h(the)h(p)s(orous)g(medium)75 2056 y(equation)32 b(\(1.1\).)43 b(W)-8 b(e)33 b(ha)m(v)m(e)75 2281 y Fd(Theorem)k(6.1)49 b Fc(Assume)33 b(that)g Fk(v)j Fc(is)d(a)f(nonne)-5 b(gative)31 b(solution)i(of)f(\(1.1\))g(in)g Fk(L)3006 2245 y Fr(1)3046 2281 y Fl(\(I)-20 b(R)3171 2240 y Fp(N)3238 2281 y Fl(\))p Fc(,)33 b(c)-5 b(orr)g(esp)g(ond-)75 2402 y(ing)32 b(to)h(a)f(nonne)-5 b(gative)32 b(initial)g(datum)g Fk(v)1595 2417 y Fr(0)1667 2402 y Fc(of)h(b)-5 b(ounde)g(d)32 b(mass)f Fk(M)44 b Fc(and)32 b(entr)-5 b(opy)32 b Fk(H)8 b Fl(\()p Fk(v)3236 2417 y Fr(0)3276 2402 y Fl(\))p Fc(,)32 b(for)h Fk(m)28 b(>)f Fl(1)p Fc(.)75 2522 y(Then,)42 b(if)g Fk(M)10 b(V)22 b Fl(\()p Fj(j)p Fk(x)p Fj(j)p Fk(;)17 b(t)p Fl(\))41 b Fc(is)h(the)f(Bar)-5 b(enblatt-Pattle)42 b(pr)-5 b(o\014le)41 b(of)g(mass)g Fk(M)10 b Fc(,)44 b(ther)-5 b(e)42 b(exists)f(a)g(c)-5 b(onstant)75 2643 y Fk(C)35 b Fl(=)27 b Fk(C)7 b Fl(\()p Fk(m;)17 b(H)8 b Fl(\()p Fk(v)701 2658 y Fr(0)740 2643 y Fl(\)\))35 b Fc(such)g(that)717 2946 y Fj(k)p Fk(v)t Fl(\()p Fk(x;)17 b(t)p Fl(\))22 b Fj(\000)h Fk(M)10 b(V)22 b Fl(\()p Fj(j)p Fk(x)p Fj(j)p Fk(;)17 b(t)p Fl(\))p Fj(k)1649 2965 y Fp(L)1697 2946 y Fg(1)1731 2965 y Fr(\()p Fg(I)-10 b(R)1815 2946 y Ff(N)1872 2965 y Fr(\))1932 2946 y Fj(\024)28 b Fk(C)2130 2800 y Fo( )2373 2879 y Fl(1)p 2206 2923 383 4 v 2206 3015 a(1)22 b(+)g(\()p Fk(t=k)s Fl(\))2599 2800 y Fo(!)2665 2823 y Fr(1)p Fp(=)p Fr(\(\()p Fp(N)7 b Fr(+2\))p Fp(m)p Fn(\000)p Fp(N)g Fr(\))221 3292 y Fl(The)45 b(case)f Fk(N)57 b Fl(=)46 b(1)d(giv)m(es)h(stronger)g(con)m(v)m(ergence)i(results,)g(in)d(that)g (w)m(e)i(obtain)d(the)i(rate)f(of)75 3413 y(con)m(v)m(ergence)35 b(b)s(oth)e(in)e Fk(L)1024 3377 y Fr(1)1097 3413 y Fl(and)h Fk(L)1352 3377 y Fn(1)1427 3413 y Fl(.)44 b(W)-8 b(e)33 b(ha)m(v)m(e)75 3639 y Fd(Theorem)k(6.2)49 b Fc(Assume)30 b(that)h Fk(v)j Fc(is)29 b(a)h(nonne)-5 b(gative)29 b(solution)h(of)g (\(1.1\))f(in)h Fk(L)2981 3603 y Fr(1)3021 3639 y Fl(\(I)-20 b(R\))p Fc(,)31 b(c)-5 b(orr)g(esp)g(onding)75 3759 y(to)42 b(a)g(nonne)-5 b(gative)41 b(initial)h(datum)g Fk(v)1481 3774 y Fr(0)1563 3759 y Fc(of)g(b)-5 b(ounde)g(d)41 b(mass)g Fk(M)53 b Fc(and)42 b(entr)-5 b(opy)42 b Fk(H)8 b Fl(\()p Fk(v)3189 3774 y Fr(0)3228 3759 y Fl(\))p Fc(,)44 b(for)e Fk(m)f(>)g Fl(1)p Fc(.)75 3879 y(Then,)h(if)g Fk(M)10 b(V)22 b Fl(\()p Fj(j)p Fk(x)p Fj(j)p Fk(;)17 b(t)p Fl(\))41 b Fc(is)h(the)f(Bar)-5 b(enblatt-Pattle)42 b(pr)-5 b(o\014le)41 b(of)g(mass)g Fk(M)10 b Fc(,)44 b(ther)-5 b(e)42 b(exists)f(a)g(c)-5 b(onstant)75 4000 y Fk(C)35 b Fl(=)27 b Fk(C)7 b Fl(\()p Fk(m;)17 b(H)8 b Fl(\()p Fk(v)701 4015 y Fr(0)740 4000 y Fl(\)\))35 b Fc(such)g(that)771 4303 y Fj(k)p Fk(v)t Fl(\()p Fk(x;)17 b(t)p Fl(\))22 b Fj(\000)g Fk(M)10 b(V)23 b Fl(\()p Fj(j)p Fk(x)p Fj(j)p Fk(;)17 b(t)p Fl(\))p Fj(k)1703 4320 y Fp(L)1751 4301 y Fg(1)1785 4320 y Fr(\()p Fg(I)-10 b(R)p Fr(\))1927 4303 y Fj(\024)29 b Fk(C)2126 4157 y Fo( )2445 4236 y Fl(1)p 2202 4280 535 4 v 2202 4372 a(1)22 b(+)g(\()p Fk(m)g Fl(+)g(1\))p Fk(t)2746 4157 y Fo(!)2812 4180 y Fr(1)p Fp(=)p Fr(\(3)p Fp(m)p Fn(\000)p Fr(1\))75 4584 y Fc(Mor)-5 b(e)g(over,)54 b(for)d(al)5 b(l)50 b Fk(t)57 b Fj(\025)g Fk(t)1141 4599 y Fr(0)1237 4584 y Fk(>)g Fl(0)p Fc(,)d Fk(v)t Fl(\()p Fk(x;)17 b(t)p Fl(\))57 b Fj(2)g Fk(L)2010 4547 y Fn(1)2085 4584 y Fl(\(I)-20 b(R\))p Fc(,)55 b(and)50 b(ther)-5 b(e)50 b(exists)g(a)h(c)-5 b(onstant)50 b Fk(C)3653 4599 y Fr(1)3749 4584 y Fl(=)75 4704 y Fk(C)145 4719 y Fr(1)201 4704 y Fl(\()p Fk(m;)17 b(H)8 b Fl(\()p Fk(v)542 4719 y Fr(0)581 4704 y Fl(\))p Fk(;)17 b(I)8 b Fl(\()p Fk(u)808 4719 y Fp(t)833 4728 y Fg(0)871 4704 y Fl(\)\))35 b Fc(such)f(that)253 5008 y Fl([1)22 b(+)g(\()p Fk(m)h Fl(+)f(1\))p Fk(t)p Fl(])842 4966 y Fr(1)p Fp(=)p Fr(\()p Fp(m)p Fr(+1\))1124 5008 y Fj(k)p Fk(v)t Fl(\()p Fk(x;)17 b(t)p Fl(\))22 b Fj(\000)g Fk(M)10 b(V)23 b Fl(\()p Fj(j)p Fk(x)p Fj(j)p Fk(;)17 b(t)p Fl(\))p Fj(k)2056 5023 y Fp(L)2104 5004 y Fe(1)2168 5023 y Fr(\()p Fg(I)-10 b(R)q Fr(\))2311 5008 y Fj(\024)28 b Fk(C)2486 5023 y Fr(1)2542 4861 y Fo( )2860 4940 y Fl(1)p 2618 4984 V 2618 5076 a(1)22 b(+)g(\()p Fk(m)g Fl(+)g(1\))p Fk(t)3162 4861 y Fo(!)3228 4884 y Fr(1)p Fp(=)p Fr([)p Fp(m)p Fr(\(3)p Fp(m)p Fn(\000)p Fr(1\)])1901 5356 y Fl(23)p eop %%Page: 24 24 24 23 bop 221 227 a Fl(Next,)33 b(theorem)f(3.2,)f(whic)m(h)i(giv)m(es) f(the)g(exp)s(onen)m(tial)f(deca)m(y)j(of)d(the)h(en)m(trop)m(y)h(pro)s (duction,)f(can)75 348 y(b)s(e)h(translated)f(in)m(to)g(a)g(deca)m(y)i (result)f(for)f(the)h(di\013erence)g(b)s(et)m(w)m(een)i(the)e(gradien)m (ts)f(of)g(the)h(solution)75 468 y(and)j(of)f(the)h(Baren)m(blatt-P)m (attle)e(pro\014le.)52 b(One)36 b(has)g(only)f(to)h(remark)f(that,)h (whenev)m(er)i Fk(x)e Fl(b)s(elongs)75 588 y(to)c(the)h(supp)s(ort)g (of)f Fk(u)890 603 y Fn(1)965 588 y Fl(,)1519 726 y Fk(x)c Fl(=)f Fj(\000)1878 659 y Fk(m)p 1792 703 256 4 v 1792 795 a(m)c Fj(\000)g Fl(1)2059 726 y Fj(r)p Fk(u)2198 685 y Fp(m)p Fn(\000)p Fr(1)2198 751 y Fn(1)2354 726 y Fk(:)1243 b Fl(\(6.4\))75 925 y(Hence)666 928 y Fo(Z)712 1117 y Fp(S)t(upp)p Fn(f)p Fp(u)948 1125 y Fe(1)1012 1117 y Fr(\()p Fn(j)p Fp(x)p Fn(j)p Fr(\))p Fn(g)1202 1045 y Fk(u)p Fl(\()p Fk(x;)17 b(t)p Fl(\))1468 946 y Fo(\014)1468 995 y(\014)1468 1045 y(\014)o Fj(r)p Fl(\()p Fk(u)1672 1004 y Fp(m)p Fn(\000)p Fr(1)1829 1045 y Fl(\()p Fk(x;)g(t)p Fl(\))22 b Fj(\000)h Fk(u)2217 1004 y Fp(m)p Fn(\000)p Fr(1)2217 1070 y Fn(1)2373 1045 y Fl(\()p Fk(x)p Fl(\)\))2542 946 y Fo(\014)2542 995 y(\014)2542 1045 y(\014)2570 972 y Fr(2)2626 1045 y Fk(dx)32 b(;)17 b Fj(\024)28 b Fk(I)8 b Fl(\()p Fk(u)p Fl(\()p Fk(t)p Fl(\)\))p Fk(:)390 b Fl(\(6.5\))75 1262 y(Applying)32 b(\(2.8\),)g(w)m(e)h (obtain)75 1468 y Fd(Theorem)k(6.3)49 b Fc(Assume)33 b(that)g Fk(v)j Fc(is)d(a)f(nonne)-5 b(gative)31 b(solution)i(of)f (\(1.1\))g(in)g Fk(L)3006 1432 y Fr(1)3046 1468 y Fl(\(I)-20 b(R)3171 1426 y Fp(N)3238 1468 y Fl(\))p Fc(,)33 b(c)-5 b(orr)g(esp)g(ond-)75 1588 y(ing)32 b(to)h(a)f(nonne)-5 b(gative)32 b(initial)g(datum)g Fk(v)1595 1603 y Fr(0)1667 1588 y Fc(of)h(b)-5 b(ounde)g(d)32 b(mass)f Fk(M)44 b Fc(and)32 b(entr)-5 b(opy)32 b Fk(H)8 b Fl(\()p Fk(v)3236 1603 y Fr(0)3276 1588 y Fl(\))p Fc(,)32 b(for)h Fk(m)28 b(>)f Fl(1)p Fc(.)75 1709 y(Then,)42 b(if)g Fk(M)10 b(V)22 b Fl(\()p Fj(j)p Fk(x)p Fj(j)p Fk(;)17 b(t)p Fl(\))41 b Fc(is)h(the)f(Bar)-5 b(enblatt-Pattle)42 b(pr)-5 b(o\014le)41 b(of)g(mass)g Fk(M)10 b Fc(,)44 b(ther)-5 b(e)42 b(exists)f(a)g(c)-5 b(onstant)75 1829 y Fk(C)35 b Fl(=)27 b Fk(C)7 b Fl(\()p Fk(m;)17 b(H)8 b Fl(\()p Fk(v)701 1844 y Fr(0)740 1829 y Fl(\)\))35 b Fc(such)g(that)g(for)f(al)5 b(l)35 b Fk(t)28 b Fj(\025)g Fk(t)1770 1844 y Fr(0)1837 1829 y Fk(>)g Fl(0)287 1970 y Fo(Z)333 2159 y Fp(S)t(upp)p Fn(f)p Fp(M)7 b(V)14 b Fr(\()p Fn(j)p Fp(x)p Fn(j)p Fp(;t)p Fr(\))p Fn(g)893 2087 y Fk(v)t Fl(\()p Fk(x;)j(t)p Fl(\))1154 1988 y Fo(\014)1154 2038 y(\014)1154 2087 y(\014)p Fj(r)p Fl(\()p Fk(v)1354 2046 y Fp(m)p Fn(\000)p Fr(1)1510 2087 y Fl(\()p Fk(x;)g(t)p Fl(\))23 b Fj(\000)f Fk(M)10 b(V)2026 2046 y Fp(m)p Fn(\000)p Fr(1)2183 2087 y Fl(\()p Fj(j)p Fk(x)p Fj(j)p Fk(;)17 b(t)p Fl(\)\))2486 1988 y Fo(\014)2486 2038 y(\014)2486 2087 y(\014)2514 2014 y Fr(2)2570 2087 y Fk(dx)35 b(;)17 b Fj(\024)28 b Fk(C)2953 1966 y Fo(\022)3014 2087 y Fl(1)22 b(+)3203 2020 y Fk(t)p 3193 2064 55 4 v 3193 2156 a(k)3257 1966 y Fo(\023)3318 1985 y Fn(\000)p Fr(2)3624 2087 y Fl(\(6.6\))221 2338 y(Theorem)k(6.1)f(is)g(the)h (\014rst)g(result,)h(as)e(far)g(as)h(w)m(e)g(are)f(concerned,)k(ab)s (out)c(general)g(rates)h(of)f(deca)m(y)75 2459 y(of)34 b(solutions)g(for)g(the)h(p)s(orous)g(medium)e(equation)i(in)f Fk(N)41 b(>)32 b Fl(1.)49 b(Theorem)35 b(6.2)g(impro)m(v)m(es)f (previous)75 2579 y(results)45 b(of)f(Kamin)f([7],)48 b([8)o(],)g(as)d(far)f(as)h(the)g(asymptotic)f(b)s(eha)m(viour)g(in)g Fk(L)2982 2543 y Fn(1)3057 2579 y Fl(\(I)-20 b(R)q(\))44 b(is)g(concerned.)75 2700 y(In)c([1,)g(18,)g(19])g(the)g(rate)g(of)g (con)m(v)m(ergence)i(in)e Fk(L)1913 2663 y Fn(1)1988 2700 y Fl(\(I)-20 b(R\))40 b(has)h(b)s(een)f(studied)h(for)e(initial)e (data)i(with)75 2820 y(compact)34 b(supp)s(ort.)51 b(In)35 b(this)g(case)h(one)f(obtain)f(b)s(etter)h(rates)g(of)g(con)m(v)m (ergence)i(but)e(the)h(constan)m(ts)75 2940 y(dep)s(end)48 b(on)e(the)h(supp)s(ort)g(of)f(the)h(solution.)84 b(Also,)50 b(the)d(sp)s(ecial)e(case)j(of)e(radially)e(symmetric)75 3061 y(compact)29 b(supp)s(ort)h(solutions)e(is)h(studied)h(and)f(it)f (is)h(pro)m(v)m(ed)i(that)e(the)h(rates)g(obtained)e(in)h(this)g(case) 75 3181 y(are)h(optimal.)40 b(Again,)30 b(the)h(constan)m(ts)g(on)f (the)h(estimates)f(dep)s(end)h(on)g(the)f(symmetry)h(h)m(yp)s(othesis.) 75 3301 y(F)-8 b(rom)24 b(the)i(rate)g(of)f(con)m(v)m(ergence)k(in)c Fk(L)1489 3265 y Fn(1)1564 3301 y Fl(\(I)-20 b(R\))26 b(and)g(taking)f(in)m(to)g(accoun)m(t)h(that)g(the)g(supp)s(ort)g (remains)75 3422 y(compact)g(for)h(all)d(times)i(one)h(obtains)g (directly)f(rates)h(of)f(con)m(v)m(ergence)k(in)c Fk(L)2892 3386 y Fr(1)2932 3422 y Fl(\(I)-20 b(R\).)41 b(The)28 b(situation)d(is)75 3542 y(summarized)i(in)g(table)g(6,)h(in)f(whic)m (h)i(w)m(e)f(write)g(the)g(orders)h(of)e(con)m(v)m(ergence)j Fk(a)e Fl(in)f(the)h(cases)i(studied)75 3663 y(in)42 b([1,)h(18,)g(19)o(])g(and)g(Theorem)g(6.2)g(in)f(the)h(form)f Fk(t)2048 3626 y Fn(\000)p Fp(a)2145 3663 y Fl(.)74 b(Our)43 b(rates)h(of)e(con)m(v)m(ergence)k(are)d(w)m(orser)75 3783 y(than)38 b(the)h(rates)f(for)g(compact)f(supp)s(ort)i(initial)34 b(data)k(but)g(are)g(v)-5 b(alid)36 b(for)i(non)g(compact)g(supp)s(ort) 75 3903 y(initial)e(data)k(and)h(w)m(e)g(reco)m(v)m(er)h(the)f(same)f (b)s(eha)m(vior)g(for)g(the)g(heat)h(equation)f(when)h Fk(m)g Fj(!)g Fl(1.)66 b(It)75 4024 y(is)36 b(in)m(teresting)g(to)h (remark)f(that)g(in)g(this)g(case)i(only)-8 b(,)37 b(the)g(rate)g(of)f (asymptotic)g(deca)m(y)i(in)e Fk(L)3555 3988 y Fr(1)3594 4024 y Fl(\(I)-20 b(R)3720 3982 y Fp(N)3787 4024 y Fl(\))75 4144 y(to)m(w)m(ards)34 b(the)f(similarit)m(y)c(solution)i(is)h(indep)s (enden)m(t)i(of)e(the)h(dimension)e Fk(N)10 b Fl(.)221 4265 y(Our)45 b(analysis)g(extends)i(to)d(co)m(v)m(er)j(the)e(case)2032 4225 y Fp(N)p 1987 4241 154 4 v 1987 4299 a(N)7 b Fr(+2)2200 4265 y Fk(<)48 b(m)i(<)e Fl(1,)g(since)e(in)e(this)h(range)g(of)g(the) 75 4385 y(parameter,)51 b(one)c(can)h(easily)e(sho)m(w)j(that)e(the)h (en)m(trop)m(y)h(of)d Fk(u)2474 4400 y Fn(1)2596 4385 y Fl(is)h(b)s(ounded.)88 b(If)48 b Fk(N)63 b Fl(=)53 b(1)47 b(and)75 4505 y Fj(\000)162 4466 y Fr(1)p 162 4482 36 4 v 162 4540 a(2)250 4505 y Fk(<)c(m)g Fj(\024)626 4466 y Fr(1)p 626 4482 V 626 4540 a(3)672 4505 y Fl(,)g(the)f(F)-8 b(okk)m(er-Planc)m(k)42 b(t)m(yp)s(e)g(equation)f(admits)g(a)g(unique)g (stationary)g(state)h Fk(u)3724 4520 y Fn(1)3798 4505 y Fl(,)75 4626 y(but)37 b(w)m(e)g(can)g(no)f(more)g(apply)g(our)h(en)m (trop)m(y-en)m(trop)m(y)h(metho)s(d)e(in)g(this)g(general)g(form,)g (since)h(no)m(w)75 4746 y(\(for)f(example\))g(the)i(energy)f(of)g(the)g (stationary)f(state)h(is)g(un)m(b)s(ounded.)57 b(On)37 b(the)g(other)g(hand,)h(for)75 4866 y(p)s(erturbations)31 b(of)g(the)h(steady)h(state)f(suc)m(h)h(that)f(the)g(relativ)m(e)f(en)m (trop)m(y)i(is)e(b)s(ounded,)i(the)f(analysis)75 4987 y(still)k(w)m(ork.)60 b(This)38 b(sho)m(ws)i(that)e(when)h Fk(m)e(<)f Fl(1)p Fk(=)p Fl(2,)j(the)f(domain)f(of)g(attraction)g(of)g (the)i(stationary)75 5107 y(solution)31 b(is)h(giv)m(en)h(only)f(b)m(y) h(the)g(class)g(of)f(functions)h(that)f(ha)m(v)m(e)i(a)e(suitable)g (deca)m(y)i(at)e(in\014nit)m(y)-8 b(.)1901 5356 y(24)p eop %%Page: 25 25 25 24 bop 75 159 a Fl(T)-8 b(able)41 b(1:)60 b(Comparison)40 b(of)g(rates)i(of)e(con)m(v)m(ergence)k(in)c(1-D.)g(RS)h(is)f(radially) f(symmetric.)68 b(CS)41 b(is)75 280 y(compact)32 b(supp)s(ort.)p 456 419 2988 4 v 1017 503 a(RSCS)208 b(CS)33 b(1)27 b Fk(<)h(m)g(<)f Fl(2)139 b(CS)33 b Fk(m)28 b(>)f Fl(2)179 b(Theorem)33 b(6.2)p 456 542 V 524 747 a Fk(L)590 711 y Fr(1)629 747 y Fl(\(I)-20 b(R)q(\))325 b(1)1758 708 y Fr(1)p 1700 724 153 4 v 1700 781 a Fp(m)p Fr(+1)2409 708 y(1)p 2333 724 188 4 v 2333 781 a Fp(m)2395 762 y Fg(2)2430 781 y Fn(\000)p Fr(1)3068 708 y(1)p 2991 724 V 2991 781 a(3)p Fp(m)p Fn(\000)p Fr(1)p 456 912 2988 4 v 506 1117 a Fk(L)572 1081 y Fn(1)647 1117 y Fl(\(I)-20 b(R\))979 1077 y Fr(1)p 920 1093 153 4 v 920 1151 a Fp(m)p Fr(+1)1105 1117 y Fl(+)1271 1077 y Fr(1)p 1213 1093 V 1213 1151 a Fp(m)p Fn(\000)p Fr(1)1758 1077 y(2)p 1700 1093 V 1700 1151 a Fp(m)p Fr(+1)2246 1077 y(1)p 2187 1093 V 2187 1151 a Fp(m)p Fr(+1)2372 1117 y Fl(+)2555 1077 y Fr(1)p 2480 1093 188 4 v 2480 1151 a Fp(m)2542 1132 y Fg(2)2577 1151 y Fn(\000)p Fr(1)2845 1077 y(1)p 2786 1093 153 4 v 2786 1151 a Fp(m)p Fr(+1)2971 1117 y Fl(+)3214 1077 y Fr(1)p 3079 1093 306 4 v 3079 1151 a Fp(m)p Fr(\(3)p Fp(m)p Fn(\000)p Fr(1\))p 456 1291 2988 4 v 75 1785 a Fd(Ac)m(kno)m(wledgemen)m(ts)221 1906 y Fl(The)29 b(\014rst)g(author)f(thanks)h(Dipartimen)m(to)d(di)h (Matematica)g(di)g(Roma)g("La)h(Sapienza")g(and)g(Di-)75 2026 y(partimen)m(to)i(di)g(Matematica)g(di)h(P)m(a)m(via)g(for)g (their)f(hospitalit)m(y)-8 b(.)41 b(P)m(artially)30 b(supp)s(orted)i(b) m(y)g(GNFM-)75 2146 y(CNR)h(\(Italy\),)e(TMR)i(Europ)s(ean)g(Union)e (ERB)i(FMBX-CT97-0157)e(and)h(DGES)g(\(Spain\))f(PB95-)75 2267 y(1203.)75 2720 y Fm(References)124 2939 y Fl([1])49 b(Aronson,)36 b(D.)g(G.,)g(V\023)-49 b(azquez,)38 b(J.)d(L.,)i Fc(Eventual)g Fk(C)2168 2903 y Fn(1)2243 2939 y Fc(-r)-5 b(e)g(gularity)38 b(and)f(c)-5 b(onc)g(avity)37 b(for)h(\015ows)f(in) 276 3059 y(one-dimensional)32 b(p)-5 b(or)g(ous)34 b(me)-5 b(dia)p Fl(,)32 b(Arc)m(h.)i(Rat.)e(Mec)m(h.)i(Anal.,)e Fd(99)p Fl(,)h(329-348)d(\(1987\).)124 3263 y([2])49 b(Arnold,)e(A.,)i(Mark)m(o)m(wic)m(h,)h(P)-8 b(.,)49 b(T)-8 b(oscani,)48 b(G.,)g(Un)m(terreiter,)h(A.,)g Fc(On)d(lo)-5 b(garithmic)45 b(Sob)-5 b(olev)276 3383 y(ine)g(qualities,)31 b(Csiszar-Kul)5 b(lb)-5 b(ack)31 b(ine)-5 b(qualities)32 b(and)f(the)i(r)-5 b(ate)32 b(of)g(c)-5 b(onver)g(genc)g(e)31 b(to)i(e)-5 b(quilibrium)276 3504 y(for)34 b(F)-7 b(okker-Planck)33 b(typ)-5 b(e)35 b(e)-5 b(quations)p Fl(,)32 b(preprin)m(t.)124 3707 y([3])49 b(Carrillo,)25 b(J.)j(A.,)h(T)-8 b(oscani,)29 b(G.,)f Fc(Exp)-5 b(onential)29 b(c)-5 b(onver)g(genc)g(e)29 b(towar)-5 b(d)30 b(e)-5 b(quilibrium)30 b(for)g(homo)-5 b(ge-)276 3827 y(ne)g(ous)34 b(F)-7 b(okker-Planck)33 b(typ)-5 b(e)35 b(e)-5 b(quations)p Fl(,)32 b(Math.)h(Meth.)g(Appl.)g (Sci.,)f Fd(21)p Fl(,)h(1269-1286)d(\(1998\).)124 4031 y([4])49 b(I.)38 b(Csiszar.)61 b Fc(Information-typ)-5 b(e)39 b(me)-5 b(asur)g(es)39 b(of)h(di\013er)-5 b(enc)g(e)39 b(of)h(pr)-5 b(ob)g(ability)40 b(distributions)g(and)276 4151 y(indir)-5 b(e)g(ct)34 b(observations)p Fl(.)d(Stud.)i(Sci.)g (Math.)g(Hung.,)g Fd(2)f Fl(:)44 b(299{318)31 b(,)h(1967.)124 4355 y([5])49 b(F)-8 b(riedman,)23 b(A.,)i(Kamin,)e(S.,)j Fc(The)f(asymptotic)h(b)-5 b(ehavior)25 b(of)h(gas)f(in)h(an)g (N-dimensional)e(p)-5 b(or)g(ous)276 4475 y(me)g(dium)p Fl(,)31 b(T)-8 b(rans.)34 b(Amer.)e(Math.)h(So)s(c.,)g Fd(262)p Fl(,)f(551-563)f(\(1980\).)124 4678 y([6])49 b(Gross,)30 b(L.)f Fc(L)-5 b(o)g(garithmic)31 b(Sob)-5 b(olev)31 b(ine)-5 b(qualities)p Fl(,)29 b(Amer.)g(J.)h(of)f(Math.,)h Fd(97)g Fl(1061{1083)d(\(1975\).)124 4882 y([7])49 b(Kamin,)28 b(S.,)j Fc(The)g(asymptotic)h(b)-5 b(ehaviour)32 b(of)g(the)g(solution) g(of)g(the)g(\014ltr)-5 b(ation)32 b(e)-5 b(quation)p Fl(,)30 b(Israel)276 5002 y(J.)i(Math.,)h Fd(14)p Fl(,)g(76-87)e (\(1973\).)1901 5356 y(25)p eop %%Page: 26 26 26 25 bop 124 107 a Fl([8])49 b(Kamin,)31 b(S.,)j Fc(Similar)h (solutions)g(and)g(the)g(asymptotics)g(of)h(\014ltr)-5 b(ation)35 b(e)-5 b(quations)p Fl(,)33 b(Arc)m(h.)h(Rat.)276 227 y(Mec)m(h.)g(Anal.,)e Fd(60)p Fl(,)g(171-183)f(\(1976\).)124 431 y([9])49 b(Kamin,)30 b(S.,)i(V\023)-49 b(azquez,)34 b(J.)e(L.,)h Fc(F)-7 b(undamental)32 b(solutions)i(and)g(asymptotic)g (b)-5 b(ehaviour)34 b(for)g(the)276 551 y(p-L)-5 b(aplacian)33 b(e)-5 b(quation)p Fl(,)32 b(Rev.)h(Mat.)g(Ib)s(eroamericana,)e Fd(4)p Fl(,)i(339-354)e(\(1988\).)75 754 y([10])49 b(Kullbac)m(k,)39 b(S.)61 b Fc(A)41 b(lower)f(b)-5 b(ound)39 b(for)i(discrimination)d (information)i(in)g(terms)g(of)g(variation.)276 875 y Fl(IEEE)33 b(T)-8 b(rans.)34 b(Inf.)e(The.,)i Fd(4)k Fl(:)43 b(126{127,)64 b(1967.)75 1078 y([11])49 b(Newman,)41 b(W.J.)f Fc(A)h(Liapunov)g(functional)f(for)h(the)g(evolution)g(of)g (solutions)g(to)g(the)g(p)-5 b(or)g(ous)276 1199 y(me)g(dium)34 b(e)-5 b(quation)34 b(to)h(self-similarity,)f(I)p Fl(,)e(J.)h(Math.)g (Ph)m(ys.,)h Fd(25)p Fl(,)f(3120-3123)d(\(1984\).)75 1402 y([12])49 b(Ralston,)k(J.,)i Fc(A)c(Lyapunov)g(functional)f(for)h (the)g(evolution)g(of)f(solutions)h(to)g(the)g(p)-5 b(or)g(ous)276 1522 y(me)g(dium)34 b(e)-5 b(quation)34 b(to)h(self-similarity,)f(II)p Fl(,)e(J.)h(Math.)f(Ph)m(ys.,)j Fd(25)p Fl(,)e(3124-3127)d(\(1984\).)75 1726 y([13])49 b(Samarskii,)29 b(A.)i(A.,)g(Galaktiono)m(v,)f(V.)h(A.,) g(Kurdyumo)m(v,)h(S.)f(P)-8 b(.,)32 b(Mikhailo)m(v,)d(A.)i(P)-8 b(.,)32 b Fc(Blow-up)276 1846 y(in)i(quasiline)-5 b(ar)34 b(p)-5 b(ar)g(ab)g(olic)34 b(e)-5 b(quations)p Fl(,)32 b(W)-8 b(alter)32 b(de)h(Gruyter,)g Fd(19)p Fl(,)f(Berlin-New)g(Y)-8 b(ork,)33 b(1995.)75 2050 y([14])49 b(T)-8 b(oscani,)26 b(G.,)f Fc(Kinetic)i(appr)-5 b(o)g(ach)27 b(to)g(the)h(asymptotic)f(b) -5 b(ehaviour)27 b(of)g(the)g(solution)g(to)h(di\013usion)276 2170 y(e)-5 b(quations)p Fl(,)32 b(Rend.)h(Mat.)g(Serie)f(VI)s(I,)h Fd(16)p Fl(,)g(329-346)d(\(1996\).)75 2373 y([15])49 b(T)-8 b(oscani,)29 b(G.)36 b(Sur)28 b(l'in)m(\023)-46 b(egalit)m(\023)g(e)26 b(logarithmique)f(de)k(Sob)s(olev)f(.)g Fc(C.R.)j(A)-5 b(c)g(ad.)30 b(Sc.)h(Paris)g Fl(,)e Fd(A)j(324)276 2494 y Fl(s)m(\023)-46 b(erie)32 b(1)h(689-694,)d(1997)75 2697 y([16])49 b(T)-8 b(oscani,)38 b(G.,)f Fc(Entr)-5 b(opy)39 b(pr)-5 b(o)g(duction)39 b(and)f(the)h(r)-5 b(ate)39 b(of)g(c)-5 b(onver)g(genc)g(e)37 b(to)i(e)-5 b(quilibrium)38 b(for)h(the)276 2817 y(F)-7 b(okker-Planck)32 b(e)-5 b(quation)p Fl(,)32 b(to)h(app)s(ear)f(in)g(Quart.)g(Appl.)g (Math.)75 3021 y([17])49 b(T)-8 b(routman)23 b(J.)h(L.,)h Fc(V)-7 b(ariational)26 b(c)-5 b(alculus)27 b(with)f(elementary)h(c)-5 b(onvexity)p Fl(,)25 b(Springer)e(V)-8 b(erlag)23 b(New)276 3141 y(Y)-8 b(ork)32 b(\(1983\).)75 3345 y([18])49 b(V\023)-49 b(azquez,)78 b(J.)68 b(L.,)77 b Fc(Asymptotic)68 b(b)-5 b(ehavior)66 b(and)h(pr)-5 b(op)g(agation)66 b(pr)-5 b(op)g(erties)67 b(of)g(the)h(one-)276 3465 y(dimensional)32 b(\015ow)j(of)g(a)f(gas)h(in)f(a)h(p)-5 b(or)g(ous)35 b(me)-5 b(dium)p Fl(,)32 b(T)-8 b(rans.)33 b(AMS,)g Fd(277)p Fl(,)g(507-527)e(\(1983\).)75 3668 y([19])49 b(V\023)-49 b(azquez,)37 b(J.)e(L.,)h Fc(Asymptotic)h(b)-5 b(ehaviour)37 b(for)g(the)g(p)-5 b(or)g(ous)37 b(me)-5 b(dium)36 b(e)-5 b(quation)37 b(in)g(the)g(whole)276 3789 y(sp)-5 b(ac)g(e)p Fl(,)28 b(Notas)g(del)g(curso)h(de)g(do)s(ctorado)e(M)m(\023)-46 b(eto)s(dos)29 b(asin)m(t\023)-49 b(oticos)28 b(en)g(ecuaciones)i(de)e (ev)m(oluci\023)-49 b(on.)1901 5356 y(26)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------9811301801365--