Content-Type: multipart/mixed; boundary="-------------0005120326675" This is a multi-part message in MIME format. ---------------0005120326675 Content-Type: text/plain; name="00-222.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-222.keywords" marginal stability, semilinear PDE's ---------------0005120326675 Content-Type: application/postscript; name="export.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="export.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: paper.dvi %%Pages: 16 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: Times-Bold Times-Italic Times-Roman %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -t a4 -D 600 -Z -f paper %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2000.05.12:1004 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3 1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: 8r.enc % @@psencodingfile@{ % author = "S. 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Fo(1)i Fn(;)1300 b Fm(\()p Fo(1)p Fn(:)p Fo(3)p Fm(\))p eop %%Page: 2 2 2 1 bop 94 -45 a Fh(Mar)n(ginal)38 b(St)-7 b(ability)38 b(Bound)2382 b Fk(2)94 194 y Fo(since)40 b(the)f(non\255linear)f(term)h (is)g(irrele)n(v)n(ant)f(at)h Fn(\030)52 b Fm(=)c Fl(1)39 b Fo(in)g(this)f(case.)80 b(F)o(or)39 b(\002x)o(ed)g Fn(\014)45 b Fo(we)40 b(clearly)f(\002nd)94 314 y Fn(c)28 b Fm(=)g(\()p Fn(\014)370 271 y FA(2)429 314 y Fm(+)18 b Fo(1)p Fm(\))p Fn(=\014)5 b Fo(,)23 b(and)g(since)g(functions)f (which)g(are)i(\(in)e(absolute)g(v)n(alue\))h(bounded)f(by)h Fn(C)g Fo(e)o(xp)p Fm(\()p Fl(\000)p Fn(\014)5 b(x)p Fm(\))23 b Fo(are)94 433 y(also)i(bounded)f(by)g Fn(C)850 390 y Fi(0)895 433 y Fo(e)o(xp)p Fm(\()p Fl(\000)p Fn(\014)1215 390 y Fi(0)1243 433 y Fn(x)p Fm(\))h Fo(for)g(0)j Fn(<)g(\014)1749 390 y Fi(0)1805 433 y Fn(<)g(\014)j Fo(one)24 b(\002nds)h(in)g(this)e (case)j(an)f(upper)g(bound)1523 725 y Fn(c)1566 682 y FA(GL)1566 752 y Fz(\014)1708 725 y Fm(=)114 b Fo(inf)1830 790 y FA(0)p Fz(<\015)t(<\014)2113 658 y Fn(\015)2171 615 y FA(2)2232 658 y Fm(+)22 b Fo(1)p 2113 702 269 4 v 2218 794 a Fn(\015)2418 725 y(;)1223 b Fm(\()p Fo(1)p Fn(:)p Fo(4)p Fm(\))94 993 y Fo(and)29 b(this)g(is)f(equal)h(to)g(2)g (for)g Fn(\014)39 b Fl(\025)34 b Fo(1.)48 b(Using)29 b(the)g(maximum)e(principle)h(for)h(parabolic)g(PDE')-5 b(s,)30 b(Aronson)94 1112 y(and)21 b(W)-8 b(einber)n(ger)21 b(were)f(able)h(to)f(sho)n(w)f([A)-9 b(W])21 b(that)e(no)h(positi)n(v)o (e)e(solution)h(starting)g(from)h(initial)f(conditions)94 1232 y(with)36 b(compact)g(support)f(can)h(mo)o(v)o(e)f(f)o(aster)h (than)g(the)g(speed)g Fn(c)2374 1189 y FA(GL)2374 1259 y(2)2515 1232 y Fm(=)44 b Fo(2.)69 b(Using)36 b(essentially)e(the)i (same)94 1351 y(ar)n(gument,)29 b(it)f(w)o(as)h(also)f(sho)n(wn)f(in)h ([CE])h(that)f(if)g(the)h(initial)e(condition)g(decays)h(lik)o(e)g Fn(e)3199 1309 y Fi(\000)p Fz(\014)t(x)3389 1351 y Fo(with)g Fn(\014)38 b(<)33 b Fo(1,)94 1471 y(then)23 b(the)f(solution)f(cannot)i (mo)o(v)o(e)e(f)o(aster)i(than)g Fn(c)1788 1428 y FA(GL)1788 1498 y Fz(\014)1886 1471 y Fo(.)35 b(Ho)n(we)n(v)o(er)l(,)22 b(in)g(cases)h(where)g(the)g(maximum)e(principle)94 1591 y(does)26 b(not)g(apply)-6 b(,)25 b(such)h(as)g(in)g(\(1.5\),)g(the)g (maximum)f(possible)g(speed)h(w)o(as)g(only)f(conjectured,)h(and)h (tested)94 1710 y(numerically)-6 b(,)24 b(b)n(ut)g(no)h(rigorous)e (result)i(w)o(as)g(obtained,)f(see)h FB(e)o(.g)o(.)p Fo(,)e([LMK,)i(DL,)f(BBDKL].)316 1830 y(In)31 b(a)g(some)n(what)f(dif)n (ferent)g(direction,)i(there)f(is)g(the)g(important,)g(and)g(dif)n (\002cult,)g(issue)f(on)h(whether)94 1949 y(there)22 b(is)f(actually)g(a)h(solution)e(mo)o(ving)f(with)i(the)g(maximal)g (allo)n(wed)f(v)o(elocity)-6 b(.)33 b(In)21 b(general,)i(its)d (realization)94 2069 y(depends)29 b(on)g(the)h(details)e(of)h(the)h (nonlinearity)-6 b(,)28 b(and)h(this)g(question)f(has)h(been)g(e)o (xtensi)n(v)o(ely)e(discussed)i(in)94 2188 y(the)c(literature)g([A)-9 b(W)g(,)25 b(B,)g(DL,)f(BBDKL,)i(vS].)f(It)g(will)f(not)g(be)h(treated) g(here.)316 2308 y(The)j(main)g(result)g(of)g(our)g(paper)h(is)f(an)g FB(upper)g(bound)f Fo(on)h(the)g(speed)h(of)f(propagation)f(of)h (solutions)94 2427 y(to)d(the)g(Swift\255Hohenber)n(g)f(equation)1339 2647 y Fn(@)1392 2674 y Fz(t)1427 2647 y Fn(u)44 b Fm(=)1650 2566 y Fs(\000)1696 2647 y Fn(")22 b Fl(\000)h Fm(\()p Fo(1)e Fm(+)i Fn(@)2133 2604 y FA(2)2127 2674 y Fz(x)2177 2647 y Fm(\))2216 2604 y FA(2)2256 2566 y Fs(\001)2301 2647 y Fn(u)g Fl(\000)f Fn(u)2537 2604 y FA(3)2602 2647 y Fn(:)1039 b Fm(\()p Fo(1)p Fn(:)p Fo(5)p Fm(\))94 2866 y Fo(The)25 b(polynomial)e(equation)h(analogous)g(to)g(\(1.3\))h(turns) f(out)h(to)f(be)1421 3086 y(0)44 b Fm(=)g Fn(")1682 3043 y FA(2)1745 3086 y Fm(+)23 b Fo(4)p Fn(\014)1956 3043 y FA(2)2018 3086 y Fm(+)g Fo(8)p Fn(\014)2229 3043 y FA(4)2291 3086 y Fl(\000)g Fn(c\014)30 b(;)1121 b Fm(\()p Fo(1)p Fn(:)p Fo(6)p Fm(\))94 3305 y Fo(and)25 b(we)g(de\002ne)h(in)e (this)g(case)1216 3574 y 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2230 4801 a Fn(\015)2861 4733 y(:)316 4989 y Fo(In)25 b(an)g(appendix,)f(we)h(sho)n(w)f(that)g(the)h(e)o (xpression)e(\(1.6\))i(is)f(nothing)g(b)n(ut)1502 5198 y(sup)1526 5285 y Fz(k)1570 5261 y Ff(\003)1568 5312 y Fe(\014)1657 5198 y Fo(Re)17 b Fn(P)d Fm(\()p Fn(z)t Fm(\))p Fl(j)2018 5225 y Fz(z)s Fj(=)p Fi(\000)p Fz(\014)t Fj(+)p Fz(ik)2364 5201 y Ff(\003)2362 5252 y Fe(\014)2439 5198 y Fn(;)p eop %%Page: 4 4 4 3 bop 94 -45 a Fh(Mar)n(ginal)38 b(St)-7 b(ability)38 b(Bound)2382 b Fk(4)94 194 y Fo(where)26 b(the)e(sup)h(is)f(o)o(v)o(er) g(the)h(solutions)d Fn(k)1549 151 y Fi(\003)1546 221 y Fz(\014)1625 194 y Fo(of)1509 382 y(dRe)17 b Fn(P)d Fm(\()p Fl(\000)p Fn(\014)28 b Fm(+)22 b Fn(ik)s Fm(\))p 1509 426 683 4 v 1789 517 a Fo(d)p Fn(k)2248 449 y Fm(=)44 b Fo(0)25 b Fn(:)94 658 y Fo(W)-8 b(e)26 b(also)e(sho)n(w)g(that)g (these)h(conditions)e(are)i(the)g(same)g(as)g(those)f(found)g(in)h ([BBDKL].)316 777 y(Finally)-6 b(,)21 b(it)f(should)g(be)i(noted)e (that)h(the)g(method)f(is)h(not)f(restricted)h(to)g(1\255dimensional)e (problems,)i(and)94 897 y(can)30 b(also)e(be)h(applied)g(to)f (questions)f(of)i(gro)n(ws)f(of)h(\223b)n(ubbles\224)g(in)f(the)h (2\255dimensional)e(Swift\255Hohenber)n(g)94 1016 y(equation.)94 1337 y Fp(2.)j(A)g(pointwise)h(bound)h(on)e(the)g(Gr)n(een')l(s)f (function)94 1526 y Fo(Here)19 b(we)f(bound)f(the)g(Green')-5 b(s)18 b(function)f(of)h(the)f(operator)h Fn(")2160 1484 y FA(2)2206 1526 y Fl(\000)6 b Fm(\()p Fo(1)g Fm(+)g Fn(@)2526 1484 y FA(2)2520 1553 y Fz(x)2572 1526 y Fm(\))2611 1484 y FA(2)2668 1526 y Fo(by)18 b(a)g(method)f(which)g(generalizes)94 1646 y(immediately)22 b(to)g(other)h(problems)f(of)h(similar)f(type.)35 b(Let)23 b Fn(P)37 b Fo(be)23 b(a)g(polynomial)e(in)i Fn(k)j Fo(which)d(is)f(of)h(the)g(form)916 1922 y Fn(P)14 b Fm(\()p Fn(ik)s Fm(\))44 b(=)h Fl(\000)p Fn(a)1457 1948 y Fz(n)1511 1922 y Fn(k)1566 1879 y Fz(n)1643 1922 y Fm(+)1752 1797 y Fz(n)p Fi(\000)p FA(1)1754 1827 y Fs(X)1742 2040 y Fz(m)p Fj(=)p FA(0)1926 1922 y Fn(a)1979 1948 y Fz(m)2054 1922 y Fn(k)2109 1879 y Fz(m)2254 1922 y Fl(\021)f(\000)p Fn(a)2505 1948 y Fz(n)2560 1922 y Fn(k)2615 1879 y Fz(n)2691 1922 y Fm(+)23 b Fn(R)q Fm(\()p Fn(k)s Fm(\))h Fn(;)94 2213 y Fo(and)k(assume)e Fn(n)h Fo(e)n(v)o(en)f(and)h Fn(a)1108 2240 y Fz(n)1193 2213 y Fn(>)k Fo(0.)43 b(\(F)o(or)27 b(the)g(Swift\255Hohenber)n(g)g (equation,)g Fn(P)14 b Fm(\()p Fn(z)t Fm(\))31 b(=)g Fn(")3269 2170 y FA(2)3333 2213 y Fl(\000)24 b Fm(\()p Fo(1)f Fm(+)h Fn(z)3697 2170 y FA(2)3737 2213 y Fm(\))3776 2170 y FA(2)3816 2213 y Fo(.\))94 2333 y(Then)h(the)g(Green')-5 b(s)25 b(function)1409 2479 y Fn(G)1487 2506 y Fz(t)1523 2479 y Fm(\()p Fn(x)p Fm(\))43 b(=)1823 2344 y Fs(Z)1939 2479 y Fo(d)p Fn(k)20 b(e)2107 2436 y Fz(ik)r(x)2229 2479 y Fn(e)2275 2436 y Fz(P)11 b Fj(\()p Fz(ik)r Fj(\))p Fz(t)2532 2479 y Fn(;)94 2687 y Fo(satis\002es:)94 2878 y Fk(Lemma)24 b(2.1.)35 b Fd(Gi)n(v)o(en)22 b Fo(0)27 b Fn(<)h(\014)34 b(<)28 b Fl(1)p Fd(,)23 b(there)g(is)g(a)g(constant)f Fn(C)7 b Fm(\()p Fn(\014)e Fm(\))24 b Fd(such)f(that)f(for)i(all)f Fn(t)k Fl(2)h Fm(\()p Fo(0)p Fn(;)17 b Fo(1])22 b Fd(one)h(has)g(the)94 2998 y(bound)1246 3131 y Fn(t)1282 3089 y FA(1)p Fz(=n)1411 3131 y Fl(j)p Fn(G)1517 3158 y Fz(t)1552 3131 y Fm(\()p Fn(x)p Fm(\))p Fl(j)p Fn(e)1761 3089 y Fj(\()p Fz(\014)1841 3046 y Ff(0)1868 3089 y Fj(+)p FA(2)p Fz(t)1994 3046 y Ff(\000)p Fc(1)p Fe(=n)2158 3089 y Fj(\))p Fi(j)p Fz(x)p Fi(j)2331 3131 y Fl(\024)45 b Fn(C)7 b Fm(\()p Fn(\014)e Fm(\))25 b Fn(;)946 b Fm(\()p Fo(2)p Fn(:)p Fo(1)p Fm(\))94 3295 y Fd(for)26 b(all)e Fn(\014)421 3253 y Fi(0)477 3295 y Fl(2)k Fo([0)p Fn(;)17 b(\014)5 b Fo(])p Fd(.)94 3509 y Fk(Remark.)26 b Fo(This)e(clearly)h(also)f(implies,)g(for)h(all) f Fn(t)k Fl(2)g Fm(\()p Fo(0)p Fn(;)17 b Fo(1])24 b(and)g(all)h Fn(\014)2538 3466 y Fi(0)2594 3509 y Fl(2)j Fo([0)p Fn(;)17 b(\014)5 b Fo(]:)1385 3606 y Fs(Z)1501 3742 y Fo(d)p Fn(x)17 b Fl(j)p Fn(G)1731 3769 y Fz(t)1766 3742 y Fm(\()p Fn(x)p 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Fz(\015)t(t)2064 4301 y Fc(1)p Fe(=n)2174 4344 y Fz(z)2264 4387 y Fl(\024)45 b Fn(C)7 b Fm(\()p Fn(\014)e Fm(\))25 b Fn(;)1013 b Fm(\()p Fo(2)p Fn(:)p Fo(3)p Fm(\))94 4582 y Fo(with)20 b Fn(\015)32 b Fm(=)c Fn(\014)16 b Fm(+)11 b Fo(2)p Fn(t)727 4539 y Fi(\000)p FA(1)p Fz(=n)921 4582 y Fo(,)20 b(and)g(it)f(clearly)h(suf)n(\002ces)g(to)g(consider)f Fn(z)32 b(>)d Fo(0.)34 b(Pro)o(ving)19 b(\(2.3\))g(is)h(a)g (straightforw)o(ard)94 4701 y(calculation)e(which)f(is)h(probably)f (well\255kno)n(wn.)32 b(Indeed,)19 b(the)f(l.h.s.)f(of)h(\(2.3\))g (equals)g(\(without)e(the)i(absolute)94 4821 y(v)n(alues\))978 4849 y Fs(Z)1094 4984 y Fo(d)p Fn(k)i(t)1252 4941 y FA(1)p Fz(=n)1398 4984 y Fo(e)o(xp)1540 4903 y Fs(\000)1586 4984 y Fn(\015)6 b(t)1680 4941 y FA(1)p Fz(=n)1809 4984 y Fn(z)27 b Fm(+)c Fn(ik)s(t)2107 4941 y FA(1)p Fz(=n)2237 4984 y Fn(z)k Fl(\000)22 b Fn(a)2462 5011 y Fz(n)2516 4984 y Fn(k)2571 4941 y Fz(n)2625 4984 y Fn(t)g Fm(+)h Fn(R)q Fm(\()p Fn(k)s Fm(\))p Fn(t)3029 4903 y Fs(\001)939 5245 y Fm(=)1061 5110 y Fs(Z)1177 5245 y Fo(d)p Fn(`)33 b Fo(e)o(xp)1444 5165 y Fs(\000)1490 5245 y Fn(\015)6 b(t)1584 5203 y FA(1)p Fz(=n)1713 5245 y Fn(z)27 b Fm(+)22 b Fn(i`z)27 b Fl(\000)22 b Fn(a)2186 5272 y Fz(n)2240 5245 y Fn(`)2282 5203 y Fz(n)2358 5245 y Fm(+)h Fn(R)q Fm(\()p Fn(`t)2652 5203 y Fi(\000)p FA(1)p Fz(=n)2843 5245 y Fm(\))p Fn(t)2918 5165 y Fs(\001)2988 5245 y Fn(:)p eop %%Page: 5 5 5 4 bop 94 -45 a Fh(Mar)n(ginal)38 b(St)-7 b(ability)38 b(Bound)2382 b Fk(5)94 194 y Fo(Since)26 b(the)f(inte)o(grand)f(is)h (an)g(entire)g(function)f(in)h Fn(`)g Fo(we)g(can)h(shift)e(the)h (contour)g(from)f Fn(`)h Fo(to)g Fn(`)3292 151 y Fi(0)3347 194 y Fm(=)k Fn(`)22 b Fl(\000)h Fn(i\015)6 b(t)3745 151 y FA(1)p Fz(=n)94 314 y Fo(and)25 b(the)g(last)f(e)o(xpression)g (is)g(seen)h(to)f(be)h(equal)g(to)814 445 y Fs(Z)930 580 y Fo(d)p Fn(`)1022 537 y Fi(0)1083 580 y Fo(e)o(xp)1225 499 y Fs(\000)1271 580 y Fn(i`)1347 537 y Fi(0)1374 580 y Fn(z)i Fl(\000)c Fn(a)1600 607 y Fz(n)1654 580 y Fm(\()p Fn(`)1735 537 y Fi(0)1784 580 y Fm(+)g Fn(i\015)6 b(t)2012 537 y FA(1)p Fz(=n)2141 580 y Fm(\))2180 537 y Fz(n)2256 580 y Fm(+)23 b Fn(R)q Fm(\()p Fn(`)2514 537 y Fi(0)2540 580 y Fn(t)2576 537 y Fi(\000)p FA(1)p Fz(=n)2791 580 y Fm(+)f Fn(i\015)6 b Fm(\))p Fn(t)3057 499 y Fs(\001)3127 580 y Fn(:)94 842 y Fo(Note)25 b(no)n(w)f(that)923 977 y Fs(\014)923 1036 y(\014)973 1061 y Fo(e)o(xp)1115 981 y Fs(\000)1161 1061 y Fn(i`)1237 1019 y Fi(0)1265 1061 y Fn(z)i Fl(\000)d Fn(a)1490 1088 y Fz(n)1544 1061 y Fm(\()p Fn(`)1625 1019 y Fi(0)1674 1061 y Fm(+)g Fn(i\015)6 b(t)1902 1019 y FA(1)p Fz(=n)2031 1061 y Fm(\))2070 1019 y Fz(n)2146 1061 y Fm(+)23 b Fn(R)q Fm(\()p Fn(`)2404 1019 y Fi(0)2431 1061 y Fn(t)2467 1019 y Fi(\000)p FA(1)p Fz(=n)2681 1061 y Fm(+)g Fn(i\015)6 b Fm(\))p Fn(t)2948 981 y Fs(\001)2992 977 y(\014)2992 1036 y(\014)984 1231 y Fm(=)1106 1147 y Fs(\014)1106 1206 y(\014)1139 1231 y Fo(e)o(xp)1281 1151 y Fs(\000)1327 1231 y Fl(\000)p Fn(a)1457 1258 y Fz(n)1511 1231 y Fm(\()p Fn(`)1592 1188 y Fi(0)1642 1231 y Fm(+)22 b Fn(i\015)6 b(t)1869 1188 y FA(1)p Fz(=n)1999 1231 y Fm(\))2038 1188 y Fz(n)2114 1231 y Fm(+)22 b Fn(R)q Fm(\()p Fn(`)2371 1188 y Fi(0)2398 1231 y Fn(t)2434 1188 y Fi(\000)p FA(1)p Fz(=n)2648 1231 y Fm(+)h Fn(i\015)6 b Fm(\))p Fn(t)2915 1151 y Fs(\001)2960 1147 y(\014)2960 1206 y(\014)3018 1231 y Fn(;)623 b Fm(\()p Fo(2)p Fn(:)p Fo(4)p Fm(\))94 1471 y Fo(and)29 b(for)f(bounded)g Fn(\014)34 b Fo(and)28 b Fn(t)k Fl(2)h Fm(\()p Fo(0)p Fn(;)17 b Fo(1])27 b(we)i(\002nd)f(that)g Fn(\015)6 b(t)2056 1428 y FA(1)p Fz(=n)2217 1471 y Fm(=)33 b(\()p Fn(\014)d Fm(+)24 b Fo(2)p Fn(t)2639 1428 y Fi(\000)p FA(1)p Fz(=n)2831 1471 y Fm(\))p Fn(t)2906 1428 y FA(1)p Fz(=n)3068 1471 y Fl(\024)33 b Fn(\014)d Fm(+)25 b Fo(2,)j(and)h(hence)94 1590 y(\(2.4\))c(is)g(uniformly)e(inte)o(grable)h(in)g Fn(`)1391 1547 y Fi(0)1419 1590 y Fo(,)h(since)f Fn(a)1751 1617 y Fz(n)1833 1590 y Fn(>)k Fo(0.)36 b(The)24 b(proof)h(of)g(Lemma)f (2.1)h(is)f(complete.)p 3812 1536 63 2 v 3812 1594 2 59 v 3872 1594 V 3812 1596 63 2 v 94 2046 a Fp(3.)30 b(Exponential)h(decay)f(of)f(solutions)94 2235 y Fo(In)h(this)e (section,)h(we)h(pro)o(v)o(e)e(a)h(bound)g(in)f(the)h(laboratory)g (frame,)i(sho)n(wing)c(that)i(if)g(the)g(initial)f(condition)94 2355 y(goes)d(e)o(xponentially)e(to)h(0)h(then)f(the)h(solution)e(at)i (time)f Fn(t)h Fo(goes)f(to)h(zero)g(as)g(well,)f FB(with)h(the)g(same) f(r)o(ate)p Fo(.)94 2546 y Fk(Theor)n(em)j(3.1.)35 b Fd(Assume)24 b(that)h Fn(u)1268 2573 y FA(0)1332 2546 y Fd(is)g(bounded)f(in)g Fl(C)1952 2504 y FA(3)2017 2546 y Fd(and)h(that)1535 2765 y Fo(lim)1500 2825 y Fz(x)p Fi(!1)1721 2765 y Fn(e)1767 2723 y Fz(\014)t(x)1866 2765 y Fn(@)1925 2723 y Fz(j)1919 2792 y(x)1969 2765 y Fn(u)2026 2792 y FA(0)2066 2765 y Fm(\()p Fn(x)p Fm(\))43 b(=)i Fo(0)25 b Fn(;)1200 b Fm(\()p Fo(3)p Fn(:)p Fo(1)p Fm(\))94 3019 y Fd(for)24 b Fn(j)33 b Fm(=)28 b Fo(0)p Fn(;)17 b(:)g(:)g(:)d(;)j Fo(3)22 b Fd(and)h(some)g Fn(\014)33 b(>)28 b Fo(0)p Fd(.)35 b(Then)23 b(the)g(solution)e Fn(u)p Fm(\()p Fn(x;)c(t)p Fm(\))22 b Fd(of)h(\(1.5\))g(with)f(initial) g(data)h Fn(u)3496 3046 y FA(0)3558 3019 y Fd(satis\002es)94 3139 y(for)j(all)e Fn(t)k(>)g Fo(0)p Fd(:)1515 3258 y Fo(lim)1480 3318 y Fz(x)p Fi(!1)1700 3258 y Fn(e)1746 3216 y Fz(\014)t(x)1846 3258 y Fn(@)1905 3216 y Fz(j)1899 3285 y(x)1949 3258 y Fn(u)p Fm(\()p Fn(x;)17 b(t)p Fm(\))43 b(=)h Fo(0)25 b Fn(;)1180 b Fm(\()p Fo(3)p Fn(:)p Fo(2)p Fm(\))94 3466 y Fd(for)26 b Fn(j)33 b Fm(=)28 b Fo(0)p Fn(;)17 b(:)g(:)g(:)d(;)j Fo(3)p Fd(.)94 3696 y Fk(Pr)n(oof)o(.)25 b Fo(The)g(proof)f(is)h(in)f(steps)g(of)h(some)f(\(\002x)o(ed\))i(time) e Fn(\034)2096 3723 y Fi(\003)2141 3696 y Fo(.)36 b(W)-8 b(e)25 b(de\002ne)g(\002rst)1522 3927 y Fn(g)1570 3954 y Fz(\030)1613 3927 y Fm(\()p Fn(x)p Fm(\))44 b(=)h Fo(1)22 b Fm(+)g Fn(e)2131 3885 y Fz(\014)t Fj(\()p Fz(x)p Fi(\000)p Fz(\030)r Fj(\))2419 3927 y Fn(:)94 4146 y Fo(The)i(assumption)e(means) h(that)g Fn(u)1259 4173 y FA(0)1322 4146 y Fo(satis\002es)g(\(1.8\))h (for)g(some)f Fn(K)7 b Fo(.)35 b(From)23 b(\(3.1\),)h(and)g(because)g Fn(L)p Fm(\()p Fn(K)7 b Fm(\))26 b Fl(\025)i Fn(K)7 b Fo(,)94 4266 y(we)26 b(conclude)e(that)g(there)i(is)e(a)h Fn(\030)32 b(>)c Fo(0)d(for)g(which)1346 4485 y(sup)1341 4570 y Fz(x)p Fi(2)p Fg(R)1507 4485 y Fn(g)1555 4512 y Fz(\030)1598 4485 y Fm(\()p Fn(x)p Fm(\))p Fl(j)p Fn(@)1820 4442 y Fz(j)1814 4512 y(x)1863 4485 y Fn(u)1920 4512 y FA(0)1960 4485 y Fm(\()p Fn(x)p Fm(\))p Fl(j)43 b(\024)i Fo(2)p Fn(L)p Fm(\()p Fn(K)7 b Fm(\))24 b Fn(;)1041 b Fm(\()p Fo(3)p Fn(:)p Fo(3)p Fm(\))94 4764 y Fo(for)24 b Fn(j)34 b Fm(=)28 b Fo(0)p Fn(;)17 b(:)g(:)g(:)d(;)j Fo(3.)34 b(Note)23 b(that)g(we)h(do)f(not)g(ha)n(v)o(e)g(an)o(y)g (control)f(on)h(the)h(size)f(of)h Fn(\030)5 b Fo(,)22 b(b)n(ut)h(such)g(a)h(control)f(is)g(not)94 4884 y(needed.)316 5003 y(From)i(\(1.8\))f(we)i(also)e(conclude)g(\(see)i(\(1.9\)\))f (that)1381 5222 y(sup)1387 5308 y Fz(t)p Fi(\025)p FA(0)1541 5222 y Fo(sup)1536 5308 y Fz(x)p Fi(2)p Fg(R)1702 5222 y Fl(j)p Fn(@)1789 5180 y Fz(j)1783 5249 y(x)1832 5222 y Fn(u)p Fm(\()p Fn(x;)17 b(t)p Fm(\))p Fl(j)43 b(\024)i Fn(L)p Fm(\()p Fn(K)7 b Fm(\))24 b Fn(;)1081 b Fm(\()p Fo(3)p Fn(:)p Fo(4)p Fm(\))p eop %%Page: 6 6 6 5 bop 94 -45 a Fh(Mar)n(ginal)38 b(St)-7 b(ability)38 b(Bound)2382 b Fk(6)94 194 y Fo(for)26 b Fn(j)33 b Fm(=)28 b Fo(0)p Fn(;)17 b(:)g(:)g(:)d(;)j Fo(3.)316 314 y(The)25 b(crucial)g(step)f(in)h(the)f(proof)h(of)g(Theorem)f(3.1)h(is)94 505 y Fk(Lemma)h(3.2.)35 b Fd(There)26 b(are)f(a)g Fn(\034)1148 532 y Fi(\003)1221 505 y Fn(>)j Fo(0)d Fd(and)g(a)g Fn(\032)p Fd(,)f(independent)g(of)h Fn(\030)5 b Fd(,)24 b(such)h(that)f(for)h Fn(t)j Fl(2)g Fo([0)p Fn(;)17 b(\034)3311 532 y Fi(\003)3355 505 y Fo(])25 b Fd(one)g(has)1353 762 y Fo(sup)1279 847 y Fz(j)t Fj(=)p FA(0)p Fz(;:::)n(;)p FA(3)1604 762 y Fo(sup)1598 848 y Fz(x)p Fi(2)p Fg(R)1764 762 y Fn(g)1812 789 y Fz(\030)1855 762 y Fm(\()p Fn(x)p Fm(\))p Fl(j)p Fn(@)2077 720 y Fz(j)2071 789 y(x)2120 762 y Fn(u)p Fm(\()p Fn(x;)17 b(t)p Fm(\))p Fl(j)43 b(\024)i Fn(\032)24 b(:)979 b Fm(\()p Fo(3)p Fn(:)p Fo(5)p Fm(\))94 1230 y Fk(Pr)n(oof)o(.)45 b Fo(W)-8 b(e)45 b(use)g(the)f(estimates)g(on)g(the)h(con)l(v)n (olution)e(k)o(ernel)h Fn(G)2530 1257 y Fz(t)2611 1230 y Fo(associated)g(with)g(the)g(semigroup)94 1350 y Fn(t)28 b Fl(7!)f Fo(e)o(xp)428 1269 y Fs(\000)473 1350 y Fn(t)p Fm(\()p Fn(")594 1307 y FA(2)657 1350 y Fl(\000)22 b Fm(\()p Fo(1)g Fm(+)h Fn(@)1026 1307 y FA(2)1020 1377 y Fz(x)1070 1350 y Fm(\))1109 1307 y FA(2)1148 1269 y Fs(\001)1219 1350 y Fo(which)h(were)i(pro)o(v)o(en)d(in)i(Section)f(2.) 36 b(One)25 b(has)1280 1668 y Fn(u)1337 1695 y Fz(t)1416 1668 y Fm(=)45 b Fn(G)1616 1695 y Fz(t)1674 1668 y Fn(?)22 b(u)1803 1695 y FA(0)1865 1668 y Fl(\000)1964 1532 y Fs(Z)2064 1575 y Fz(t)2020 1759 y FA(0)2116 1668 y Fo(d)p Fn(s)17 b(G)2308 1695 y Fz(t)p Fi(\000)p Fz(s)2464 1668 y Fn(?)22 b(u)2593 1625 y FA(3)2593 1695 y Fz(s)2661 1668 y Fn(;)94 1982 y Fo(where)34 b Fn(u)428 2009 y Fz(s)470 1982 y Fm(\()p Fn(x)p Fm(\))39 b(=)g Fn(u)p Fm(\()p Fn(x;)17 b(s)p Fm(\))p Fo(.)58 b(W)-8 b(e)34 b(de\002ne)f Fb(B)1643 2009 y Fz(\030)1719 1982 y Fo(as)g(the)g(space)g(of)g(uniformly)e (continuous)h(functions)f Fn(f)44 b Fo(for)94 2102 y(which)1320 2239 y Fl(k)p Fn(f)11 b Fl(k)1480 2266 y Fz(\030)1567 2239 y Fm(=)51 b Fo(sup)1689 2325 y Fz(x)p Fi(2)p Fg(R)1855 2239 y Fn(g)1903 2266 y Fz(\030)1946 2239 y Fm(\()p Fn(x)p Fm(\))p Fl(j)p Fn(f)11 b Fm(\()p Fn(x)p Fm(\))p Fl(j)42 b Fn(<)j Fl(1)24 b Fn(:)94 2508 y Fo(Using)f(this)f(quantity)g(as)h(a)h (norm)e(mak)o(es)h Fb(B)1633 2535 y Fz(\030)1700 2508 y Fo(a)g(Banach)h(space.)36 b(Consider)23 b(ne)o(xt)f(the)i(space)f Fl(K)29 b Fm(=)f Fl(K)3623 2535 y Fz(\030)r(;\034)3722 2553 y Ff(\003)3797 2508 y Fm(=)94 2652 y Fl(C)152 2609 y FA(0)193 2652 y Fm(\()p Fo([0)p Fn(;)17 b(\034)404 2679 y Fi(\003)448 2652 y Fo(])p Fn(;)g Fb(B)602 2679 y Fz(\030)645 2652 y Fm(\))25 b Fo(of)f(functions)g Fn(h)k Fo(:)g Fm(\()p Fn(x;)17 b(t)p Fm(\))26 b Fl(7!)h Fn(h)p Fm(\()p Fn(x;)17 b(t)p Fm(\))p Fo(,)24 b(with)g(the)h(norm)1389 2909 y Fl(k)p Fn(h)p Fl(k)1546 2936 y Fz(\030)r(;\034)1645 2954 y Ff(\003)1736 2909 y Fm(=)109 b Fo(sup)1857 2995 y Fz(t)p Fi(2)p FA([0)p Fz(;\034)2060 3012 y Ff(\003)2101 2995 y FA(])2141 2909 y Fl(k)p Fn(h)p Fm(\()p Fl(\001)p Fn(;)17 b(t)p Fm(\))p Fl(k)2485 2936 y Fz(\030)2552 2909 y Fn(:)94 3238 y Fo(This)25 b(is)f(again)g(a)h(Banach)h(space.)36 b(F)o(or)25 b Fn(v)31 b Fl(2)d(K)f Fo(we)e(de\002ne)g(the)g(map)f Fn(v)32 b Fl(7!)27 b(Q)p Fn(v)i Fo(by)885 3462 y Fs(\000)930 3543 y Fl(Q)p Fn(v)1063 3462 y Fs(\001)1109 3543 y Fm(\()p Fn(x;)17 b(t)p Fm(\))43 b(=)1490 3462 y Fs(\000)1536 3543 y Fn(G)1614 3570 y Fz(t)1671 3543 y Fn(?)22 b(u)1800 3570 y FA(0)1840 3462 y Fs(\001)1886 3543 y Fm(\()p Fn(x)p Fm(\))f Fl(\000)2142 3407 y Fs(Z)2242 3450 y Fz(t)2197 3634 y FA(0)2293 3543 y Fo(d)p Fn(s)2407 3462 y Fs(\000)2452 3543 y Fn(G)2530 3570 y Fz(t)p Fi(\000)p Fz(s)2687 3543 y Fn(?)h(v)2811 3500 y FA(3)2807 3570 y Fz(s)2851 3462 y Fs(\001)2897 3543 y Fm(\()p Fn(x)p Fm(\))i Fn(:)585 b Fm(\()p Fo(3)p Fn(:)p Fo(6)p Fm(\))94 3852 y Fo(Note)30 b(that)f(if)g Fl(Q)p Fn(v)38 b Fm(=)d Fn(v)t Fo(,)30 b(then)f Fn(v)34 b Fo(is)28 b(a)i(solution)e(to)h(\(1.5\))g(with)g (initial)f(condition)g Fn(u)3035 3879 y FA(0)3075 3852 y Fo(.)49 b(T)-8 b(o)29 b(\002nd)h Fn(v)t Fo(,)g(we)g(will)94 3972 y(sho)n(w)e(that)f(for)i(suf)n(\002ciently)e(small)g Fn(\034)1415 3998 y Fi(\003)1493 3972 y Fn(>)33 b Fo(0)28 b(the)g(operator)g Fl(Q)h Fo(contracts)f(a)g(small)f(ball)h(of)g Fl(K)3370 3998 y Fz(\030)r(;\034)3469 4016 y Ff(\003)3544 3972 y Fo(to)g(itself.)94 4091 y(The)d(center)h(of)f(this)e(ball)i(is)f (the)h(function)f Fm(\()p Fn(x;)17 b(t)p Fm(\))26 b Fl(7!)h Fo(0.)316 4211 y(First)d(we)i(bound)e Fn(G)1017 4238 y Fz(t)1074 4211 y Fn(?)e(u)1203 4238 y FA(0)1243 4211 y Fo(.)36 b(Note)24 b(that)h(from)f(the)h(de\002nition)f(of)h Fn(g)2625 4238 y Fz(\030)2693 4211 y Fo(we)g(\002nd)1613 4473 y Fn(g)1661 4500 y Fz(\030)1704 4473 y Fm(\()p Fn(x)p Fm(\))p 1613 4530 226 4 v 1615 4621 a Fn(g)1663 4648 y Fz(\030)1707 4621 y Fm(\()p Fn(y)t Fm(\))1895 4553 y Fl(\024)45 b Fn(e)2063 4510 y Fz(\014)t Fi(j)p Fz(x)p Fi(\000)p Fz(y)s Fi(j)2340 4553 y Fn(;)94 4876 y Fo(since)31 b(for)g Fn(x)36 b(<)h(y)e Fo(the)30 b(quotient)g(is)g(bounded)g(by)h(1) g(and)g(for)g Fn(x)36 b(>)h(y)d Fo(we)d(ha)n(v)o(e)g(the)g(\(v)o(ery)f (rough\))h(bound)94 4996 y Fn(e)140 4953 y Fz(\014)t Fj(\()p Fz(x)p Fi(\000)p Fz(y)s Fj(\))407 4996 y Fo(.)36 b(From)25 b(Lemma)f(2.1,)h(we)g(ha)n(v)o(e)f(for)h(all)g Fn(t)i Fl(2)h Fm(\()p Fo(0)p Fn(;)17 b Fo(1])24 b(and)h(all)f Fn(x)k Fl(2)g Fk(R)p Fo(:)1186 5291 y Fl(j)p Fn(G)1292 5318 y Fz(t)1327 5291 y Fm(\()p Fn(x)p Fm(\))p Fl(j)p Fn(e)1536 5248 y FA(2)p Fz(\014)t Fi(j)p Fz(x)p Fi(j)1761 5291 y Fl(\024)44 b Fn(C)7 b Fm(\()p Fn(\014)e Fm(\))p Fn(t)2135 5248 y Fi(\000)p FA(1)p Fz(=)p FA(4)2314 5291 y Fn(e)2360 5248 y Fi(\000)p FA(2)p Fi(j)p Fz(x)p Fi(j)p Fz(t)2580 5206 y Ff(\000)p Fc(1)p Fe(=)p Fc(4)2755 5291 y Fn(;)886 b Fm(\()p Fo(3)p Fn(:)p Fo(7)p Fm(\))p eop %%Page: 7 7 7 6 bop 94 -45 a Fh(Mar)n(ginal)38 b(St)-7 b(ability)38 b(Bound)2382 b Fk(7)94 194 y Fo(and,)25 b(clearly)-6 b(,)25 b Fn(C)7 b Fm(\()p Fn(\014)e Fm(\))25 b Fo(can)g(be)g(chosen)g (the)f(same)h(v)n(alue)f(for)h(all)g(smaller)f Fn(\014)5 b Fo(.)36 b(Using)24 b(this,)g(we)h(\002nd)739 474 y Fl(j)767 393 y Fs(\000)813 474 y Fn(G)891 501 y Fz(t)948 474 y Fn(?)d(u)1077 501 y FA(0)1117 393 y Fs(\001)1163 474 y Fm(\()p Fn(x)p Fm(\))17 b Fn(g)1363 501 y Fz(\030)1405 474 y Fm(\()p Fn(x)p Fm(\))p Fl(j)43 b(\024)1733 338 y Fs(Z)1849 474 y Fo(d)p Fn(y)20 b Fl(j)p Fn(G)2074 501 y Fz(t)2109 474 y Fm(\()p Fn(x)i Fl(\000)h Fn(y)t Fm(\))p Fn(u)2476 501 y FA(0)2515 474 y Fm(\()p Fn(y)t Fm(\))p Fl(j)17 b Fn(g)2739 501 y Fz(\030)2780 474 y Fm(\()p Fn(y)t Fm(\))2923 394 y Fn(g)2971 421 y Fz(\030)3013 394 y Fm(\()p Fn(x)p Fm(\))p 2923 451 226 4 v 2925 542 a Fn(g)2973 569 y Fz(\030)3015 542 y Fm(\()p Fn(y)t Fm(\))1611 758 y Fl(\024)1733 622 y Fs(Z)1849 758 y Fo(d)p Fn(y)j Fl(j)p Fn(G)2074 785 y Fz(t)2109 758 y Fm(\()p Fn(x)i Fl(\000)h Fn(y)t Fm(\))p Fn(u)2476 785 y FA(0)2515 758 y Fm(\()p Fn(y)t Fm(\))p Fl(j)17 b Fn(g)2739 785 y Fz(\030)2780 758 y Fm(\()p Fn(y)t Fm(\))p Fn(e)2957 715 y Fz(\014)t Fi(j)p Fz(x)p Fi(\000)p Fz(y)s Fi(j)1611 1019 y Fl(\024)1733 884 y Fs(Z)1849 1019 y Fo(d)p Fn(z)k Fl(j)p Fn(G)2072 1046 y Fz(t)2107 1019 y Fm(\()p Fn(z)t Fm(\))p Fn(e)2281 976 y Fz(\014)t Fi(j)p Fz(z)s Fi(j)2424 1019 y Fl(j)33 b Fo(sup)2469 1105 y Fz(z)2510 1085 y Ff(0)2536 1105 y Fi(2)p Fg(R)2657 1019 y Fl(j)p Fn(u)2742 1046 y FA(0)2781 1019 y Fm(\()p Fn(z)2870 976 y Fi(0)2899 1019 y Fm(\))p Fl(j)17 b Fn(g)3031 1046 y Fz(\030)3073 1019 y Fm(\()p Fn(z)3162 976 y Fi(0)3190 1019 y Fm(\))1611 1245 y Fl(\024)45 b Fn(C)7 b Fm(\()p Fn(\014)e Fm(\))34 b Fo(sup)1967 1330 y Fz(z)2008 1311 y Ff(0)2035 1330 y Fi(2)p Fg(R)2156 1245 y Fl(j)p Fn(u)2241 1272 y FA(0)2280 1245 y Fm(\()p Fn(z)2369 1202 y Fi(0)2398 1245 y Fm(\))p Fl(j)17 b Fn(g)2530 1272 y Fz(\030)2572 1245 y Fm(\()p Fn(z)2661 1202 y Fi(0)2689 1245 y Fm(\))25 b Fn(:)3669 858 y Fm(\()p Fo(3)p Fn(:)p Fo(8)p Fm(\))94 1513 y Fo(Combining)f(these)g(bounds)g(with)g(\(3.3\))h (we)g(get)1287 1724 y Fl(j)1315 1643 y Fs(\000)1360 1724 y Fn(G)1438 1751 y Fz(t)1496 1724 y Fn(?)d(u)1625 1751 y FA(0)1665 1643 y Fs(\001)1710 1724 y Fm(\()p Fn(x)p Fm(\))17 b Fn(g)1910 1751 y Fz(\030)1953 1724 y Fm(\()p Fn(x)p Fm(\))p Fl(j)43 b(\024)i Fn(C)2352 1751 y FA(2)2392 1724 y Fn(L)p Fm(\()p Fn(K)7 b Fm(\))24 b Fn(:)94 1960 y Fo(In)h(f)o(act,)g(we)f(can)h(do)f(a)h(little)e(better)h(in)g (\(3.8\))g(by)g(e)o(xtracting)f(a)i(f)o(actor)g(of)f Fn(e)2694 1917 y Fi(\000)p Fz(\014)t Fi(j)p Fz(x)p Fi(\000)p Fz(y)s Fi(j)3008 1960 y Fo(.)36 b(The)24 b(last)g(tw)o(o)g(lines)g(in) 94 2080 y(\(3.8\))h(are)h(replaced)f(by)966 2287 y Fl(j)994 2206 y Fs(\000)1039 2287 y Fn(G)1117 2314 y Fz(t)1175 2287 y Fn(?)d(u)1304 2314 y FA(0)1344 2206 y Fs(\001)1389 2287 y Fm(\()p Fn(x)p Fm(\))17 b Fn(g)1589 2314 y Fz(\030)1632 2287 y Fm(\()p Fn(x)p Fm(\))p Fl(j)1067 2503 y(\024)1189 2367 y Fs(Z)1305 2503 y Fo(d)p Fn(z)k Fl(j)p Fn(G)1528 2530 y Fz(t)1563 2503 y Fm(\()p Fn(z)t Fm(\))p Fn(e)1737 2460 y FA(2)p Fz(\014)t Fi(j)p Fz(z)s Fi(j)1915 2503 y Fl(j)f Fo(sup)1960 2588 y Fz(y)s Fi(2)p Fg(R)2123 2503 y Fl(j)p Fn(u)2208 2530 y FA(0)2247 2503 y Fm(\()p Fn(y)t Fm(\))p Fl(j)d Fn(g)2471 2530 y Fz(\030)2512 2503 y Fm(\()p Fn(y)t Fm(\))p Fn(e)2689 2460 y Fi(\000)p Fz(\014)t Fi(j)p Fz(x)p Fi(\000)p Fz(y)s Fi(j)1067 2747 y Fl(\024)45 b Fn(C)7 b Fm(\()p Fo(2)p Fn(\014)e Fm(\))21 b Fo(sup)1473 2832 y Fz(y)s Fi(2)p Fg(R)1636 2747 y Fl(j)p Fn(u)1721 2774 y FA(0)1761 2747 y Fm(\()p Fn(y)t Fm(\))p Fl(j)c Fn(g)1985 2774 y Fz(\030)2026 2747 y Fm(\()p Fn(y)t Fm(\))p Fn(e)2203 2704 y Fi(\000)p Fz(\014)t Fi(j)p Fz(x)p Fi(\000)p Fz(y)s Fi(j)2541 2747 y Fn(:)3669 2554 y Fm(\()p Fo(3)p Fn(:)p Fo(9)p Fm(\))94 3031 y Fo(Since)28 b Fl(j)p Fn(u)428 3058 y FA(0)468 3031 y Fm(\()p Fn(y)t Fm(\))p Fl(j)17 b Fn(g)692 3058 y Fz(\030)733 3031 y Fm(\()p Fn(y)t Fm(\))27 b Fo(is)g(bounded)g(and)g(con)l(v)o(er)n(ges)g(to)h(0)f(as)h Fn(y)35 b Fl(!)c Fm(+)p Fl(1)p Fo(,)d(we)g(conclude)f(that)g(the)g (quantity)94 3151 y(in)e(\(3.9\))g(tends)f(to)g(0)h(as)g Fn(x)j Fl(!)f Fm(+)p Fl(1)p Fo(.)36 b(Thus,)24 b(we)h(also)f(ha)n(v)o (e)1361 3361 y(lim)1326 3421 y Fz(x)p Fi(!1)1547 3361 y Fl(j)1575 3280 y Fs(\000)1620 3361 y Fn(G)1698 3388 y Fz(t)1755 3361 y Fn(?)e(u)1884 3388 y FA(0)1924 3280 y Fs(\001)1970 3361 y Fm(\()p Fn(x)p Fm(\))17 b Fn(g)2170 3388 y Fz(\030)2212 3361 y Fm(\()p Fn(x)p Fm(\))p Fl(j)43 b Fm(=)i Fo(0)25 b Fn(:)976 b Fm(\()p Fo(3)p Fn(:)p Fo(10)p Fm(\))316 3606 y Fo(W)-8 b(e)24 b(ne)o(xt)f(bound)g(the)h (non\255linear)f(term.)35 b(Assume)23 b Fn(v)31 b Fl(2)e(K)2354 3633 y Fz(\030)r(;\034)2453 3651 y Ff(\003)2523 3606 y Fo(and)24 b Fl(k)p Fn(v)t Fl(k)2843 3633 y Fz(\030)r(;\034)2942 3651 y Ff(\003)3016 3606 y Fn(<)k(\032)p Fo(.)35 b(Then)23 b(an)o(y)h(po)n(wer)94 3726 y(\()p Fl(\025)29 b Fo(1\))c(of)g Fn(v)j Fo(is)d(also)f(in)h Fl(K)981 3753 y Fz(\030)r(;\034)1080 3771 y Ff(\003)1151 3726 y Fo(and)g(one)g(has)f(a)h(bound)f(of)h(the)g (form)1605 3961 y Fl(k)p Fn(v)1707 3918 y FA(3)1747 3961 y Fl(k)1797 3988 y Fz(\030)r(;\034)1896 4006 y Ff(\003)1987 3961 y Fl(\024)44 b Fn(C)2179 3988 y FA(3)2219 3961 y Fn(\032)2271 3918 y FA(3)2336 3961 y Fn(:)94 4171 y Fo(Therefore,)26 b(the)f(method)e(leading)i(to)f(\(3.8\))h(no)n(w)f(yields)1165 4294 y Fs(\014)1165 4354 y(\014)1165 4413 y(\014)1165 4473 y(\014)1198 4303 y(Z)1298 4345 y Fz(t)1254 4529 y FA(0)1350 4438 y Fo(d)p Fn(s)1464 4358 y Fs(\000)1508 4438 y Fn(G)1586 4465 y Fz(t)p Fi(\000)p Fz(s)1744 4438 y Fn(?)e(v)1868 4395 y FA(3)1864 4465 y Fz(s)1908 4358 y Fs(\001)1953 4438 y Fm(\()p Fn(x)p Fm(\))p Fn(g)2136 4465 y Fz(\030)2179 4438 y Fm(\()p Fn(x)p Fm(\))2314 4294 y Fs(\014)2314 4354 y(\014)2314 4413 y(\014)2314 4473 y(\014)2391 4438 y Fl(\024)44 b Fn(C)2583 4465 y FA(4)2623 4438 y Fn(\032)2675 4395 y FA(3)2715 4438 y Fn(t)25 b(;)94 4708 y Fo(and)g(if)g(also)f Fl(k)p Fn(w)s Fl(k)708 4735 y Fz(\030)r(;\034)807 4753 y Ff(\003)881 4708 y Fn(<)29 b(\032)p Fo(,)24 b(then)g(a)h(v)n(ariant)g(of)f(that)h (method)f(gi)n(v)o(es:)756 4854 y Fs(\014)756 4914 y(\014)756 4973 y(\014)756 5033 y(\014)806 4863 y(Z)905 4906 y Fz(t)861 5089 y FA(0)957 4998 y Fo(d)p Fn(s)1071 4918 y Fs(\000)1116 4998 y Fn(G)1194 5025 y Fz(t)p Fi(\000)p Fz(s)1351 4998 y Fn(?)e(v)1475 4955 y FA(3)1471 5025 y Fz(s)1515 4918 y Fs(\001)1561 4998 y Fm(\()p Fn(x)p Fm(\))p Fn(g)1744 5025 y Fz(\030)1786 4998 y Fm(\()p Fn(x)p Fm(\))g Fl(\000)2042 4863 y Fs(Z)2142 4906 y Fz(t)2098 5089 y FA(0)2194 4998 y Fo(d)p Fn(s)2308 4918 y Fs(\000)2352 4998 y Fn(G)2430 5025 y Fz(t)p Fi(\000)p Fz(s)2588 4998 y Fn(?)g(w)2734 4955 y FA(3)2731 5025 y Fz(s)2774 4918 y Fs(\001)2819 4998 y Fm(\()p Fn(x)p Fm(\))p Fn(g)3002 5025 y Fz(\030)3045 4998 y Fm(\()p Fn(x)p Fm(\))3180 4854 y Fs(\014)3180 4914 y(\014)3180 4973 y(\014)3180 5033 y(\014)1357 5223 y Fl(\024)45 b Fn(C)1550 5250 y FA(5)1590 5223 y Fn(\032)1642 5180 y FA(2)1681 5223 y Fn(t)61 b Fo(sup)1734 5308 y Fz(s)p Fi(2)p FA([0)p Fz(;t)p FA(])1982 5223 y Fo(sup)1977 5308 y Fz(x)p Fi(2)p Fg(R)2126 5138 y Fs(\014)2126 5198 y(\014)2159 5223 y Fn(v)2207 5250 y Fz(s)2250 5223 y Fm(\()p Fn(x)p Fm(\))21 b Fl(\000)i Fn(w)2577 5250 y Fz(s)2620 5223 y Fm(\()p Fn(x)p Fm(\))2755 5138 y Fs(\014)2755 5198 y(\014)2804 5223 y Fn(g)2852 5250 y Fz(\030)2895 5223 y Fm(\()p Fn(x)p Fm(\))i Fn(:)p eop %%Page: 8 8 8 7 bop 94 -45 a Fh(Mar)n(ginal)38 b(St)-7 b(ability)38 b(Bound)2382 b Fk(8)94 194 y Fo(T)-8 b(aking)48 b(the)g(center)h(of)f (the)g(ball)g(at)g Fm(\()p Fn(x;)17 b(t)p Fm(\))59 b Fl(7!)h Fo(0)48 b(and)g(the)g(radius)g Fn(\032)59 b Fm(=)i Fo(2)p Fn(C)3055 221 y FA(2)3095 194 y Fn(K)55 b Fo(and)48 b(then)g Fn(\034)3691 221 y Fi(\003)3797 194 y Fn(<)94 314 y Fo(min)p Fl(f)p Fm(\()p Fo(4)p Fn(C)460 341 y FA(4)499 314 y Fn(\032)551 271 y FA(3)590 314 y Fm(\))629 271 y Fi(\000)p FA(1)731 314 y Fn(;)17 b Fm(\()p Fo(4)p Fn(C)936 341 y FA(5)975 314 y Fn(\032)1027 271 y FA(2)1066 314 y Fm(\))1105 271 y Fi(\000)p FA(1)1207 314 y Fl(g)p Fo(,)35 b(we)e(ha)n(v)o(e)g(a)g(contraction)f(and)h(hence)g(a)g(unique)f(\002x) o(ed)h(point)f Fn(v)37 b Fo(for)c Fl(Q)p Fo(.)94 433 y(F)o(or)22 b Fn(j)34 b Fm(=)28 b Fo(1)p Fn(;)17 b Fo(2)p Fn(;)g Fo(3,)j(we)i(use)g(the)g(same)f(methods)g(since)h(we)g(can)g (push)f(all)g(deri)n(v)n(ati)n(v)o(es)f(from)h(the)h(operator)g Fn(G)3839 460 y Fz(t)94 553 y Fo(to)i(the)f(function)g Fn(v)t Fo(,)g(because)h Fn(G)1214 580 y Fz(t)1250 553 y Fn(?)f Fo(is)g(a)h(con)l(v)n(olution.)34 b(The)23 b(details)g(are)h (left)g(to)f(the)g(reader)-5 b(.)36 b(The)24 b(e)o(xistence)94 672 y(of)h(this)f(\002x)o(ed)h(point)f(clearly)h(sho)n(ws)e(Lemma)h (3.2.)p 3812 618 63 2 v 3812 677 2 59 v 3872 677 V 3812 679 63 2 v 316 911 a(W)-8 b(e)25 b(come)g(back)g(to)f(the)h(proof)g(of) f(Theorem)h(3.1.)35 b(W)-8 b(e)26 b(de\002ne)1359 1150 y Fm(\000\()p Fn(t)p Fm(\))44 b(=)g Fo(lim)17 b(sup)1742 1231 y Fz(x)p Fi(!1)2005 1150 y Fl(j)p Fn(u)p Fm(\()p Fn(x;)g(t)p Fm(\))p Fn(g)2354 1177 y Fz(\030)2395 1150 y Fm(\()p Fn(x)p Fm(\))p Fl(j)24 b Fn(:)94 1445 y Fo(By)i(assumption,)c (we)j(ha)n(v)o(e)g Fm(\000\()p Fo(0)p Fm(\))i(=)h Fo(0)d(and)g(by)f (Lemma)h(3.2)f(we)h(ha)n(v)o(e)1548 1684 y Fl(j)p Fn(u)p Fm(\()p Fn(x;)17 b(t)p Fm(\))p Fl(j)42 b(\024)j Fn(\032=g)2191 1711 y Fz(\030)2234 1684 y Fm(\()p Fn(x)p Fm(\))24 b Fn(;)94 1923 y Fo(so)k(that)f Fm(\000\()p Fn(t)p Fm(\))32 b Fl(\024)h Fn(\032)27 b Fo(for)h Fn(t)k Fl(\024)h Fn(\034)1151 1950 y Fi(\003)1196 1923 y Fo(.)45 b(W)-8 b(e)28 b(no)n(w)f(sho)n(w)g (it)h(is)f(actually)g(0)h(for)g(those)g Fn(t)p Fo(.)45 b(Consider)27 b Fl(Q)i Fo(as)f(in)f(\(3.6\).)94 2042 y(Note)e(that)419 2258 y Fm(\000\()p Fn(t)p Fm(\))44 b(=)h Fo(lim)17 b(sup)803 2339 y Fz(x)p Fi(!1)1065 2258 y Fl(j)p Fn(u)p Fm(\()p Fn(x;)g(t)p Fm(\))p Fn(g)1414 2285 y Fz(\030)1456 2258 y Fm(\()p Fn(x)p Fm(\))p Fl(j)639 2524 y Fm(=)45 b Fo(lim)17 b(sup)803 2606 y Fz(x)p Fi(!1)1065 2524 y Fn(g)1113 2551 y Fz(\030)1156 2524 y Fm(\()p Fn(x)p Fm(\))p Fl(j)1319 2444 y Fs(\000)1364 2524 y Fn(G)1442 2551 y Fz(t)1500 2524 y Fn(?)22 b(u)1629 2551 y FA(0)1668 2444 y Fs(\001)1714 2524 y Fm(\()p Fn(x)p Fm(\))p Fl(j)f Fm(+)i Fo(lim)17 b(sup)2040 2606 y Fz(x)p Fi(!1)2302 2524 y Fn(g)2350 2551 y Fz(\030)2394 2524 y Fm(\()p Fn(x)p Fm(\))p Fl(j)2574 2389 y Fs(Z)2672 2432 y Fz(t)2628 2615 y FA(0)2724 2524 y Fo(d)p Fn(s)2838 2444 y Fs(\000)2882 2524 y Fn(G)2960 2551 y Fz(t)p Fi(\000)p Fz(s)3118 2524 y Fn(?)22 b(u)3247 2482 y FA(3)3247 2551 y Fz(s)3289 2444 y Fs(\001)3335 2524 y Fm(\()p Fn(x)p Fm(\))p Fl(j)i Fn(:)94 2813 y Fo(The)j(\002rst)g(term)g(v)n(anishes)e(by)h(\(3.9\).)42 b(Thus,)26 b Fm(\000)i Fo(only)e(depends)g(on)g(the)h(nonlinear)f (part.)42 b(Using)25 b(\(3.7\),)j(that)94 2933 y(part)d(can)h(be)e (bounded)g(as)204 3238 y Fn(g)252 3265 y Fz(\030)295 3238 y Fm(\()p Fn(x)p Fm(\))447 3093 y Fs(\014)447 3153 y(\014)447 3213 y(\014)447 3273 y(\014)479 3102 y(Z)579 3145 y Fz(t)535 3328 y FA(0)631 3238 y Fo(d)p Fn(s)17 b(G)823 3265 y Fz(t)p Fi(\000)p Fz(s)979 3238 y Fn(?)22 b(u)1108 3195 y FA(3)1108 3265 y Fz(s)1151 3238 y Fm(\()p Fn(x)p Fm(\))1286 3093 y Fs(\014)1286 3153 y(\014)1286 3213 y(\014)1286 3273 y(\014)1363 3238 y Fl(\024)1484 3102 y Fs(Z)1584 3145 y Fz(t)1540 3328 y FA(0)1636 3238 y Fo(d)p Fn(s)1765 3102 y Fs(Z)1882 3238 y Fo(d)p Fn(y)2013 3158 y(g)2061 3185 y Fz(\030)2104 3158 y Fm(\()p Fn(x)p Fm(\))p 2013 3215 226 4 v 2015 3309 a Fn(g)2067 3274 y FA(3)2063 3339 y Fz(\030)2106 3309 y Fm(\()p Fn(y)t Fm(\))2283 3153 y Fs(\014)2283 3213 y(\014)2316 3238 y Fn(G)2394 3265 y Fz(t)p Fi(\000)p Fz(s)2530 3238 y Fm(\()p Fn(x)g Fl(\000)g Fn(y)t Fm(\))2839 3153 y Fs(\014)2839 3213 y(\014)2888 3153 y(\014)2888 3213 y(\014)2921 3238 y Fn(g)2969 3265 y Fz(\030)3012 3238 y Fm(\()p Fn(y)t Fm(\))p Fn(u)3200 3265 y Fz(s)3242 3238 y Fm(\()p Fn(y)t Fm(\))3373 3153 y Fs(\014)3373 3213 y(\014)3405 3195 y FA(3)1363 3528 y Fl(\024)1484 3392 y Fs(Z)1584 3435 y Fz(t)1540 3619 y FA(0)1636 3528 y Fo(d)p Fn(s)1765 3392 y Fs(Z)1882 3528 y Fo(d)p Fn(y)2017 3443 y Fs(\014)2017 3503 y(\014)2050 3528 y Fn(G)2128 3555 y Fz(t)p Fi(\000)p Fz(s)2264 3528 y Fm(\()p Fn(x)f Fl(\000)i Fn(y)t Fm(\))2573 3443 y Fs(\014)2573 3503 y(\014)2622 3528 y Fn(e)2668 3485 y Fz(\014)t Fi(j)p Fz(x)p Fi(\000)p Fz(y)s Fi(j)2953 3443 y Fs(\014)2953 3503 y(\014)2986 3528 y Fn(g)3034 3555 y Fz(\030)3077 3528 y Fm(\()p Fn(y)t Fm(\))p Fn(u)3265 3555 y Fz(s)3307 3528 y Fm(\()p Fn(y)t Fm(\))3438 3443 y Fs(\014)3438 3503 y(\014)3470 3485 y FA(3)1363 3795 y Fl(\024)44 b Fn(C)7 b Fm(\()p Fn(\014)e Fm(\))1718 3660 y Fs(Z)1818 3702 y Fz(t)1774 3886 y FA(0)1870 3795 y Fo(d)p Fn(s)2000 3660 y Fs(Z)2116 3795 y Fo(d)p Fn(z)21 b Fm(\()p Fn(t)h Fl(\000)g Fn(s)p Fm(\))2515 3752 y Fi(\000)p FA(1)p Fz(=)p FA(4)2693 3795 y Fn(e)2739 3752 y Fi(\000)p FA(2)p Fi(j)p Fz(z)s Fi(j)p Fj(\()p Fz(t)p Fi(\000)p Fz(s)p Fj(\))3117 3709 y Ff(\000)p Fc(1)p Fe(=)p Fc(4)1756 4016 y Fl(\001)1805 3932 y Fs(\014)1805 3991 y(\014)1839 4016 y Fn(g)1887 4043 y Fz(\030)1930 4016 y Fm(\()p Fn(x)g Fl(\000)g Fn(z)t Fm(\))p Fn(u)2293 4043 y Fz(s)2336 4016 y Fm(\()p Fn(x)g Fl(\000)h Fn(z)t Fm(\))2643 3932 y Fs(\014)2643 3991 y(\014)2676 3974 y FA(3)2758 4016 y Fn(:)3619 3595 y Fm(\()p Fo(3)p Fn(:)p Fo(11)p Fm(\))94 4253 y Fo(W)-8 b(e)29 b(need)f(an)g(upper)g(bound)f(for)h(the)g(lim)17 b(sup)1705 4280 y Fz(x)p Fi(!1)1942 4253 y Fo(of)28 b(this)f(e)o (xpression.)44 b(Fix)28 b(an)g Fn(")k(>)h Fo(0.)45 b(F)o(or)28 b Fn(s)j Fl(2)i Fo([0)p Fn(;)17 b(t)p Fo(],)94 4373 y(we)26 b(can)f(\002nd)g(an)g Fn(\021)t Fm(\()p Fn(s;)17 b(")p Fm(\))26 b Fn(>)i Fo(0)d(such)g(that)1382 4611 y(sup)1297 4702 y Fz(y)s Fi(\025)p Fz(\021)s Fj(\()p Fz(s;")p Fj(\))1623 4611 y Fl(j)p Fn(g)1699 4638 y Fz(\030)1742 4611 y Fm(\()p Fn(y)t Fm(\))p Fn(u)1930 4638 y Fz(s)1971 4611 y Fm(\()p Fn(y)t Fm(\))p Fl(j)i(\024)h Fm(\000\()p Fn(s)p Fm(\))21 b(+)i Fn(")28 b(:)94 4933 y Fo(There)e(is)e(also)g(a)i(number)e Fn(\020)7 b Fm(\()p Fn(")p Fm(\))27 b Fn(>)i Fo(0)24 b(such)h(that)f(for)h(an)o(y)g Fn(s)i Fl(2)h Fo([0)p Fn(;)17 b(t)p Fo(]:)1109 5087 y Fs(Z)1164 5314 y Fi(j)p Fz(z)s Fi(j)p Fz(>\020)5 b Fj(\()p Fz(")p Fj(\))1477 5223 y Fo(d)p Fn(z)21 b Fm(\()p Fn(t)h Fl(\000)g Fn(s)p Fm(\))1876 5180 y Fi(\000)p FA(1)p Fz(=)p FA(4)2054 5223 y Fn(e)2100 5180 y Fi(\000)p FA(2)p Fi(j)p Fz(z)s Fi(j)p Fj(\()p Fz(t)p Fi(\000)p Fz(s)p Fj(\))2478 5137 y Ff(\000)p Fc(1)p Fe(=)p Fc(4)2656 5223 y Fl(\024)28 b Fn(")d(:)p eop %%Page: 9 9 9 8 bop 94 -45 a Fh(Mar)n(ginal)38 b(St)-7 b(ability)38 b(Bound)2382 b Fk(9)94 194 y Fo(If)26 b Fn(x)h(>)i(\020)7 b Fm(\()p Fn(")p Fm(\))22 b(+)g Fn(\021)t Fm(\()p Fn(s;)17 b(")p Fm(\))p Fo(,)24 b(we)h(ha)n(v)o(e)442 332 y Fs(Z)558 467 y Fo(d)p Fn(z)c Fm(\()p Fn(t)h Fl(\000)h Fn(s)p Fm(\))958 424 y Fi(\000)p FA(1)p Fz(=)p FA(4)1135 467 y Fn(e)1181 424 y Fi(\000)p FA(2)p Fi(j)p Fz(z)s Fi(j)p Fj(\()p Fz(t)p Fi(\000)p Fz(s)p Fj(\))1559 381 y Ff(\000)p Fc(1)p Fe(=)p Fc(4)1710 467 y Fl(j)p Fn(g)1786 494 y Fz(\030)1828 467 y Fm(\()p Fn(x)f Fl(\000)h Fn(z)t Fm(\))p Fn(u)2192 494 y Fz(s)2235 467 y Fm(\()p Fn(x)f Fl(\000)g Fn(z)t Fm(\))p Fl(j)2569 424 y FA(3)2637 467 y Fl(\024)28 b Fm(\(\000\()p Fn(s)p Fm(\))22 b(+)g Fn(")p Fm(\))3174 424 y FA(3)3236 467 y Fm(+)h Fn(\032)3388 424 y FA(3)3427 467 y Fn(")j(;)94 732 y Fo(by)35 b(Lemma)f(3.2.)64 b(W)-8 b(e)35 b(cannot)f(conclude)h (directly)f(by)g(inte)o(gration)f(o)o(v)o(er)g Fn(s)i Fo(because)f Fn(\021)39 b Fo(depends)34 b(on)g Fn(s)p Fo(.)94 852 y(Ho)n(we)n(v)o(er)l(,)d Fn(\021)t Fm(\()p Fn(s;)17 b(")p Fm(\))29 b Fo(is)h(\002nite)g(for)g(almost)f(e)n(v)o (ery)h Fn(s)g Fo(\(in)g(reality)g(for)g(e)n(v)o(ery)g Fn(s)p Fo(\).)52 b(Therefore,)32 b(we)f(can)f(\002nd)h(a)94 972 y(\002nite)25 b(number)f Fm(\002\()p Fn(")p Fm(\))i Fo(such)e(that)h(the)f(set)1200 1194 y Fn(E)6 b Fm(\()p Fn(")p Fm(\))43 b(=)1569 1113 y Fs(\010)1627 1194 y Fn(s)28 b Fl(2)g Fo([0)p Fn(;)17 b(t)p Fo(])27 b Fl(j)g Fn(\021)t Fm(\()p Fn(s;)17 b(")p Fm(\))43 b Fn(>)i Fm(\002\()p Fn(")p Fm(\))2710 1113 y Fs(\011)94 1431 y Fo(has)22 b(Lebesgue)f(measure)h(at)f(most)f Fn(")1360 1388 y FA(2)1422 1431 y Fo(\(note)i(that)e Fn(E)6 b Fm(\()p Fn(")p Fm(\))21 b Fo(is)g(measurable\).)35 b(Therefore,)23 b(if)e Fn(x)28 b(>)g Fm(\002\()p Fn(")p Fm(\))15 b(+)g Fn(\020)7 b Fm(\()p Fn(")p Fm(\))94 1550 y Fo(we)26 b(ha)n(v)o(e)347 1664 y Fs(Z)447 1707 y Fz(t)402 1891 y FA(0)482 1800 y Fo(d)p Fn(s)611 1664 y Fs(Z)728 1800 y Fo(d)p Fn(z)21 b Fm(\()p Fn(t)h Fl(\000)g Fn(s)p Fm(\))1127 1757 y Fi(\000)p FA(1)p Fz(=)p FA(4)1305 1800 y Fn(e)1351 1757 y Fi(\000)p FA(2)p Fi(j)p Fz(z)s Fi(j)p Fj(\()p Fz(t)p Fi(\000)p Fz(s)p Fj(\))1729 1714 y Ff(\000)p Fc(1)p Fe(=)p Fc(4)1879 1800 y Fl(j)p Fn(g)1955 1827 y Fz(\030)1998 1800 y Fm(\()p Fn(x)g Fl(\000)g Fn(z)t Fm(\))p Fn(u)2361 1827 y Fz(s)2404 1800 y Fm(\()p Fn(x)g Fl(\000)h Fn(z)t Fm(\))p Fl(j)2739 1757 y FA(3)509 2067 y Fm(=)631 1931 y Fs(Z)687 2158 y Fj(\()p FA([0)p Fz(;t)p FA(])p Fi(n)p Fz(E)t Fj(\()p Fz(")p Fj(\)\))p Fi([)p Fz(E)t Fj(\()p Fz(")p Fj(\))1324 2067 y Fo(d)p Fn(s)1454 1931 y Fs(Z)1570 2067 y Fo(d)p Fn(z)f Fm(\()p Fn(t)f Fl(\000)i Fn(s)p Fm(\))1970 2024 y Fi(\000)p FA(1)p Fz(=)p FA(4)2147 2067 y Fn(e)2193 2024 y Fi(\000)p FA(2)p Fi(j)p Fz(z)s Fi(j)p Fj(\()p Fz(t)p Fi(\000)p Fz(s)p Fj(\))2571 1981 y Ff(\000)p Fc(1)p Fe(=)p Fc(4)2722 2067 y Fl(j)p Fn(g)2798 2094 y Fz(\030)2840 2067 y Fm(\()p Fn(x)f Fl(\000)h Fn(z)t Fm(\))p Fn(u)3204 2094 y Fz(s)3247 2067 y Fm(\()p Fn(x)f Fl(\000)g Fn(z)t Fm(\))p Fl(j)3581 2024 y FA(3)509 2351 y Fl(\024)45 b Fn(C)702 2377 y FA(6)759 2215 y Fs(Z)858 2258 y Fz(t)814 2441 y FA(0)910 2351 y Fo(d)p Fn(s)1024 2270 y Fs(\000)1069 2351 y Fm(\(\000\()p Fn(s)p Fm(\))21 b(+)i Fn(")p Fm(\))1501 2308 y FA(3)1563 2351 y Fm(+)g Fn(\032)1715 2308 y FA(3)1754 2351 y Fn(")1800 2270 y Fs(\001)1868 2351 y Fm(+)g Fn(C)2039 2377 y FA(7)2079 2351 y Fn(\032)2131 2308 y FA(3)2187 2215 y Fs(Z)2242 2441 y Fz(E)t Fj(\()p Fz(")p Fj(\))2426 2351 y Fo(d)p Fn(s)17 b Fm(\()p Fn(t)k Fl(\000)i Fn(s)p Fm(\))2822 2308 y Fi(\000)p FA(1)p Fz(=)p FA(4)3024 2351 y Fn(:)94 2666 y Fo(The)i(last)g(inte)o(gral)e(is)i(of)g(order)g Fn(")1246 2624 y FA(1)p Fz(=)p FA(2)1387 2666 y Fo(by)f(the)h(Schw)o (arz)h(inequality)-6 b(.)34 b(Since)25 b Fn(")j(>)g Fo(0)d(is)f (arbitrary)-6 b(,)25 b(we)g(get)1478 2936 y Fm(\000\()p Fn(t)p Fm(\))44 b Fl(\024)h Fn(C)1891 2963 y FA(8)1947 2801 y Fs(Z)2047 2844 y Fz(t)2003 3027 y FA(0)2099 2936 y Fo(d)p Fn(s)17 b Fm(\000\()p Fn(s)p Fm(\))2400 2893 y FA(3)2463 2936 y Fn(:)94 3215 y Fo(Since)22 b Fm(\000)f Fo(is)f(bounded)g(by)g(what)g(we)h(said)g(abo)o(v)o(e)e(and)i Fm(\000\()p Fo(0)p Fm(\))27 b(=)h Fo(0,)22 b(it)e(follo)n(ws)f(from)h (Gronw)o(all')-5 b(s)20 b(lemma)g(that)94 3335 y Fm(\000\()p Fn(t)p Fm(\))33 b(=)g Fo(0)28 b(for)h Fn(t)k Fl(\024)g Fn(\034)859 3362 y Fi(\003)904 3335 y Fo(.)47 b(One)28 b(then)g(repeats)h(the)f(ar)n(gument)g(for)h(all)f(consecuti)n(v)o(e)f (interv)n(als)g(of)h(length)g Fn(\034)3804 3362 y Fi(\003)3849 3335 y Fo(.)94 3454 y(The)d(proof)g(of)g(the)g(corresponding)e(bounds)h (on)h(the)f(deri)n(v)n(ati)n(v)o(es)e(is)j(similar)e(and)i(is)g(left)f (to)h(the)f(reader)-5 b(.)p 3812 3400 63 2 v 3812 3459 2 59 v 3872 3459 V 3812 3461 63 2 v 94 3914 a Fp(4.)30 b(Bound)h(on)f(the)g(speed)94 4103 y Fo(W)-8 b(e)26 b(de\002ne)f Fn(J)573 4130 y Fz(\030)642 4103 y Fo(by)1292 4289 y Fn(J)1347 4316 y Fz(\030)1390 4289 y Fm(\()p Fn(t)p Fm(\))44 b(=)1670 4153 y Fs(Z)1769 4196 y Fi(1)1725 4380 y Fz(\030)1791 4289 y Fo(d)p Fn(x)17 b Fl(j)p Fn(u)p Fm(\()p Fn(x;)g(t)p Fm(\))p Fl(j)2244 4246 y FA(2)2281 4289 y Fn(e)2327 4246 y FA(2)p Fz(\014)t Fj(\()p Fz(x)p Fi(\000)p Fz(\030)r Fj(\))2649 4289 y Fn(;)992 b Fm(\()p Fo(4)p Fn(:)p Fo(1)p Fm(\))94 4540 y Fo(where)25 b Fn(u)p Fm(\()p Fn(x;)17 b(t)p Fm(\))23 b Fo(is)h(the)h(solution)d(of)i(the)h(Swift\255Hohenber) n(g)f(equation.)35 b(The)24 b(main)g(result)g(of)g(this)f(paper)i(is)94 4732 y Fk(Theor)n(em)36 b(4.1.)64 b Fd(Let)34 b Fn(u)p Fm(\()p Fn(x;)17 b(t)p Fm(\))33 b Fd(be)h(a)h(solution)d(of)i(the)g (Swift\255Hohenber)n(g)g(equation)g(\(1.5\))g(for)g(an)g(initial)94 4851 y(condition)21 b Fn(u)550 4878 y FA(0)590 4851 y Fm(\()p Fn(x)p Fm(\))27 b(=)i Fn(u)p Fm(\()p Fn(x;)17 b Fo(0)p Fm(\))k Fd(which)h(is)g(in)g Fb(B)p Fd(,)g(which)g (satis\002es)g Fn(J)2403 4878 y FA(0)2444 4851 y Fm(\()p Fo(0)p Fm(\))27 b Fn(<)h Fl(1)22 b Fd(for)h(some)e Fn(\014)34 b(>)28 b Fo(0)22 b Fd(and)h(which)94 4971 y(satis\002es)i(the)g (assumptions)d(of)j(Theorem)g(3.1.)35 b(Then)25 b(one)f(has)1245 5240 y Fo(lim)1217 5300 y Fz(t)p Fi(!1)1422 5105 y Fs(Z)1522 5148 y Fi(1)1477 5331 y Fz(ct)1623 5240 y Fo(d)p Fn(x)17 b Fl(j)p Fn(u)p Fm(\()p Fn(x;)g(t)p Fm(\))p Fl(j)2076 5198 y FA(2)2113 5240 y Fn(e)2159 5198 y FA(2)p Fz(\014)t Fj(\()p Fz(x)p Fi(\000)p Fz(ct)p Fj(\))2528 5240 y Fm(=)45 b Fo(0)24 b Fn(;)917 b Fm(\()p Fo(4)p Fn(:)p Fo(2)p Fm(\))p eop %%Page: 10 10 10 9 bop 94 -45 a Fh(Mar)n(ginal)38 b(St)-7 b(ability)38 b(Bound)2332 b Fk(10)94 194 y Fd(for)26 b(all)e Fn(c)k(>)536 113 y Fs(\000)582 194 y Fn(")628 151 y FA(2)690 194 y Fm(+)23 b Fo(4)p Fn(\014)901 151 y FA(2)963 194 y Fm(+)g Fo(8)p Fn(\014)1174 151 y FA(4)1214 113 y Fs(\001)1260 194 y Fn(=\014)5 b Fd(.)94 432 y Fk(Remark.)26 b Fo(If)f(one)g(is)g (willing)e(to)i(pay)g(a)g(price)g(of)g(slightly)f(more)g(complicated)g (formulations)g(and)h(proofs,)94 551 y(one)33 b(can)g(omit)f(the)h (condition)e(on)h Fn(J)1409 578 y FA(0)1449 551 y Fm(\()p Fo(0)p Fm(\))g Fo(in)h(Theorem)f(4.1.)60 b(One)32 b(w)o(ould)g(then)h (assume)f(the)h(pointwise)94 671 y(bounds)c(of)h(Theorem)f(3.1)h(fore)g (some)f Fn(\014)40 b(>)35 b Fo(0)29 b(and)h(w)o(ork)g(throughout)e(the) h(proof)h(with)f(a)h Fn(J)3394 698 y Fz(\030)3437 671 y Fm(\()p Fn(t)p Fm(\))g Fo(de\002ned)94 803 y(with)25 b(some)f Fn(\014)593 760 y Fi(0)649 803 y Fn(<)k(\014)5 b Fo(,)26 b(b)n(ut)e(arbitrarily)g(close)h(to)f(it,)g(since)h(the)g (condition)e(on)h Fn(c)i Fo(is)e(open.)94 982 y Fk(Pr)n(oof)o(.)31 b Fo(W)-8 b(e)32 b(de\002ne)f Fn(v)869 1009 y Fz(\030)913 982 y Fm(\()p Fn(x;)17 b(t)p Fm(\))35 b(=)i Fn(u)p Fm(\()p Fn(x;)17 b(t)p Fm(\))p Fn(e)1597 939 y Fz(\014)t Fj(\()p Fz(x)p Fi(\000)p Fz(\030)r Fj(\))1858 982 y Fo(,)33 b(so)d(that)h Fn(J)2271 1009 y Fz(\030)2315 982 y Fm(\()p Fn(t)p Fm(\))k(=)2578 901 y Fs(R)2645 939 y Fi(1)2625 1017 y Fz(\030)2746 982 y Fo(d)p Fn(x)17 b Fl(j)p Fn(v)2946 1009 y Fz(\030)2989 982 y Fm(\()p Fn(x;)g(t)p Fm(\))p Fl(j)3233 939 y FA(2)3271 982 y Fo(,)32 b(and)f Fn(v)3551 1009 y Fz(\030)3626 982 y Fo(solv)o(es)94 1101 y(the)25 b(equation)394 1337 y Fn(@)447 1364 y Fz(t)482 1337 y Fn(v)530 1364 y Fz(\030)574 1337 y Fm(\()p Fn(x;)17 b(t)p Fm(\))43 b(=)i Fn(")1001 1295 y FA(2)1041 1337 y Fn(v)1089 1364 y Fz(\030)1133 1337 y Fm(\()p Fn(x;)17 b(t)p Fm(\))k Fl(\000)1470 1257 y Fs(\000)1515 1337 y Fo(1)h Fm(+)h(\()p Fn(@)1779 1364 y Fz(x)1851 1337 y Fl(\000)f Fn(\014)5 b Fm(\))2050 1295 y FA(2)2091 1257 y Fs(\001)2136 1295 y FA(2)2176 1337 y Fn(v)2224 1364 y Fz(\030)2268 1337 y Fm(\()p Fn(x;)17 b(t)p Fm(\))k Fl(\000)i Fn(v)2657 1295 y FA(3)2653 1364 y Fz(\030)2696 1337 y Fm(\()p Fn(x;)17 b(t)p Fm(\))p Fn(e)2958 1295 y Fi(\000)p FA(2)p Fz(\014)t Fj(\()p Fz(x)p Fi(\000)p Fz(\030)r Fj(\))3342 1337 y Fn(:)299 b Fm(\()p Fo(4)p Fn(:)p Fo(3)p Fm(\))94 1573 y Fo(Since)34 b Fn(u)f Fo(is)g(real,)i(the)e(absolute)f(v)n(alues)h(in)f(the)h(de\002nition)f (of)h Fn(J)2405 1600 y Fz(\030)2449 1573 y Fm(\()p Fn(t)p Fm(\))g Fo(can)g(be)g(omitted.)59 b(Dif)n(ferentiating)94 1693 y(\(4.1\))25 b(with)f(respect)h(to)g(time,)f(we)h(get)1235 1938 y FA(1)p 1235 1954 35 4 v 1235 2011 a(2)1281 1977 y Fn(@)1334 2004 y Fz(t)1369 1977 y Fn(J)1424 2004 y Fz(\030)1468 1977 y Fm(\()p Fn(t)p Fm(\))44 b(=)1748 1841 y Fs(Z)1847 1884 y Fi(1)1803 2068 y Fz(\030)1869 1977 y Fo(d)p Fn(x)17 b(v)2041 2004 y Fz(\030)2084 1977 y Fm(\()p Fn(x;)g(t)p Fm(\))p Fn(@)2353 2004 y Fz(t)2387 1977 y Fn(v)2435 2004 y Fz(\030)2479 1977 y Fm(\()p Fn(x;)g(t)p Fm(\))23 b Fn(:)94 2277 y Fo(Since)f Fn(\030)k Fo(is)20 b(\002x)o(ed)h(throughout)f(the)h(calculation,)g(we)g(omit)f(the)h (inde)o(x)g(of)g Fn(v)2677 2304 y Fz(\030)2721 2277 y Fo(.)34 b(W)-8 b(e)22 b(also)f(omit)f(the)h(ar)n(guments)94 2416 y Fm(\()p Fn(x;)c(t)p Fm(\))p Fo(.)60 b(Note)33 b(that)f(by)h(Theorem)g(3.1,)i(lim)1648 2443 y Fz(x)p Fi(!1)1873 2416 y Fn(@)1932 2374 y Fz(j)1926 2443 y(x)1976 2416 y Fn(v)2024 2443 y Fz(\030)2068 2416 y Fm(\()p Fn(x;)17 b(t)p Fm(\))38 b(=)i Fo(0,)35 b(for)e Fn(j)45 b Fm(=)40 b Fo(0)p Fn(;)17 b(:)g(:)g(:)d(;)j Fo(3,)34 b(so)f(that)f(we)i(can)94 2548 y(freely)26 b(inte)o(grate)e(by)g(parts)h(in)f(the)h(follo)n(wing) e(calculation.)35 b(W)-8 b(e)25 b(\002nd,)g(using)f Fn(@)2878 2575 y Fz(x)2928 2548 y Fn(v)31 b Fm(=)d Fn(v)3164 2506 y Fi(0)3192 2548 y Fo(:)133 2807 y FA(1)p 133 2823 V 133 2881 a(2)180 2846 y Fn(@)233 2873 y Fz(t)267 2846 y Fn(J)322 2873 y Fz(\030)366 2846 y Fm(\()p Fn(t)p Fm(\))44 b(=)646 2711 y Fs(Z)745 2754 y Fi(1)701 2937 y Fz(\030)767 2846 y Fo(d)p Fn(x)17 b(v)943 2706 y Fs(\022)1015 2846 y Fn(")1061 2803 y FA(2)1102 2846 y Fn(v)26 b Fl(\000)1275 2766 y Fs(\000)1321 2846 y Fo(1)c Fm(+)h(\()p Fn(@)1585 2873 y Fz(x)1656 2846 y Fl(\000)g Fn(\014)5 b Fm(\))1856 2803 y FA(2)1896 2766 y Fs(\001)1942 2803 y FA(2)1982 2846 y Fn(v)26 b Fl(\000)c Fn(v)2207 2803 y FA(3)2247 2846 y Fn(e)2293 2803 y Fi(\000)p FA(2)p Fz(\014)t Fj(\()p Fz(x)p Fi(\000)p Fz(\030)r Fj(\))2653 2706 y Fs(\023)524 3135 y Fm(=)646 2999 y Fs(Z)745 3042 y Fi(1)701 3226 y Fz(\030)767 3135 y Fo(d)p Fn(x)17 b(v)943 2994 y Fs(\022)1015 3135 y Fn(")1061 3092 y FA(2)1102 3135 y Fn(v)26 b Fl(\000)1275 3054 y Fs(\000)1321 3135 y Fo(1)c Fm(+)h Fn(@)1552 3092 y FA(2)1546 3162 y Fz(x)1618 3135 y Fl(\000)f Fo(2)p Fn(\014)5 b(@)1881 3162 y Fz(x)1954 3135 y Fm(+)23 b Fn(\014)2115 3092 y FA(2)2155 3054 y Fs(\001)2201 3092 y FA(2)2241 3135 y Fn(v)j Fl(\000)c Fn(v)2466 3092 y FA(3)2506 3135 y Fn(e)2552 3092 y Fi(\000)p FA(2)p Fz(\014)t Fj(\()p Fz(x)p Fi(\000)p Fz(\030)r Fj(\))2911 2994 y Fs(\023)524 3424 y Fm(=)646 3288 y Fs(Z)745 3331 y Fi(1)701 3514 y Fz(\030)767 3424 y Fo(d)p Fn(x)17 b(v)943 3283 y Fs(\022)1015 3424 y Fn(")1061 3381 y FA(2)1102 3424 y Fn(v)26 b Fl(\000)c Fn(@)1334 3381 y FA(4)1328 3450 y Fz(x)1378 3424 y Fn(v)k Fm(+)d Fo(4)p Fn(\014)5 b(@)1722 3381 y FA(3)1716 3450 y Fz(x)1766 3424 y Fn(v)26 b Fl(\000)d Fo(2)p Fm(\()p Fo(1)f Fm(+)g Fo(3)p Fn(\014)2311 3381 y FA(2)2351 3424 y Fm(\))p Fn(@)2449 3381 y FA(2)2443 3450 y Fz(x)2493 3424 y Fn(v)k Fm(+)d Fo(4)p Fn(\014)5 b Fm(\()p Fo(1)22 b Fm(+)h Fn(\014)3050 3381 y FA(2)3090 3424 y Fm(\))p Fn(@)3182 3450 y Fz(x)3232 3424 y Fn(v)j Fl(\000)d Fm(\()p Fo(1)e Fm(+)i Fn(\014)3677 3381 y FA(2)3717 3424 y Fm(\))3756 3381 y FA(2)3796 3424 y Fn(v)1042 3712 y Fl(\000)f Fn(v)1193 3669 y FA(3)1233 3712 y Fn(e)1279 3669 y Fi(\000)p FA(2)p Fz(\014)t Fj(\()p Fz(x)p Fi(\000)p Fz(\030)r Fj(\))1639 3572 y Fs(\023)524 3991 y Fm(=)646 3856 y Fs(Z)745 3898 y Fi(1)701 4082 y Fz(\030)767 3991 y Fo(d)p Fn(x)891 3851 y Fs(\022)963 3910 y(\000)1009 3991 y Fn(")1055 3948 y FA(2)1118 3991 y Fl(\000)g Fm(\()p Fo(1)g Fm(+)h Fn(\014)1489 3948 y FA(2)1529 3991 y Fm(\))1568 3948 y FA(2)1608 3910 y Fs(\001)1653 3991 y Fn(v)1705 3948 y FA(2)1767 3991 y Fl(\000)g Fn(e)1913 3948 y Fi(\000)p FA(2)p Fz(\014)t Fj(\()p Fz(x)p Fi(\000)p Fz(\030)r Fj(\))2272 3991 y Fn(v)2324 3948 y FA(4)2386 3991 y Fl(\000)g Fo(2)p Fm(\()p Fo(1)e Fm(+)i Fo(3)p Fn(\014)2857 3948 y FA(2)2897 3991 y Fm(\))p Fn(v)t(v)3040 3948 y Fi(00)1042 4280 y Fm(+)47 b Fn(v)1218 4237 y Fi(0)1246 4280 y Fn(v)1298 4237 y Fi(000)1394 4280 y Fl(\000)22 b Fo(4)p Fn(\014)5 b(v)1656 4237 y Fi(0)1685 4280 y Fn(v)1737 4237 y Fi(00)1787 4139 y Fs(\023)1042 4504 y Fm(+)1141 4423 y Fs(\000)1187 4504 y Fn(v)t(v)1291 4461 y Fi(000)1387 4504 y Fl(\000)22 b Fo(4)p Fn(\014)5 b(v)t(v)1701 4461 y Fi(00)1774 4504 y Fl(\000)23 b Fo(2)p Fn(\014)5 b Fm(\()p Fo(1)22 b Fm(+)h Fn(\014)2257 4461 y FA(2)2297 4504 y Fm(\))p Fn(v)2388 4461 y FA(2)2428 4423 y Fs(\001)2473 4419 y(\014)2473 4479 y(\014)2507 4544 y Fz(x)p Fj(=)p 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Fn(v)t(v)1359 5235 y Fi(000)1455 5278 y Fl(\000)g Fo(4)p Fn(\014)5 b(v)t(v)1770 5235 y Fi(00)1843 5278 y Fl(\000)23 b Fo(2)p Fn(\014)5 b Fm(\()p Fo(1)22 b Fm(+)g Fn(\014)2325 5235 y FA(2)2366 5278 y Fm(\))p Fn(v)2457 5235 y FA(2)2518 5278 y Fl(\000)h Fn(v)2670 5235 y Fi(0)2698 5278 y Fn(v)2750 5235 y Fi(00)2823 5278 y Fm(+)f Fo(2)p Fn(\014)5 b Fm(\()p Fn(v)3124 5235 y Fi(0)3152 5278 y Fm(\))3191 5235 y FA(2)3231 5197 y Fs(\001)3277 5193 y(\014)3277 5253 y(\014)3310 5318 y Fz(x)p Fj(=)p Fz(\030)r(;t)3539 5278 y Fn(:)3669 5140 y Fm(\()p Fo(4)p Fn(:)p Fo(4)p Fm(\))p eop %%Page: 11 11 11 10 bop 94 -45 a Fh(Mar)n(ginal)38 b(St)-7 b(ability)38 b(Bound)2332 b Fk(11)94 194 y Fo(W)-8 b(e)26 b(write)e Fn(B)555 221 y Fz(\030)599 194 y Fm(\()p Fn(t)p Fm(\))g Fo(for)h(the)g(boundary)f(term)g(obtained)g(abo)o(v)o(e:)607 450 y Fn(B)683 477 y Fz(\030)726 450 y Fm(\()p Fn(t)p Fm(\))44 b(=)1005 369 y Fs(\000)1051 450 y Fn(v)t(v)1155 407 y Fi(000)1251 450 y Fl(\000)22 b Fo(4)p Fn(\014)5 b(v)t(v)1565 407 y Fi(00)1639 450 y Fl(\000)22 b Fo(2)p Fn(\014)5 b Fm(\()p Fo(1)22 b Fm(+)h Fn(\014)2121 407 y FA(2)2161 450 y Fm(\))p Fn(v)2252 407 y FA(2)2314 450 y Fl(\000)g Fn(v)2466 407 y Fi(0)2494 450 y Fn(v)2546 407 y Fi(00)2618 450 y Fm(+)g Fo(2)p Fn(\014)5 b Fm(\()p Fn(v)2920 407 y Fi(0)2948 450 y Fm(\))2987 407 y FA(2)3027 369 y Fs(\001)3072 365 y(\014)3072 425 y(\014)3106 489 y Fz(x)p Fj(=)p Fz(\030)r(;t)3334 450 y Fn(:)94 695 y Fo(Finally)-6 b(,)24 b(we)h(re)n(write)g(\(4.4\))g(by)f(completing)f(a) i(square:)594 948 y FA(1)p 594 964 35 4 v 594 1021 a(2)640 987 y Fn(@)693 1014 y Fz(t)728 987 y Fn(J)783 1014 y Fz(\030)827 987 y Fm(\()p Fn(t)p Fm(\))44 b(=)1107 851 y Fs(Z)1206 894 y Fi(1)1162 1078 y Fz(\030)1228 987 y Fo(d)p Fn(x)1352 846 y Fs(\022)1424 906 y(\000)1470 987 y Fn(")1516 944 y FA(2)1578 987 y Fl(\000)23 b Fm(\()p Fo(1)f Fm(+)g Fn(\014)1949 944 y FA(2)1990 987 y Fm(\))2029 944 y FA(2)2091 987 y Fm(+)g(\()p Fo(1)g Fm(+)h Fo(3)p Fn(\014)2512 944 y FA(2)2552 987 y Fm(\))2591 944 y 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Fl(\024)h Fn(C)2173 2795 y FA(9)2238 2768 y Fn(:)1403 b Fm(\()p Fo(4)p Fn(:)p Fo(8)p Fm(\))94 3147 y Fk(Pr)n(oof)o(.)25 b Fo(Recall)g(that)g Fn(v)881 3174 y Fz(\030)925 3147 y Fm(\()p Fn(x;)17 b(t)p Fm(\))26 b(=)i Fn(e)1318 3104 y Fz(\014)t Fj(\()p Fz(x)p Fi(\000)p Fz(\030)r Fj(\))1581 3147 y Fn(u)p Fm(\()p Fn(x;)17 b(t)p Fm(\))p Fo(.)34 b(Using)24 b(elementary)h(calculus,)f(we)h(\002nd)1024 3498 y Fn(@)1083 3455 y Fz(j)1077 3525 y(x)1127 3498 y Fn(v)1175 3525 y Fz(\030)1218 3498 y Fm(\()p Fn(x;)17 b(t)p Fm(\))43 b(=)1653 3369 y Fz(j)1599 3403 y Fs(X)1601 3618 y Fz(k)r Fj(=)p FA(0)1760 3357 y Fs(\022)1849 3430 y Fn(j)1845 3566 y(k)1912 3357 y Fs(\023)2002 3498 y Fn(\014)2063 3455 y Fz(j)2105 3498 y Fn(e)2151 3455 y Fz(\014)t Fj(\()p Fz(x)p Fi(\000)p Fz(\030)r Fj(\))2414 3498 y Fn(@)2473 3455 y Fz(j)t Fi(\000)p Fz(k)2467 3525 y(x)2621 3498 y Fn(u)p Fm(\()p Fn(x;)17 b(t)p Fm(\))23 b Fn(:)94 3821 y Fo(Therefore,)1093 4030 y Fn(@)1152 3987 y Fz(j)1146 4057 y(x)1196 4030 y Fn(v)1244 4057 y Fz(\030)1288 4030 y Fm(\()p Fn(\030)5 b(;)17 b(t)p Fm(\))42 b(=)1714 3901 y Fz(j)1660 3935 y Fs(X)1662 4150 y Fz(k)r Fj(=)p FA(0)1821 3890 y Fs(\022)1910 3963 y Fn(j)1906 4098 y(k)1973 3890 y Fs(\023)2063 4030 y Fn(\014)2124 3987 y Fz(j)2167 4030 y Fn(@)2226 3982 y Fz(j)t Fi(\000)p Fz(k)2220 4060 y(\030)2373 4030 y Fn(u)p Fm(\()p Fn(x;)17 b(t)p Fm(\))p Fl(j)2674 4057 y Fz(x)p Fj(=)p Fz(\030)2847 4030 y Fn(;)94 4308 y Fo(and)25 b(the)g(assertion)f(follo)n(ws)f (because)j Fn(u)h Fl(2)h Fb(B)p Fo(.)316 4428 y(Using)c(Lemma)g(4.2,)g (we)h(conclude)g(from)g(\(4.6\))f(that)1344 4663 y Fn(@)1397 4690 y Fz(t)1432 4663 y Fn(J)1487 4690 y Fz(\030)1531 4663 y Fm(\()p Fn(t)p Fm(\))43 b Fl(\024)i Fo(2)p Fn(G)p Fm(\()p Fn(\014)5 b Fm(\))p Fn(J)2132 4690 y Fz(\030)2176 4663 y Fm(\()p Fn(t)p Fm(\))22 b(+)g Fo(2)p Fn(C)2532 4690 y FA(9)2597 4663 y Fn(:)94 4910 y Fo(Solving)i(the)h(dif)n (ferential)f(inequality)g(from)g Fn(t)h Fo(to)f Fn(t)1882 4867 y Fi(0)1910 4910 y Fo(,)h(we)g(obtain)f(for)h Fn(t)2552 4867 y Fi(0)2608 4910 y Fn(>)j(t)p Fo(,)939 5238 y Fn(J)994 5265 y Fz(\030)1037 5238 y Fm(\()p Fn(t)1112 5195 y Fi(0)1140 5238 y Fm(\))44 b Fl(\024)h Fn(e)1391 5195 y FA(2)p Fz(G)p Fj(\()p Fz(\014)t Fj(\)\()p Fz(t)1660 5152 y Ff(0)1687 5195 y Fi(\000)p Fz(t)p Fj(\))1816 5238 y Fn(J)1871 5265 y Fz(\030)1915 5238 y Fm(\()p Fn(t)p Fm(\))21 b(+)i Fo(2)2212 5171 y Fn(e)2258 5128 y FA(2)p Fz(G)p Fj(\()p Fz(\014)t Fj(\)\()p Fz(t)2527 5085 y Ff(0)2554 5128 y Fi(\000)p Fz(t)p Fj(\))2705 5171 y Fl(\000)f Fo(1)p 2212 5215 643 4 v 2399 5306 a(2)p Fn(G)p Fm(\()p Fn(\014)5 b Fm(\))2866 5238 y Fn(C)2937 5265 y FA(9)3002 5238 y Fn(:)639 b Fm(\()p Fo(4)p Fn(:)p Fo(9)p Fm(\))p eop %%Page: 12 12 12 11 bop 94 -45 a Fh(Mar)n(ginal)38 b(St)-7 b(ability)38 b(Bound)2332 b Fk(12)94 194 y Fo(W)-8 b(e)26 b(need)f(this)f (inequality)f(in)h(a)i(slightly)c(dif)n(ferent)j(form.)35 b(Note)25 b(that)f(for)h Fn(\030)2753 151 y Fi(0)2808 194 y Fn(>)j(\030)5 b Fo(,)24 b(one)h(has)358 500 y Fn(J)413 527 y Fz(\030)451 507 y Ff(0)483 500 y Fm(\()p Fn(t)p Fm(\))44 b(=)763 365 y Fs(Z)863 408 y Fi(1)818 591 y Fz(\030)856 571 y Ff(0)964 500 y Fo(d)p Fn(x)17 b(u)1145 458 y FA(2)1184 500 y Fm(\()p Fn(x;)g(t)p Fm(\))p Fn(e)1446 458 y FA(2)p Fz(\014)t Fj(\()p Fz(x)p Fi(\000)p Fz(\030)1706 415 y Ff(0)1733 458 y Fj(\))1813 500 y Fm(=)45 b Fn(e)1981 458 y Fi(\000)p FA(2)p Fz(\014)t Fj(\()p Fz(\030)2196 415 y Ff(0)2224 458 y Fi(\000)p Fz(\030)r Fj(\))2377 365 y Fs(Z)2477 408 y Fi(1)2433 591 y Fz(\030)2471 571 y Ff(0)2578 500 y Fo(d)p Fn(x)17 b(e)2748 458 y FA(2)p Fz(\014)t Fj(\()p Fz(x)p Fi(\000)p Fz(\030)r Fj(\))3045 500 y Fn(u)3102 458 y FA(2)3142 500 y Fm(\()p Fn(x;)g(t)p Fm(\))1813 780 y Fl(\024)45 b Fn(e)1981 737 y Fi(\000)p FA(2)p Fz(\014)t Fj(\()p Fz(\030)2196 695 y Ff(0)2224 737 y Fi(\000)p Fz(\030)r Fj(\))2377 645 y Fs(Z)2477 687 y Fi(1)2433 871 y Fz(\030)2578 780 y Fo(d)p Fn(x)17 b(e)2748 737 y FA(2)p Fz(\014)t Fj(\()p Fz(x)p Fi(\000)p Fz(\030)r Fj(\))3045 780 y Fn(u)3102 737 y FA(2)3142 780 y Fm(\()p Fn(x;)g(t)p Fm(\))1813 1038 y(=)45 b Fn(e)1981 995 y Fi(\000)p FA(2)p Fz(\014)t Fj(\()p Fz(\030)2196 952 y Ff(0)2224 995 y Fi(\000)p Fz(\030)r Fj(\))2361 1038 y Fn(J)2416 1065 y Fz(\030)2460 1038 y Fm(\()p Fn(t)p Fm(\))24 b Fn(:)3619 746 y Fm(\()p Fo(4)p Fn(:)p Fo(10)p Fm(\))94 1309 y Fo(Combining)g(this)g(with)g(\(4.9\))h(we)g(get)f(for)h Fn(\030)1632 1266 y Fi(0)1687 1309 y Fn(>)j(\030)h Fo(and)c Fn(t)2070 1266 y Fi(0)2126 1309 y Fn(>)j(t)p Fo(:)691 1660 y Fn(J)746 1687 y Fz(\030)784 1667 y Ff(0)817 1660 y Fm(\()p Fn(t)892 1618 y Fi(0)920 1660 y Fm(\))44 b Fl(\024)h Fn(e)1171 1618 y Fi(\000)p FA(2)p Fz(\014)t Fj(\()p Fz(\030)1386 1575 y Ff(0)1414 1618 y Fi(\000)p Fz(\030)r Fj(\))1551 1580 y Fs(\000)1596 1660 y Fn(e)1642 1618 y FA(2)p Fz(G)p Fj(\()p Fz(\014)t Fj(\)\()p Fz(t)1911 1575 y Ff(0)1939 1618 y Fi(\000)p Fz(t)p Fj(\))2067 1660 y Fn(J)2122 1687 y Fz(\030)2166 1660 y Fm(\()p Fn(t)p Fm(\))22 b(+)2413 1593 y Fn(e)2459 1550 y FA(2)p Fz(G)p Fj(\()p Fz(\014)t Fj(\)\()p Fz(t)2728 1507 y Ff(0)2756 1550 y Fi(\000)p Fz(t)p Fj(\))2906 1593 y Fl(\000)h Fo(1)p 2413 1637 643 4 v 2626 1729 a Fn(G)p Fm(\()p Fn(\014)5 b Fm(\))3068 1660 y Fn(C)3139 1687 y FA(9)3179 1580 y Fs(\001)3249 1660 y Fn(:)342 b Fm(\()p Fo(4)p Fn(:)p Fo(11)p Fm(\))94 1984 y Fo(T)-8 b(o)31 b(complete)f(the)g(proof)g(of)h (Theorem)f(4.1,)i(it)e(suf)n(\002ces)g(to)g(set)h Fn(\030)2425 1941 y Fi(0)2487 1984 y Fm(=)36 b Fn(c\034)11 b Fo(,)32 b Fn(t)2791 1941 y Fi(0)2855 1984 y Fm(=)k Fn(\034)11 b Fo(,)32 b Fn(\030)39 b Fm(=)e Fo(0)30 b(and)g Fn(t)36 b Fm(=)g Fo(0)31 b(in)94 2104 y(\(4.11\).)36 b(Then)25 b(we)g(get)1144 2290 y Fn(J)1199 2317 y Fz(c\034)1284 2290 y Fm(\()p Fn(\034)11 b Fm(\))44 b Fl(\024)h Fn(e)1629 2248 y FA(2)p Fj(\()p Fz(G)p Fj(\()p Fz(\014)t Fj(\))p Fi(\000)p Fz(\014)t(c)p Fj(\))p Fz(\034)2096 2210 y Fs(\000)2141 2290 y Fn(J)2196 2317 y FA(0)2236 2290 y Fm(\()p Fo(0)p Fm(\))22 b(+)2551 2223 y Fn(C)2622 2250 y FA(9)p 2497 2268 218 4 v 2497 2359 a Fn(G)p Fm(\()p Fn(\014)5 b Fm(\))2727 2210 y Fs(\001)2797 2290 y Fn(:)794 b Fm(\()p Fo(4)p Fn(:)p Fo(12)p Fm(\))94 2558 y Fo(Clearly)-6 b(,)26 b(if)f Fn(c)j(>)g(G)p Fm(\()p Fn(\014)5 b Fm(\))p Fn(=\014)g Fo(,)26 b(then)f Fn(J)1325 2585 y Fz(c\034)1411 2558 y Fm(\()p Fn(\034)11 b Fm(\))27 b Fl(!)h Fo(0)d(as)g Fn(\034)39 b Fl(!)28 b(1)p Fo(.)36 b(Thus,)25 b(if)g Fn(J)2645 2585 y FA(0)2685 2558 y Fm(\()p Fo(0)p Fm(\))i Fn(<)i Fl(1)p Fo(,)c(and)g(the)g(assertion)f(of)94 2677 y(Theorem)h(4.1)f(follo)n(ws.)p 3812 2623 63 2 v 3812 2681 2 59 v 3872 2681 V 3812 2683 63 2 v 94 3015 a Fk(Remark.)j Fo(One)e(can)h(do)f(a)h(little)f(better)g(than)g(\(4.12\).)38 b(Namely)-6 b(,)25 b(consider)g(the)g(case)h(where)g Fn(c)j Fm(=)g Fn(G)p Fm(\()p Fn(\014)5 b Fm(\))p Fn(=\014)g Fo(,)94 3135 y(that)25 b(is,)f(the)h(case)g(of)g(a)g(critical)g(speed.) 35 b(Then)25 b(one)g(\002nds)g(from)f(\(4.11\))h(that)1224 3441 y Fn(J)1279 3468 y Fz(c\034)8 b Fj(+)p Fz(\025)1473 3441 y Fm(\()p Fn(\034)j Fm(\))43 b Fl(\024)i Fn(e)1817 3398 y Fi(\000)p FA(2)p Fz(\014)t(\025)2016 3360 y Fs(\000)2061 3441 y Fn(J)2116 3468 y FA(0)2156 3441 y Fm(\()p Fo(0)p Fm(\))22 b(+)2470 3373 y Fn(C)2541 3400 y FA(9)p 2417 3418 218 4 v 2417 3509 a Fn(G)p Fm(\()p Fn(\014)5 b Fm(\))2647 3360 y Fs(\001)2717 3441 y Fn(;)94 3758 y Fo(and)23 b(in)f(particular)h (lim)899 3785 y Fz(\025)p Fi(!1)1127 3758 y Fn(J)1182 3785 y Fz(c\034)8 b Fj(+)p Fz(\025)1376 3758 y Fm(\()p Fn(\034)j Fm(\))27 b(=)h Fo(0,)23 b(if)f Fn(J)1877 3785 y FA(0)1917 3758 y Fm(\()p Fo(0)p Fm(\))g Fo(is)h(\002nite.)35 b(This)22 b(means)g(that)g(in)g(the)h(frame)g(mo)o(ving)94 3894 y(with)34 b(e)o(xactly)g(the)h(critical)f(speed,)j(no)e(amplitude) e(\223leaks\224)i(f)o(ar)g(ahead)g(in)g(that)f(frame)h(in)f(L)3428 3851 y FA(2)3468 3894 y Fm(\()p Fn(e)3553 3851 y FA(2)p Fz(\014)t(x)3687 3894 y Fo(d)p Fn(x)17 b Fm(\))p Fo(.)94 4014 y(One)32 b(can)g(compare)g(this)f(with)g(the)g(results)g(of)h (Bramson)f([B])i(who)e(sho)n(wed)g(\(for)h(positi)n(v)o(e)d(solutions)h (of)94 4133 y(the)25 b(Ginzb)n(ur)n(g\255Landau)e(equation\))h(that)g (such)g(a)g(leakage)h(is)e(only)h(possible)f(if)h(the)g(initial)f(data) h(decay)h(lik)o(e)94 4253 y Fn(e)140 4210 y Fi(\000)p Fz(x)253 4253 y Fn(x)310 4210 y Fz(\013)398 4253 y Fo(with)30 b Fn(\013)35 b(>)i Fo(1.)53 b(In)31 b(that)f(case,)i(he)f(gets)f (positi)n(v)o(e)f(amplitudes)g(at)h Fn(ct)c Fm(+)g(\()p Fn(\013)f Fl(\000)h Fo(1)p Fm(\))17 b Fo(log)e Fn(t)p Fo(.)53 b(Note)31 b(that)94 4372 y(the)25 b(condition)e Fn(J)697 4399 y FA(0)737 4372 y Fm(\()p Fo(0)p Fm(\))k Fn(<)h Fl(1)d Fo(can)g(only)f(hold)f(for)i Fn(\013)j(<)g Fl(\000)2115 4333 y FA(1)p 2116 4349 35 4 v 2116 4407 a(2)2163 4372 y Fo(,)c(and)h(then)f(the)h(correction)f(term)h(will)e (push)h(the)94 4492 y(amplitude)31 b(behind)g(the)g(position)f(of)i Fn(ct)p Fo(.)57 b(Thus,)33 b(in)e(the)h(case)g(of)g(the)f(Ginzb)n(ur)n (g\255Landau)h(equations)e(the)94 4611 y(tw)o(o)25 b(results)f(are)h (consistent.)p eop %%Page: 13 13 13 12 bop 94 -45 a Fh(Mar)n(ginal)38 b(St)-7 b(ability)38 b(Bound)2332 b Fk(13)94 194 y Fp(5.)30 b(An)h(example)f(of)f(a)h (non\255linear)h(v)o(elocity)f(bound)94 383 y Fo(Consider)25 b(the)g(semi\255linear)f(parabolic)h(equation)1503 617 y Fn(@)1556 644 y Fz(t)1591 617 y Fn(u)j Fm(=)g Fn(P)14 b Fm(\()p Fn(@)1951 644 y Fz(x)2001 617 y Fm(\))p Fn(u)22 b Fm(+)g Fn(f)11 b Fm(\()p Fn(u)p Fm(\))24 b Fn(;)1204 b Fm(\()p Fo(5)p Fn(:)p Fo(1)p Fm(\))94 850 y Fo(where)20 b Fn(P)33 b Fo(is)19 b(a)h(real)f(polynomial,)g(Re)e Fn(P)d Fm(\()p Fn(ik)s Fm(\))19 b Fo(di)n(v)o(er)n(ges)f(to)h Fl(\0001)g Fo(as)h Fl(j)p Fn(k)s Fl(j)27 b(!)g(1)19 b Fo(and)g(Im)e Fm(\()p Fn(ik)s Fm(\))i Fo(is)g(a)g(polynomial)94 969 y(of)31 b(lo)n(wer)f(order)-5 b(.)693 927 y Fi(\003)791 969 y Fo(W)d(e)31 b(also)f(assume)g(that)g Fn(f)41 b Fo(is)29 b(a)i Fl(C)1966 927 y FA(2)2037 969 y Fo(function)e(for)i (which)f Fn(f)11 b Fm(\()p Fo(0)p Fm(\))34 b(=)i Fo(0,)c(and)e Fn(f)3496 927 y Fi(0)3524 969 y Fm(\()p Fo(0)p Fm(\))k(=)i Fo(0.)94 1089 y(This)25 b(implies)e(that)h Fn(u)k Fm(=)g Fo(0)d(is)f(an)h(unstable)f(\002x)o(ed)h(point)e(of)i(\(5.1\))g(.)36 b(W)-8 b(e)25 b(also)g(assume)f(that)1589 1376 y(lim)17 b(sup)1606 1467 y Fi(j)p Fz(u)p Fi(j!1)1905 1309 y Fn(f)11 b Fm(\()p Fn(u)p Fm(\))p 1905 1353 195 4 v 1974 1445 a Fn(u)2156 1376 y(<)44 b Fo(0)25 b Fn(:)94 1689 y Fo(This)35 b(assumption)f(ensures)h(global)g(e)o(xistence)g(and)g(re)o(gularity)g (of)g(the)h(semi\003o)n(w)e(\(see)i([CE]\).)h(\(If)e Fn(~)-49 b(u)36 b Fo(is)94 1809 y(v)o(ector)25 b(v)n(alued)f(we)h (impose)f(lim)17 b(sup)1397 1835 y Fi(k)p Fz(~)-41 b(u)o Fi(k!1)1704 1809 y Fn(~)-48 b(u)22 b Fl(\001)1849 1782 y Fn(~)1835 1809 y(f)10 b Fm(\()p Fn(u)p Fm(\))p Fn(=)p Fl(k)o Fn(~)-49 b(u)o Fl(k)2235 1766 y FA(2)2303 1809 y Fn(<)28 b Fo(0.\))316 1928 y(De\002ne)1658 2081 y Fn(\033)48 b Fm(=)d Fo(sup)1930 2162 y Fz(u)2052 2014 y Fn(f)11 b Fm(\()p Fn(u)p Fm(\))p 2052 2058 V 2120 2149 a Fn(u)2283 2081 y(:)94 2320 y Fo(This)32 b(is)g(a)g(\002nite)h(positi)n(v)o(e)d (quantity)h(from)h(the)g(abo)o(v)o(e)f(assumptions)f(\(if)h Fn(~)-49 b(u)33 b Fo(is)f(v)o(ector)f(v)n(alued)h(we)h(de\002ne)94 2440 y(it)40 b(as)h(the)f(sup)g(of)f Fn(~)-49 b(u)31 b Fl(\001)938 2414 y Fn(~)924 2440 y(f)11 b Fm(\()p Fn(u)p Fm(\))p Fn(=)p Fl(k)o Fn(~)-49 b(u)o Fl(k)1325 2397 y FA(2)1405 2440 y Fo(.\))82 b(Note)41 b(that)e(one)i(can)f(ha)n(v)o(e)g Fn(\033)53 b(>)d(f)2854 2397 y Fi(0)2881 2440 y Fm(\()p Fo(0)p Fm(\))p Fo(,)44 b(and)c(if)g(this)g(happens)94 2560 y(Aronson)21 b(and)g(W)-8 b(einber)n(ger)22 b([A)-9 b(W])21 b(sho)n(wed)g(that)f(the)h(minimal)f(speed)h(is)g(bounded)f (abo)o(v)o(e)g(by)3418 2476 y Fl(p)p 3501 2476 111 4 v 84 x Fo(4)p Fn(\033)s Fo(,)i(when)94 2679 y Fn(P)14 b Fm(\()p Fn(ik)s Fm(\))28 b(=)g Fl(\000)p Fn(k)604 2636 y FA(2)644 2679 y Fo(.)36 b(In)25 b(this)e(section)h(we)g(sho)n(w)g (that)g(the)g(same)g(result)g(can)h(be)f(reco)o(v)o(ered)h(for)f(this,) g(and)g(man)o(y)94 2799 y(other)33 b(equations)e(using)h(the)g(methods) f(of)i(Section)f(4,)j(again)c(without)h(an)o(y)f(recourse)i(to)g(the)f (maximum)94 2918 y(principle.)316 3038 y(In)25 b(this)f(case,)h (Eq.\(4.7\))f(becomes)1566 3271 y Fn(G)p Fm(\()p Fn(\014)5 b Fm(\))45 b(=)g Fn(Q)p Fm(\()p Fn(\014)5 b Fm(\))22 b(+)g Fn(\033)29 b(;)94 3504 y Fo(where)d Fn(Q)e Fo(is)h(gi)n(v)o(en)e (by)1373 3624 y Fn(Q)p Fm(\()p Fn(\014)5 b Fm(\))44 b(=)h Fo(sup)1782 3710 y Fz(k)1826 3686 y Ff(\003)1824 3737 y Fe(\014)1912 3624 y Fo(Re)17 b Fn(P)d Fm(\()p Fl(\000)p Fn(\014)28 b Fm(+)23 b Fn(ik)2506 3581 y Fi(\003)2503 3651 y Fz(\014)2557 3624 y Fm(\))94 3909 y Fo(where)j(the)e Fn(k)564 3866 y Fi(\003)561 3936 y Fz(\014)640 3909 y Fo(are)h(the)g(solutions)e(of)1398 4150 y(dRe)17 b Fn(P)d Fm(\()p Fl(\000)p Fn(\014)28 b Fm(+)23 b Fn(ik)s Fm(\))p 1398 4195 683 4 v 1679 4286 a Fo(d)p Fn(k)2093 4073 y Fs(\014)2093 4133 y(\014)2093 4193 y(\014)2093 4252 y(\014)2126 4317 y Fz(k)r Fj(=)p Fz(k)2275 4293 y Ff(\003)2273 4344 y Fe(\014)2370 4218 y Fm(=)45 b Fo(0)24 b Fn(:)94 4567 y Fo(The)29 b(remainder)g(of)f(the)h(proof)f(is)h(the)f(same,)h(e)o (xcept)g(that)f(in)g(\(4.5\))h(the)f(term)h Fl(\000)17 b Fo(e)o(xp)o Fm(\()p Fl(\000)p Fo(2)p Fn(\014)5 b Fm(\()p Fn(x)25 b Fl(\000)f Fn(\030)5 b Fm(\)\))p Fn(v)3740 4524 y FA(4)3808 4567 y Fo(is)94 4686 y(replaced)26 b(by)1362 4806 y Fn(e)1408 4763 y Fz(\014)t Fj(\()p Fz(x)p Fi(\000)p Fz(\030)r Fj(\))1671 4806 y Fn(v)t(f)1783 4725 y Fs(\000)1828 4806 y Fn(e)1874 4763 y Fi(\000)p Fz(\014)t Fj(\()p Fz(x)p Fi(\000)p Fz(\030)r Fj(\))2199 4806 y Fn(v)2251 4725 y Fs(\001)2324 4806 y Fl(\024)i Fn(\033)t(v)2542 4763 y FA(2)94 4993 y Fo(After)e(this)d(modi\002cation)h(the)h(proof)f (proceeds)h(as)g(before.)p 94 5105 1200 4 v 233 5159 a Fq(\003)316 5202 y Fx(The)18 b(comple)o(x)g(Ginzb)n(ur)o(g\255Landau) f(equation)h(is)g(some)n(what)h(more)e(complicated)j(because)f(in)f (that)h(case)g Fv(P)29 b Fx(is)18 b(a)h(2)14 b Fr(\002)g Fx(2)k(matrix)94 5291 y(polynomial.)29 b(But)20 b(it)h(is)f(co)o(v)o (ered)g(by)f(our)g(methods.)p eop %%Page: 14 14 14 13 bop 94 -45 a Fh(Mar)n(ginal)38 b(St)-7 b(ability)38 b(Bound)2332 b Fk(14)94 194 y Fp(A)m(ppendix:)46 b(The)30 b(determination)h(of)e(the)h(critical)h(speed)94 383 y Fo(Let)e Fn(P)44 b Fo(be)29 b(a)g(real)h(polynomial)d(for)i(which)g (Re)17 b Fn(P)d Fm(\()p Fn(ik)s Fm(\))29 b Fo(di)n(v)o(er)n(ges)f(to)h Fl(\0001)g Fo(as)g Fl(j)p Fn(k)s Fl(j)k(!)h(1)28 b Fo(and)h(Im)17 b Fn(P)d Fm(\()p Fn(ik)s Fm(\))29 b Fo(is)94 503 y(of)j(lo)n(wer)f (order)-5 b(.)56 b(In)32 b(the)f(case)h(of)g(SH,)f(we)h(ha)n(v)o(e)g Fn(P)14 b Fm(\()p Fn(z)t Fm(\))37 b(=)g Fn(")2288 460 y FA(2)2354 503 y Fl(\000)27 b Fm(\()p Fo(1)e Fm(+)h Fn(z)2725 460 y FA(2)2766 503 y Fm(\))2805 460 y FA(2)2845 503 y Fo(.)56 b(F)o(or)31 b Fn(\014)43 b(>)37 b Fo(0)32 b(we)f(consider)94 623 y Fn(P)14 b Fm(\()p Fl(\000)p Fn(\014)28 b Fm(+)23 b Fn(ik)s Fm(\))p Fo(,)i(tak)o(e)g(the)f(real)i (part)e(and)h(look)f(for)h(an)g(e)o(xtremum)f(in)g Fn(k)s Fo(.)36 b(In)25 b(other)f(w)o(ords,)h(we)g(solv)o(e)1509 836 y(dRe)17 b Fn(P)d Fm(\()p Fl(\000)p Fn(\014)28 b Fm(+)22 b Fn(ik)s Fm(\))p 1509 880 683 4 v 1789 971 a Fo(d)p Fn(k)2248 903 y Fm(=)44 b Fo(0)25 b Fn(;)94 1155 y Fo(in)g(the)g(unkno)n(wn)e Fn(k)s Fo(.)36 b(Since)25 b Fn(P)39 b Fo(is)24 b(analytic,)g(one)h(can)g(write)g(this)f(as)1353 1465 y(0)44 b Fm(=)h Fo(Im)1712 1294 y Fs( )1815 1397 y Fo(d)p Fn(P)14 b Fm(\()p Fn(z)t Fm(\))p 1815 1442 256 4 v 1885 1533 a Fo(d)p Fn(z)2083 1320 y Fs(\014)2083 1380 y(\014)2083 1440 y(\014)2083 1500 y(\014)2116 1564 y Fz(z)s Fj(=)p Fi(\000)p Fz(\014)t Fj(+)p Fz(ik)2468 1294 y Fs(!)2588 1465 y Fn(:)1031 b Fm(\()p Fo(A)p Fn(:)p Fo(1)p Fm(\))94 1778 y Fo(F)o(or)24 b(each)h Fn(\014)k Fo(we)24 b(\002nd)g(solutions)d Fn(k)1304 1735 y Fi(\003)1301 1805 y Fz(\014)1355 1778 y Fo(.)36 b(The)23 b(v)o(elocity)g Fn(c)1981 1735 y Fi(\003)1981 1805 y Fz(\014)2058 1778 y Fo(is)g(related)h(to)g(the)f(critical)h(v)n(alue)f(of)h Fn(P)37 b Fo(in)24 b(\(A.1\))f(by)1352 2001 y Fn(c)1395 1959 y Fi(\003)1395 2028 y Fz(\014)1493 2001 y Fm(=)45 b Fo(sup)1639 2088 y Fz(k)1683 2064 y Ff(\003)1681 2115 y Fe(\014)1770 2001 y Fo(Re)17 b Fn(P)d Fm(\()p Fl(\000)p Fn(\014)28 b Fm(+)22 b Fn(ik)2363 1959 y Fi(\003)2360 2028 y Fz(\014)2414 2001 y Fm(\))p Fn(=\014)30 b(:)1030 b Fm(\()p Fo(A)p Fn(:)p Fo(2)p Fm(\))94 2315 y Fo(Then,)25 b(the)g(minimal)e(speed)h(is)1626 2434 y Fn(c)1669 2461 y Fi(\003)1760 2434 y Fm(=)134 b Fo(inf)1881 2503 y Fz(\014)t Fi(2)p Fj(\()p FA(0)p Fz(;)p Fi(1)p FA(])2193 2434 y Fn(c)2236 2392 y Fi(\003)2236 2461 y Fz(\014)2314 2434 y Fn(;)94 2670 y Fo(which)32 b(is)g(determined)f(by)h(\(A.3\).)58 b(T)-8 b(o)32 b(simplify)e(the)i(discussion,)g(we)h(will)e(assume)h (from)f(no)n(w)h(on)g(that)94 2790 y(for)i(all)f Fn(k)432 2747 y Fi(\003)429 2817 y Fz(\014)516 2790 y Fo(one)g(obtains)f(the)i (same)f(critical)g(v)n(alue.)60 b(This)33 b(is)g(the)g(case)h(for)f (the)g(Ginzb)n(ur)n(g\255Landau)g(and)94 2909 y(Swift\255Hohenber)n(g) 25 b(equations.)316 3029 y(Note)f(that)h(there)g(is)f(at)h(least)g(one) f 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Fn(k)r Fm(\))p Fl(j)1916 3884 y Fz(k)r Fj(=)2023 3866 y(\026)2021 3884 y Fz(k)r Fj(\()p Fz(c)p Fj(\))2212 3858 y Fm(=)45 b Fn(ic)25 b(;)1183 b Fm(\()p Fo(A)p Fn(:)p Fo(4)p Fm(\))94 4040 y Fo(and)25 b(then)g Fn(c)503 4067 y Fi(\003)576 4040 y Fl(2)j Fk(R)d Fo(by)g(the)f(condition)1325 4263 y(Re)1453 4182 y Fs(\000)1498 4263 y Fn(!)t Fm(\()1606 4237 y(\026)1603 4263 y Fn(k)s Fm(\()p Fn(c)1740 4290 y Fi(\003)1785 4263 y Fm(\)\))e Fl(\000)g Fn(i)2021 4237 y Fm(\026)2018 4263 y Fn(k)t Fm(\()p Fn(c)2156 4290 y Fi(\003)2201 4263 y Fm(\))p Fn(c)2283 4290 y Fi(\003)2329 4182 y Fs(\001)2419 4263 y Fm(=)45 b Fo(0)25 b Fn(:)1003 b Fm(\()p Fo(A)p Fn(:)p Fo(5)p Fm(\))316 4486 y Fo(T)-8 b(o)37 b(compare)h(the)f(tw)o(o)g (approaches,)k(note)c(that)g Fn(P)14 b Fm(\()p Fl(\000)p Fn(\014)34 b Fm(+)c Fn(ik)s Fm(\))45 b(=)h Fl(\000)p Fn(!)2918 4513 y FA(0)2958 4486 y Fm(\()p Fn(k)32 b Fm(+)e Fn(i\014)5 b Fm(\))p Fo(.)74 b(Clearly)38 b(the)94 4606 y(equations)27 b(\(A.2\))h(and)f(\(A.5\))h(are)h(equi)n(v)n(alent.)42 b(T)-8 b(o)27 b(see)h(that)g(\(A.1\))f(and)h(\(A.4\))g(say)f(the)h (same)f(thing,)g(note)94 4725 y(that)e(since)f Fn(c)h Fo(is)g(real)g(one)g(has)849 4949 y(Im)16 b Fm(\()p Fn(P)1093 4906 y Fi(0)1121 4949 y Fm(\()p Fn(z)t Fm(\)\))44 b(=)h Fo(Re)17 b Fm(\()p Fl(\000)p Fn(!)1763 4906 y Fi(0)1759 4975 y FA(0)1799 4949 y Fm(\()p Fl(\000)p Fn(iz)t Fm(\)\))45 b(=)g Fo(Re)17 b Fm(\()p Fl(\000)p Fn(!)2553 4906 y Fi(0)2549 4975 y FA(0)2589 4949 y Fm(\()p Fl(\000)p Fn(iz)t Fm(\))23 b(+)g Fn(ic)p Fm(\))i Fn(:)527 b Fm(\()p Fo(A)p Fn(:)p Fo(6)p Fm(\))94 5172 y Fo(In)21 b(particular)l(,)f(if)g Fn(!)764 5199 y FA(0)825 5172 y Fo(is)f(an)h(e)n(v)o(en)g(function,)g (the)g(relation)f(Re)e Fm(\()p Fn(!)2306 5129 y Fi(0)2302 5199 y FA(0)2342 5172 y Fm(\()2384 5145 y(\026)2381 5172 y Fn(k)s Fm(\))12 b Fl(\000)g Fn(ic)p Fm(\))27 b(=)h Fo(0)20 b(is)g(equi)n(v)n(alent)e(to)i(requiring)94 5291 y Fn(!)160 5248 y Fi(0)156 5318 y FA(0)196 5291 y Fm(\()238 5265 y(\026)235 5291 y Fn(k)s Fm(\))40 b(=)g Fn(ic)p Fo(,)c(which)d(is)g(\(A.4\).)62 b(Using)33 b(\(A.6\),)j(we)e(conclude)f (that)g(the)g(solution)3086 5265 y Fm(\026)3083 5291 y Fn(k)k Fo(of)c Fn(!)3354 5248 y Fi(0)3350 5318 y FA(0)3390 5291 y Fm(\()3432 5265 y(\026)3429 5291 y Fn(k)s Fm(\))40 b(=)g Fn(ic)34 b Fo(of)p eop %%Page: 15 15 15 14 bop 94 -45 a Fh(Mar)n(ginal)38 b(St)-7 b(ability)38 b(Bound)2332 b Fk(15)94 194 y Fo([BBDKL])28 b(is)d(the)h(same)g(as)g Fl(\000)p Fn(i)h Fo(times)e(the)h(solution)e Fn(z)31 b Fo(of)26 b(Im)16 b Fm(\()p Fn(P)2410 151 y Fi(0)2437 194 y Fm(\()p Fn(z)t Fm(\)\))30 b(=)g Fo(0,)c(which)f(is)h(\(A.1\).)39 b(Therefore)97 287 y Fm(\026)94 314 y Fn(k)31 b Fm(=)d Fn(k)337 271 y Fi(\003)334 341 y Fz(\014)379 358 y Ff(\003)448 314 y Fm(+)23 b Fn(i\014)638 341 y Fi(\003)684 314 y Fo(.)36 b(Finally)-6 b(,)24 b(to)g(\002nd)h Fn(c)1396 271 y Fi(\003)1396 341 y Fz(\014)1441 358 y Ff(\003)1513 314 y Fo(one)g(can)g(solv)o(e)377 528 y(0)44 b Fm(=)h Fo(Re)17 b Fm(\()p 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Fi(\003)932 4859 y Fz(\014)1019 4832 y Fm(=)33 b(\()p Fn(")1214 4789 y FA(2)1278 4832 y Fm(+)25 b Fo(4)p Fn(\014)1491 4789 y FA(2)1555 4832 y Fm(+)g Fo(8)p Fn(\014)1768 4789 y FA(4)1808 4832 y Fm(\))p Fn(=\014)5 b Fo(.)47 b(T)-8 b(o)28 b(\002nd)g(the)h(absolutely)e(minimal)f(speed,) j(we)g(\002nd)94 4951 y(that)c Fn(\014)30 b Fo(for)25 b(which)g Fn(c)808 4908 y Fi(\003)808 4978 y Fz(\014)887 4951 y Fo(is)f(e)o(xtremal,)g(that)g(is)h Fn(@)1694 4978 y Fz(\014)1747 4951 y Fn(c)1790 4908 y Fi(\003)1790 4978 y Fz(\014)1872 4951 y Fm(=)j Fo(0.)36 b(The)24 b(only)g(positi)n(v)o(e) f(solution)g(is)1456 5262 y Fn(\014)1512 5289 y Fi(\003)1602 5262 y Fm(=)1736 5094 y Fs(p)p 1836 5094 613 4 v 100 x Fo(3)1886 5110 y Fl(p)p 1969 5110 308 4 v 84 x Fo(1)f Fm(+)g Fo(6)p Fn(")2236 5165 y FA(2)2298 5194 y Fl(\000)h Fo(3)p 1736 5239 712 4 v 2067 5330 a(6)2485 5262 y Fn(;)p eop %%Page: 16 16 16 15 bop 94 -45 a Fh(Mar)n(ginal)27 b(St)-7 b(ability)27 b(Bound)2354 b Fk(16)94 194 y Fo(and)25 b(hence,)1315 384 y Fn(c)1358 341 y Fi(\003)1358 411 y Fz(\014)1403 429 y Ff(\003)1494 384 y Fm(=)1628 310 y Fo(4)1678 229 y Fs(\000)1723 226 y Fl(p)p 1806 226 308 4 v 84 x Fo(1)d Fm(+)h Fo(6)p Fn(")2074 281 y FA(2)2136 310 y Fl(\000)g Fo(1)f Fm(+)g Fo(6)p Fn(")2503 267 y FA(2)2543 229 y Fs(\001)p 1628 361 962 4 v 1728 481 a Fo(3)1778 381 y Fs(p)p 1877 381 613 4 v 1877 481 a Fo(3)1927 397 y Fl(p)p 2010 397 308 4 v 84 x Fo(1)g Fm(+)h Fo(6)p Fn(")2278 452 y FA(2)2340 481 y Fl(\000)f Fo(3)2626 384 y Fn(:)94 643 y Fo(This)j(quantity)e(is)h(the)h(same)g(as)g(inf)1339 670 y Fz(\014)t Fi(2)p Fg(R)1514 643 y Fn(c)1557 670 y Fz(\014)1635 643 y Fo(where)h Fn(c)1947 670 y Fz(\014)2025 643 y Fo(is)f(gi)n(v)o(en)e(by)i(\(1.7\).)94 822 y Fk(Ackno)o (wledgments.)d Fo(W)-8 b(e)21 b(thank)f(W)-9 b(.)20 b(v)n(an)g (Saarloos)g(for)h(some)e(help)h(with)g(the)g(references,)i(and)f(M.)f (Hairer)94 942 y(and)29 b(G.)g(v)n(an)f(Baalen)i(for)f(a)g(critical)g (reading)f(of)h(the)g(manuscript.)47 b(This)28 b(w)o(ork)g(w)o(as)h (partially)f(supported)94 1061 y(by)d(the)g(F)o(onds)f(National)g (Suisse.)94 1420 y Fp(Refer)n(ences)94 1699 y Fx([A)-7 b(W])184 b(Aronson,)25 b(D.)h(and)g(H.F)-6 b(.)26 b(W)-6 b(einber)o(ger:)26 b(Multidimensional)i(nonlinear)e(dif)n(fusion)g (arising)g(in)h(population)f(genetics.)458 1789 y(Adv)-5 b(.)25 b(Math.)i Fw(30)p Fx(,)f(33\22676)g(\(1978\).)94 1878 y([BBDKL])39 b(Ben\255Jacob,)26 b(E.,)h(H.)f(Brand,)g(G.)g(Dee,)h (L.)f(Kramers,)g(and)g(J.S.)h(Langer:)g(P)o(attern)g(propagation)f(in)g (non\255linear)g(dissi\255)458 1968 y(pati)n(v)o(e)h(systems.)40 b(Physica)27 b Fw(14D)p Fx(,)f(348\226364)f(\(1985\).)94 2058 y([B])257 b(Bramson,)26 b(M.:)g(Con)m(v)o(er)o(gence)h(of)f (solutions)g(of)g(the)h(K)m(olmogoro)o(v)e(equation)i(to)f(tra)n(v)o (elling)i(w)o(a)n(v)o(es..)40 b(Mem.)27 b(Amer)l(.)458 2147 y(Math.)f(Soc.)h Fw(44)p Fx(,)f(No.)g(285,)g(i)n(v+190pp)g (\(1985\).)94 2237 y([CE])208 b(Collet,)27 b(P)-9 b(.)27 b(and)f(J.\255P)-9 b(.)27 b(Eckmann:)g 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