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(constants)g(are)h(solutions)d(of)j(Eq.\(1.2\),)f(for)h(an)o(y)f Fq(u)2769 3585 y Fz(0)2809 3558 y Fr(.)38 b(Furthermore,)26 b(it)f(is)g(easy)h(to)94 3691 y(check)d(that)e(these)h(solutions)e(are)j (\(locally\))e(linearly)h(stable)f(for)h Fo(j)p Fq(u)2436 3718 y Fz(0)2476 3691 y Fo(j)27 b Fq(>)h Fr(3)2686 3649 y Fn(\000)p Fz(1)p Fy(=)p Fz(2)2864 3691 y Fr(,)22 b(and)g(linearly)g(unstable)f(for)94 3824 y Fo(j)p Fq(u)179 3851 y Fz(0)219 3824 y Fo(j)27 b Fq(<)i Fr(3)430 3781 y Fn(\000)p Fz(1)p Fy(=)p Fz(2)607 3824 y Fr(.)35 b(W)-8 b(e)23 b(concentrate)g(our)f(analysis)g(on)g(the)h(remaining)e(case,)j (namely)e Fq(u)3087 3851 y Fz(0)3154 3824 y Fp(=)28 b Fo(\006)p Fr(3)3386 3781 y Fn(\000)p Fz(1)p Fy(=)p Fz(2)3564 3824 y Fr(.)35 b(In)23 b(this)94 3957 y(case,)j(linearizing)e(about)g Fq(u)1070 3984 y Fz(0)1138 3957 y Fp(=)k Fr(3)1293 3914 y Fn(\000)p Fz(1)p Fy(=)p Fz(2)1495 3957 y Fr(leads)d(to)f(the)h(linear)g(equation)1679 4210 y Fq(@)1732 4237 y Fy(t)1767 4210 y Fq(v)48 b Fp(=)d Fo(\000)p Fp(\001)2145 4167 y Fz(2)2185 4210 y Fq(v)29 b(;)1379 b Fp(\()p Fr(1)p Fq(:)p Fr(3)p Fp(\))94 4463 y Fr(which)33 b(has)f(spectrum)g(in)g Fp(\()p Fo(\0001)p Fq(;)17 b Fr(0])32 b(and)g(corresponds)g(to)g(the)g(case)h Fq(P)14 b Fp(\()p Fq(\030)5 b Fp(\))37 b(=)i Fo(\000)p Fq(\030)3062 4420 y Fz(4)3102 4463 y Fr(.)58 b(It)33 b(is)f(ob)o(vious)f(that)94 4583 y(bounded)23 b(initial)f(data)i(lead)f(to)g(solutions)e(which)i(tend)g (to)g(0)h(as)f Fq(t)28 b Fo(!)f(1)c Fr(and)h(the)f(purpose)g(of)g(this)g (paper)h(is)94 4702 y(to)d(study)f(under)h(which)f(conditions)f(the)i (addition)f(of)h(the)f(nonlinear)h(terms)f(does)h FA(not)f Fr(change)h(the)g(stability)94 4822 y(of)26 b(the)g(solutions.)37 b(This)25 b(is)g(more)h(dif)n(\256cult,)f(for)h(tw)o(o)f(reasons:)38 b(First,)25 b(as)h(we)g(ha)n(v)o(e)g(said,)f(the)h(spectrum)f(of)94 4941 y(the)32 b(linearized)f(problem)f(e)o(xtends)g(all)h(the)g(w)o(ay)g(to)g (0,)i(and)e(second,)h(the)g(nonlinearity)d(does)i(not)g(ha)n(v)o(e)g(a)94 5061 y(sign.)p 94 5195 1200 4 v 211 5248 a Fk(\003)294 5291 y Fw(In)19 b(our)g(e)o(xample,)i(the)f(curv)n(atures)g(of)f(the)i(tw)o(o)e (minima)i(are)f(equal.)29 b(This)20 b(does)g(not)g(seem)h(to)f(be)g (necessary)g(for)f(our)g(proofs.)p eop %%Page: 3 3 bop 94 -45 a Fu(Cahn-Hilliard)39 b(Equa)-7 b(tion)2501 b Ft(3)294 194 y Fr(Another)l(,)28 b(more)g(complicated,)g(e)o(xample)f(of)h(a)h (similar)d(nature)j(is)e(pro)o(vided)g(by)h(time-independent)94 314 y(solutions)23 b(of)i(the)g(Ginzb)n(ur)n(g-Landau)f(equations)g(\(on)h Fm(R)p Fr(\))1467 557 y Fq(@)1520 584 y Fy(t)1555 557 y Fq(u)45 b Fp(=)g Fq(@)1838 514 y Fz(2)1832 584 y Fy(x)1882 557 y Fq(u)22 b Fp(+)g Fq(u)h Fo(\000)f Fq(u)p Fo(j)p Fq(u)p Fo(j)2409 514 y Fz(2)2473 557 y Fq(;)1168 b Fp(\()p Fr(1)p Fq(:)p Fr(4)p Fp(\))94 801 y Fr(which)22 b(are)g(e)o(xactly)g(on)f(the)h(borderline)f (between)h(being)f(Eckhaus)h(stable)f(and)h(Eckhaus)f(unstable.)34 b(These)94 920 y(solutions)23 b(are)1496 1044 y Fq(u)1553 1071 y Fy(q)1597 1044 y Fp(\()p Fq(x)p Fp(\))44 b(=)g Fq(e)1943 1001 y Fy(iq)s(x)2061 951 y Fl(p)p 2161 951 260 4 v 93 x Fr(1)22 b Fo(\000)g Fq(q)2380 1015 y Fz(2)2445 1044 y Fq(;)94 1260 y Fr(with)33 b Fq(q)44 b Fp(=)39 b Fr(1)p Fq(=)609 1176 y Fo(p)p 692 1176 50 4 v 84 x Fr(3,)c FA(cf)o(.)61 b Fr([EG].)34 b(W)-8 b(e)33 b(will)g(not)f(pro)o(v)o(e)h(that)g(this)f(problem)g(scales)i(lik)o(e) e(the)i(Cahn-Hilliard)94 1379 y(equations,)28 b(b)n(ut)g(only)g(describe)g(a) g(program)g(which)g(we)h(belie)n(v)o(e)e(w)o(ould)g(lead)h(to)g(a)h(proof.)46 b(The)28 b(\256rst)g(part)94 1499 y(of)h(the)g(analysis)f(of)g(this)g (problem)g(w)o(ould)g(follo)n(w)f(rather)j(closely)e(that)g(gi)n(v)o(en)f(in) h([EEW])h(for)g(the)g(Swift-)94 1618 y(Hohenber)n(g)23 b(equation.)35 b(Letting)21 b Fq(u)1329 1575 y Fn(\003)1403 1618 y Fp(=)28 b Fq(u)1565 1645 y Fy(q)1632 1618 y Fr(for)23 b(the)g(critical)g(v)n(alue)f Fq(q)32 b Fp(=)c Fr(1)p Fq(=)2733 1535 y Fo(p)p 2816 1535 V 83 x Fr(3)o(,)c(and)e(writing)g Fq(u)28 b Fp(=)g Fq(u)3638 1575 y Fn(\003)3702 1618 y Fp(+)18 b Fq(v)t Fr(,)94 1738 y(the)25 b(equation)f(for)h Fq(v)k Fr(is)1052 1981 y Fq(@)1105 2008 y Fy(t)1140 1981 y Fq(v)48 b Fp(=)d Fq(@)1417 1938 y Fz(2)1411 2008 y Fy(x)1461 1981 y Fq(v)26 b Fp(+)d Fq(v)j Fo(\000)d Fr(2)p Fq(v)t Fo(j)p Fq(u)1996 1938 y Fn(\003)2041 1981 y Fo(j)2069 1938 y Fz(2)2130 1981 y Fo(\000)k Fp(\026)-54 b Fq(v)t Fp(\()p Fq(u)2378 1938 y Fn(\003)2423 1981 y Fp(\))2462 1938 y Fz(2)2524 1981 y Fp(+)23 b Fj(O)p Fp(\()p Fq(v)2785 1938 y Fz(2)2825 1981 y Fp(\))h Fq(:)753 b Fp(\()p Fr(1)p Fq(:)p Fr(5)p Fp(\))94 2225 y Fr(It)18 b(has)g(a)h(linear)e(part)h(which)g(is)f(lik)o(e)h(a)g(Schr) 1569 2224 y(\310)1561 2225 y(odinger)g(operator)g(in)g(a)g(periodic)g (potential)e(\(the)i(inhomogeneity)94 2344 y Fq(u)151 2301 y Fn(\003)197 2344 y Fr(\).)36 b(This)24 b(can)i(be)f(handled)f(by)g(going)g (to)h(Floquet)f(v)n(ariables,)g(namely)g(setting)1351 2641 y Fq(v)t Fp(\()p Fq(x;)17 b(t)p Fp(\))43 b(=)1784 2505 y Fl(Z)1884 2530 y Fy(q)1839 2732 y Fn(\000)p Fy(q)1962 2641 y Fr(d)p Fq(k)20 b(e)2130 2598 y Fy(ik)r(x)2252 2641 y Fq(v)2300 2668 y Fy(k)2350 2641 y Fp(\()p Fq(x;)d(t)p Fp(\))23 b Fq(;)94 2954 y Fr(where)j Fq(v)411 2981 y Fy(k)485 2954 y Fr(is)e Fq(\031)t(=q)t Fr(-periodic)g(in)h Fq(x)p Fr(:)1334 3219 y Fq(v)1382 3246 y Fy(k)1431 3219 y Fp(\()p Fq(x;)17 b(t)p Fp(\))43 b(=)1829 3124 y Fl(X)1812 3338 y Fy(m)p Fn(2)p Fi(Z)2008 3219 y Fq(e)2054 3176 y Fz(2)p Fy(imq)s(x)2277 3219 y Fq(v)2325 3246 y Fy(k)r(;m)2469 3219 y Fp(\()p Fq(t)p Fp(\))24 b Fq(:)94 3557 y Fr(The)34 b(linear)f(part)g(of)h(Eq.\(1.5\))e(lea)n (v)o(es)h(the)g(subspace)g(spanned)g(by)g(the)g Fq(v)2730 3584 y Fy(k)2813 3557 y Fr(in)l(v)n(ariant,)h(and)f(has)g(discrete)94 3677 y(spectrum)21 b(in)h(each)g(such)f(subspace.)35 b(The)21 b(spectrum)g(is)g(in)g Fq(\033)32 b Fo(\024)c Fr(0)21 b(and)h(the)f(lar)n (gest)h(eigen)l(v)n(alue)f(is)g Fo(\000)p Fj(O)p Fp(\()p Fq(k)3795 3634 y Fz(4)3835 3677 y Fp(\))94 3796 y Fr(when)31 b Fq(q)j Fr(equals)c(its)g(critical)g(v)n(alue)g Fq(q)40 b Fp(=)c Fr(1)p Fq(=)1677 3713 y Fo(p)p 1759 3713 V 1759 3796 a Fr(3)31 b(\(which)f(is)g(the) g(case)h(we)g(discuss)e(here\).)53 b(In)31 b(this)e(sense,)94 3916 y(the)35 b(problem)g(of)g(the)g(mar)n(ginal)f(Eckhaus)h(instability)e (resembles)h(the)h(problem)g(of)g(the)g(Cahn-Hilliard)94 4035 y(equation.)68 b(At)35 b(this)g(point,)i(the)e(discussion)f(of)h(the)h (problem)f(follo)n(ws)f(the)h(techniques)g(we)h(de)n(v)o(eloped)94 4155 y(in)c([EWW].)g(W)-8 b(e)32 b(w)o(ould)f(lik)o(e)g(to)g(rescale)h(as)g (we)g(will)f(do)g(belo)n(w)g(for)h(the)f(Cahn-Hilliard)g(equation)g(and)94 4274 y(its)37 b(generalizations,)i(b)n(ut)e(the)f(problem)h(will)f(be)h(more) g(complicated)f(because)h(the)g(Brillouin)f(zone)i(is)94 4394 y(restricted)c(to)g Fq(k)43 b Fo(2)e Fr([)p Fo(\000)p Fq(q)t(;)17 b(q)t Fr(].)64 b(W)-8 b(e)34 b(then)g(ha)n(v)o(e)g(to)f(check)i(that)e(the)h (non-linearity)f(is)g(\252irrele)n(v)n(ant\272)h(in)f(the)94 4513 y(terminology)28 b(de)n(v)o(eloped)h(belo)n(w)-6 b(.)48 b(Again,)30 b(as)g(in)f([EWW],)h(we)g(belie)n(v)o(e)f(that)g(this)g(will)f (not)h(be)h(quite)f(the)94 4633 y(case,)38 b(b)n(ut)c(the)g(sa)n(ving)g (grace)h(will)f(be)h(that)f(the)g(projection)g(of)h(the)f(potentially)f (non-irrele)n(v)n(ant)g(modes)94 4753 y(onto)27 b(the)g(eigenstates)f (corresponding)g(to)h(the)g Fo(\000)p Fj(O)p Fp(\()p Fq(k)2004 4710 y Fz(4)2044 4753 y Fp(\))g Fr(term)g(v)n(anish)f(to)h(some)f(higher)h (de)o(grees)g(because)94 4872 y(of)e(translation)f(in)l(v)n(ariance)g(of)h (the)g(original)f(problem,)g FA(cf)o(.)35 b Fr([EWW)-9 b(,)25 b(Section)f(4],)h(and)g([S].)294 4992 y(W)-8 b(e)30 b(place)h(our)e(e)o (xamples)g(in)h(the)g(follo)n(wing)e(more)i(general)g(setting.)50 b(Consider)30 b(equations)f(of)h(the)94 5111 y(form)1217 5195 y Fq(@)6 b(u)p 1217 5240 116 4 v 1228 5331 a(@)g(t)1389 5263 y Fp(=)45 b(\()p Fo(\000)p Fr(1)p Fp(\))1716 5220 y Fy(n)p Fh(+)p Fz(1)1866 5263 y Fp(\001)1949 5220 y Fy(n)2003 5263 y Fq(u)22 b Fp(+)h Fq(F)14 b Fp(\()p Fq(u;)j Fo(f)p Fq(@)2510 5220 y Fy(\013)2504 5290 y(x)2565 5263 y Fq(u)p Fo(g)p Fp(\))24 b Fq(;)906 b Fp(\()p Fr(1)p Fq(:)p Fr(6)p Fp(\))p eop %%Page: 4 4 bop 94 -45 a Fu(Cahn-Hilliard)39 b(Equa)-7 b(tion)2501 b Ft(4)94 194 y Fr(where)37 b(the)e(multi-indices)f Fq(\013)h Fr(satisfy)g Fo(j)p Fq(\013)p Fo(j)42 b(\024)h Fr(2)p Fq(n)28 b Fo(\000)h Fr(1,)38 b(and)d Fq(x)43 b Fo(2)g Fm(R)2601 151 y Fy(d)2648 194 y Fr(,)38 b Fq(t)k Fo(\025)i Fr(1.)68 b(Furthermore,)38 b Fq(F)50 b Fr(is)35 b(a)94 314 y(polynomial)23 b(in)i Fq(u)f Fr(and)h(its)f(deri)n(v)n(ati)n(v)o(es.)33 b(W)-8 b(e)25 b(wish)f(to)h(study) f(the)g(asymptotics)f(of)i(the)f(solution)f Fq(u)i Fr(of)g(\(1.6\))94 433 y(as)g Fq(t)j Fo(!)f(1)p Fr(.)36 b(First,)24 b(one)h(introduces)f (scaling)g(v)n(ariables)g(by)h(de\256ning)1257 713 y Fq(u)p Fp(\()p Fq(x;)17 b(t)p Fp(\))43 b(=)1816 645 y Fr(1)p 1707 690 270 4 v 1707 783 a Fq(t)1743 754 y Fy(d=)p Fh(\()p Fz(2)p Fy(n)p Fh(\))2005 713 y Fq(v)2057 632 y Fl(\000)2217 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b(will)e(discuss)h(the)g(form)g(of)h(the)f(non-linear) g(terms)g(belo)n(w)-6 b(,)18 b(and)h(consider)f(\256rst)g(the)h(linear)f (operator)1395 3623 y Fj(L)45 b Fp(=)g Fo(\000)p Fp(\()p Fq(p)22 b Fo(\001)g Fq(p)p Fp(\))1960 3580 y Fy(n)2036 3623 y Fo(\000)2178 3556 y Fr(1)p 2148 3600 110 4 v 2148 3691 a(2)p Fq(n)2269 3623 y(p)h Fo(\001)e(r)2474 3650 y Fy(p)2545 3623 y Fq(:)1046 b Fp(\()p Fr(1)p Fq(:)p Fr(10)p Fp(\))94 3885 y Fr(A)25 b(straightforw)o(ard)f (calculation)g(sho)n(ws)g(that)g Fj(L)i Fr(has)e(the)h(countable)f(set)h(of)g (eigen)l(v)n(alues)1338 4163 y Fq(\025)1396 4190 y Fy(j)1482 4163 y Fp(=)45 b Fo(\000)1725 4095 y Fq(j)p 1693 4140 V 1693 4231 a Fr(2)p Fq(n)1840 4163 y(;)116 b(j)33 b Fp(=)28 b Fr(0)p Fq(;)17 b Fr(1)p Fq(;)g Fr(2)p Fq(;)g(:)g(:)g(:)37 b(;)988 b Fp(\()p Fr(1)p Fq(:)p Fr(11)p Fp(\))94 4421 y Fr(with)25 b(eigenfunctions)e(\(written)h(in)h(multi-inde)o(x)d(notation\),)1531 4654 y Fq(')1596 4681 y Fy(\013)1653 4654 y Fp(\()p Fq(p)p Fp(\))44 b(=)h Fq(p)1997 4612 y Fy(\013)2054 4654 y Fq(e)2100 4612 y Fn(\000)p Fh(\()p Fy(p)p Fn(\001)p Fy(p)p Fh(\))2330 4581 y Ff(n)2409 4654 y Fq(;)1182 b Fp(\()p Fr(1)p Fq(:)p Fr(12)p Fp(\))94 4887 y Fr(and)25 b Fo(j)p Fq(\013)p Fo(j)i Fp(=)i Fq(j)6 b Fr(.)294 5007 y(If)25 b(we)g(consider)g Fj(L)g Fr(as)g(acting)f(on)h (the)f(Sobole)n(v)g(spaces)768 5241 y Fp(~)742 5266 y Fq(H)825 5293 y Fy(`;m)1002 5266 y Fp(=)1124 5156 y Fl(n)1194 5266 y Fp(~)-54 b Fq(v)57 b Fr(:)52 b Fo(k)p Fq(p)1475 5224 y Fy(\013)1532 5266 y Fq(@)1591 5224 y Fy(\014)1585 5293 y(p)1648 5266 y Fp(~)-54 b Fq(v)t Fo(k)1746 5293 y Fz(L)1789 5273 y Fg(2)1851 5266 y Fq(<)28 b Fo(1)c Fq(;)42 b Fr(for)25 b(all)f Fo(j)p Fq(\013)p Fo(j)j(\024)i Fq(`;)17 b Fo(j)p Fq(\014)5 b Fo(j)26 b(\024)i Fq(m)3090 5156 y Fl(o)3199 5266 y Fq(;)p eop %%Page: 5 5 bop 94 -45 a Fu(Cahn-Hilliard)39 b(Equa)-7 b(tion)2501 b Ft(5)94 194 y Fr(then)29 b Fj(L)g Fr(will)e(ha)n(v)o(e)h(continuous)f(spectrum)h(in)g (the)h(half-plane)f(Re)17 b Fq(\025)33 b(<)h Fo(\000)p Fq(\033)2803 221 y Fy(m)2907 194 y Fr(in)28 b(addition)g(to)g(the)g(eigen-)94 314 y(v)n(alues)f(abo)o(v)o(e.)44 b(By)28 b(choosing)e Fq(m)i Fr(appropriately)-6 b(,)28 b(we)f(can)h(force)h(this)e(continuous)f(spectrum) h(arbitrarily)94 433 y(f)o(ar)g(into)e(the)h(left)f(half-plane,)h(and)g(the)g (dominant)e(beha)n(vior)i(of)g(the)f(linear)h(operator)g(will)f(be)h (dictated)f(by)94 553 y(the)g(eigen)l(v)n(alues)f(with)g(the)h(lar)n(gest)f (real)i(part.)94 719 y Fm(Remark.)47 b Fr(In)33 b(order)h(to)f(switch)f(back) i(and)f(forth)g(from)g(the)h(F)o(ourier)f(transform)g(representation)f(of)i Fj(L)94 839 y Fr(to)g(the)f(un-F)o(ourier)h(transformed)f(representation)g (of)h(this)f(operator)g(with)g(ease,)k(we)d(also)f(consider)g(the)94 959 y(Sobole)n(v)24 b(spaces)739 1167 y Fq(H)822 1194 y Fy(`;m)999 1167 y Fp(=)1120 1057 y Fl(n)1187 1167 y Fq(v)56 b Fr(:)d Fo(k)p Fq(@)1481 1125 y Fy(\013)1475 1194 y(x)1537 1167 y Fq(x)1594 1125 y Fy(\014)1647 1167 y Fq(v)t Fo(k)1749 1194 y Fz(L)1792 1174 y Fg(2)1854 1167 y Fq(<)28 b Fo(1)d Fq(;)41 b Fr(for)25 b(all)g Fo(j)p Fq(\013)p Fo(j)i(\024)h Fq(`;)17 b Fo(j)p Fq(\014)5 b Fo(j)26 b(\024)j Fq(m)3094 1057 y Fl(o)3202 1167 y Fq(:)94 1397 y Fr(Note)c(that)f(F)o(ourier)h(transformation)f(is)g(an)h(isomorphism)d (from)2416 1372 y Fp(~)2390 1397 y Fq(H)2473 1424 y Fy(`;m)2630 1397 y Fr(to)j Fq(H)2816 1424 y Fy(`;m)2948 1397 y Fr(.)294 1516 y(Note)32 b(that)g Fj(L)h Fr(is)e FA(not)h Fr(sectorial,)i(and)e (therefore)h(we)g(kno)n(w)e(of)i(no)f(w)o(ay)g(to)g(bound)g(the)g(semi-group) 94 1636 y(generated)23 b(by)g Fj(L)f Fr(by)h(spectral)f(information)f(alone.) 35 b(Ho)n(we)n(v)o(er)l(,)22 b(in)g(Appendix)f(A,)i(we)f(de)n(v)o(elop)g(an)g (inte)o(gral)94 1755 y(representation)j(of)h(the)f(semi-group)g(and)g(we)h (then)f(sho)n(w)g(that)g(it)g(satis\256es)g(the)g(estimates)g(needed)g(for)h (the)94 1875 y(in)l(v)n(ariant)e(manifold)g(theorem.)294 1994 y(W)-8 b(e)25 b(ne)o(xt)f(discuss)g(which)g(terms)g(in)h(the)f(non-linearity) g(are)i(\252rele)n(v)n(ant.)-7 b(\272)35 b(Consider)24 b(a)i(monomial)1521 2254 y Fq(A)44 b Fp(=)1809 2129 y Fy(s)1765 2159 y Fl(Y)1762 2372 y Fy(j)t Fh(=)p Fz(0)1911 2143 y Fl(\020)1971 2254 y Fq(@)2030 2211 y Fy(\013)2082 2181 y Fe(\()p Ff(j)s Fe(\))2024 2280 y Fy(x)2177 2254 y Fq(u)2234 2143 y Fl(\021)2294 2164 y Fy(k)2336 2174 y Ff(j)2420 2254 y Fq(;)1171 b Fp(\()p Fr(1)p Fq(:)p Fr(13)p Fp(\))94 2563 y Fr(where)36 b(the)f Fq(\013)594 2520 y Fh(\()p Fy(j)t Fh(\))734 2563 y Fr(are)h(distinct)e(multi-indices.)64 b(After)36 b(rescaling)f(and)g(taking)g(F)o(ourier)g(transforms)f(this)94 2683 y(becomes)876 2873 y Fp(~)850 2898 y Fq(A)44 b Fp(=)h Fr(e)o(xp)1250 2757 y Fl(\022)1323 2898 y Fp(\()1374 2830 y Fr(2)p Fq(n)22 b Fp(+)g Fq(d)p 1374 2875 284 4 v 1461 2966 a Fr(2)p Fq(n)1669 2898 y Fp(\))p Fq(\034)1763 2757 y Fl(\023)1886 2898 y Fr(e)o(xp)2045 2698 y Fl(0)2045 2877 y(@)2132 2898 y Fo(\000)2279 2773 y Fy(s)2226 2803 y Fl(X)2231 3016 y Fy(j)t Fh(=)p Fz(0)2370 2817 y Fl(\000)2428 2830 y Fo(j)p Fq(\013)2520 2788 y Fh(\()p Fy(j)t Fh(\))2624 2830 y Fo(j)f Fp(+)i Fq(d)p 2428 2875 398 4 v 2571 2966 a Fr(2)p Fq(n)2837 2817 y Fl(\001)2883 2898 y Fq(k)2935 2925 y Fy(j)2976 2898 y Fq(\034)3031 2698 y Fl(1)3031 2877 y(A)964 3202 y Fo(\002)1063 3122 y Fl(\000)1109 3202 y Fp(\()p Fo(\000)p Fq(ip)p Fp(\))1348 3160 y Fy(\013)1400 3129 y Fe(\()p Fg(0)p Fe(\))1493 3202 y Fp(~)-54 b Fq(v)1541 3122 y Fl(\001)1586 3142 y Fn(\003)p Fy(k)1669 3152 y Fg(0)1727 3202 y Fo(\003)22 b(\001)17 b(\001)g(\001)j(\003)2009 3122 y Fl(\000)2055 3202 y Fp(\()p Fo(\000)p Fq(ip)p Fp(\))2294 3160 y Fy(\013)2346 3129 y Fe(\()p Ff(s)p Fe(\))2447 3202 y Fp(~)-54 b Fq(v)2495 3122 y Fl(\001)2541 3142 y Fn(\003)p Fy(k)2624 3152 y Ff(s)2692 3202 y Fq(:)3619 2990 y Fp(\()p Fr(1)p Fq(:)p Fr(14)p Fp(\))94 3359 y Fr(Here,)26 b Fo(\003)f Fr(denotes)f(the)h(con)l(v)n (olution)e(product.)36 b(If)25 b(we)g(combine)f(the)h(po)n(wers)f(of)h Fq(\034)36 b Fr(in)25 b(the)g(e)o(xponential,)e(we)94 3478 y(see)j(that)e(if)1370 3654 y(2)p Fq(n)e Fp(+)h Fq(d)44 b(<)1873 3529 y Fy(s)1819 3559 y Fl(X)1825 3772 y Fy(j)t Fh(=)p Fz(0)1963 3573 y Fl(\000)2009 3654 y Fo(j)p Fq(\013)2101 3611 y Fh(\()p Fy(j)t Fh(\))2205 3654 y Fo(j)22 b Fp(+)g Fq(d)2406 3573 y Fl(\001)2452 3654 y Fq(k)2504 3681 y Fy(j)2571 3654 y Fq(;)1020 b Fp(\()p Fr(1)p Fq(:)p Fr(15)p Fp(\))94 3917 y Fr(then)35 b(the)g(coef)n(\256cient)g(of)g(this)f(term)h(will)f(go)g(to)h(zero)g(e)o (xponentially)e(f)o(ast)i(in)g Fq(\034)11 b Fr(,)37 b(and)e(hence)g(it)g (will)f(be)94 4037 y(irrele)n(v)n(ant)24 b(from)h(the)f(point)g(of)h(vie)n(w) f(of)h(the)g(long)f(time)g(beha)n(vior)g(of)h(the)g(solutions.)94 4203 y Fm(De\014nitions.)36 b Fr(A)25 b(monomial)e(lik)o(e)h(\(1.14\))g(is)g (called)h FA(irr)l(ele)o(vant)e Fr(if)i(it)f(satis\256es)g(the)g(inequality)g (\(1.15\).)35 b(It)94 4323 y(is)23 b(called)g FA(critical)f Fr(if)h(the)g(l.h.s.)f(of)h(Eq.\(1.15\))g(is)f(equal)h(to)g(the)g(r)-5 b(.h.s,)22 b(and)h FA(r)l(ele)o(vant)g 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5173 113 4 v 1182 5213 a(2)p Ff(n)1285 5189 y Fn(\000)p Fy(")p Fh(\))1437 5087 y Fl(\015)1437 5147 y(\015)1437 5207 y(\015)1437 5266 y(\015)1493 5232 y Fq(u)p Fp(\()p Fq(x;)17 b(t)p Fp(\))j Fo(\000)1970 5164 y Fq(B)2051 5121 y Fn(\003)p 1898 5209 270 4 v 1898 5302 a Fq(t)1934 5273 y Fy(d=)p Fh(\()p Fz(2)p Fy(n)p Fh(\))2180 5232 y Fq(f)2240 5189 y Fn(\003)2285 5232 y Fp(\()2439 5164 y Fq(x)p 2336 5209 264 4 v 2336 5302 a(t)2372 5273 y Fz(1)p Fy(=)p Fh(\()p Fz(2)p Fy(n)p Fh(\))2611 5232 y Fp(\))2650 5087 y Fl(\015)2650 5147 y(\015)2650 5207 y(\015)2650 5266 y(\015)2705 5331 y Fz(L)2748 5311 y Fd(1)2869 5232 y Fp(=)45 b Fr(0)25 b Fq(:)p eop %%Page: 6 6 bop 94 -45 a Fu(Cahn-Hilliard)39 b(Equa)-7 b(tion)2501 b Ft(6)94 194 y Fc(Here,)1180 354 y Fq(f)1240 311 y Fn(\003)1285 354 y Fp(\()p Fq(\030)5 b Fp(\))43 b(=)1719 286 y Fr(1)p 1589 331 310 4 v 1589 424 a Fp(\()p Fr(2)p Fq(\031)t Fp(\))1778 395 y Fy(d=)p Fz(2)1927 218 y Fl(Z)2060 354 y Fr(d)2110 311 y Fy(d)2156 354 y Fq(p)17 b(e)2269 311 y 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Fq(Q)801 5288 y Fy(k)850 5261 y Fq(v)t Fo(k)952 5288 y Fy(`;m)1128 5261 y Fo(\024)1304 5194 y Fq(C)1375 5221 y Fy(k)p 1262 5238 205 4 v 1262 5331 a Fq(t)1298 5302 y Fy(q)s(=)p Fz(2)p Fy(n)1496 5261 y Fr(e)o(xp)1638 5180 y Fl(\000)1696 5194 y Fq(k)25 b Fp(+)d Fr(1)p 1696 5238 227 4 v 1754 5329 a(2)p Fq(n)1934 5261 y(\034)1989 5180 y Fl(\001)2035 5261 y Fo(k)p Fq(v)t Fo(k)2187 5288 y Fy(`)p Fn(\000)p Fy(q)s(;m)2445 5261 y Fq(;)17 b(q)31 b Fp(=)d Fr(0)p Fq(;)17 b Fr(1)p Fq(;)g(:)g(:)g(:)c(;)k Fr(2)p Fq(n)k Fo(\000)i Fr(1)i Fq(:)3669 5253 y Fp(\()p Fr(2)p Fq(:)p Fr(1)p Fp(\))p eop %%Page: 7 7 bop 94 -45 a Fu(Cahn-Hilliard)39 b(Equa)-7 b(tion)2501 b Ft(7)94 194 y Fm(Pro)s(of.)33 b Fr(The)20 b(proof,)h(which)e(is)h(presented)f(in)h (Appendix)f(A,)g(is)h(modeled)f(on)g(the)h(proof)g(in)f([EWW])i(which)94 314 y(treats)k(the)g(case)g Fq(n)j Fp(=)g Fr(1.)294 433 y(Gi)n(v)o(en)34 b(such)h(estimates)g(on)h(the)f(linear)h(e)n(v)n(olution,)g(the)g 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Fp(\()p Fr(2)p Fq(:)p Fr(3)p Fp(\))p eop %%Page: 8 8 bop 94 -45 a Fu(Cahn-Hilliard)39 b(Equa)-7 b(tion)2501 b Ft(8)94 194 y Fc(Then)32 b(there)g(e)o(xists)e(a)i Fo(C)953 151 y Fz(1)p Fh(+)p Fy(\013)1137 194 y Fc(function)f Fq(h)p Fp(\()p Fq(\021)t(;)17 b(y)t Fp(\))p Fc(,)32 b(with)f Fq(\013)37 b(>)g Fr(0)p Fc(,)c(de\256ned)f(in) g(some)f(neighborhood)f(of)h(the)94 324 y(origin)20 b(in)g Fm(R)13 b Fo(\002)g Fm(R)726 281 y Fz(dim)25 b(range)p Fh(\()p Fy(P)1096 292 y Ff(k)1139 281 y Fh(\))1176 324 y Fc(,)c(such)f(that)g(the)h (manifold)e Fq(z)32 b Fp(=)c Fq(h)p Fp(\()p Fq(\021)t(;)17 b(y)t Fp(\))j Fc(is)g(left)g(in)l(v)n(ariant)f(by)i(the)f(semi-\257o)n(w)94 444 y(of)30 b(Eq.\(2.2\).)47 b(Furthermore,)30 b(an)o(y)f(solution)e(of)i (Eq.\(2.2\))g(which)f(remains)h(near)g(the)g(origin)f(for)h(all)g(times)94 585 y(approaches)d(a)f(solution)e(of)i(\(2.2\)\320restricted)f(to)g(the)h(in) l(v)n(ariant)e(manifold\320at)h(a)h(rate)g Fj(O)3268 504 y Fl(\000)3314 585 y Fq(e)3372 510 y Ff(k)q Fe(+)p Fg(1)p Fd(\000)p Ff(")p 3372 526 203 4 v 3439 566 a Fg(2)p Ff(n)3587 542 y Fy(\034)3636 504 y Fl(\001)3682 585 y Fc(.)94 791 y Fm(Pro)s(of.)34 b Fr(The)20 b(e)o(xistence)f(of)h(the)g(in)l(v)n(ariant)f(manifold,)h(gi)n(v)o(en)e(the)i (assumptions)e(on)h(the)h(linear)g(semi-group)94 910 y(and)25 b(the)g(non-linearity)-6 b(,)23 b(seems,)i(to)f(our)h(kno)n(wledge,)f(not)g (to)h(be)g(e)o(xplicitly)e(spelled)h(out)g(in)h(the)f(literature.)94 1030 y(The)19 b(formulation)e(which)i(comes)f(closest)g(to)g(our)h(needs)f (is)g(the)h(one)f(gi)n(v)o(en)g(in)g([H],)h(where)g(the)f(assumptions)94 1149 y(on)j(the)f(non-linearity)g(are)h(those)f(we)h(ha)n(v)o(e)g(in)f(our)h (case,)h(b)n(ut)e(the)g(semi-group)g(is)g(supposed)g(to)g(be)h(analytic.)94 1269 y(Ho)n(we)n(v)o(er)l(,)j(Henry')-5 b(s)24 b(construction)f(of)h(the)g (in)l(v)n(ariant)g(manifold)f(only)g(uses)h(certain)h(bounds)e(on)h(the)h (decay)94 1389 y(of)g(the)f(semi-group,)f(and)h(not)f(the)h(stronger)g (assumption)e(of)i(analyticity)-6 b(.)34 b(Those)24 b(bounds)f FA(ar)l(e)h Fr(true)g(in)g(our)94 1508 y(case,)36 b(by)c(Lemma)h(2.2.)59 b(Thus,)35 b(e)o(xistence)d(of)h(the)f(in)l(v)n(ariant)g(manifold)g(follo)n (ws)f(in)i(f)o(act)g(from)g(Henry')-5 b(s)94 1628 y(proof.)294 1747 y(Once)26 b(one)f(kno)n(ws)f(that)i(the)f(manifold)f(e)o(xists,)h(it)g (is)g(also)g(easy)h(to)f(sho)n(w)f(that)h(an)o(y)g(solution)f(near)i(the)94 1867 y(origin)e(must)f(approach)i(a)g(solution)d(on)i(the)h(in)l(v)n(ariant)e (manifold)g(\(see,)i FA(e)o(.g)o(.)e Fr([C]\).)j(Note)e(that)g(e)n(v)o(en)f (though)94 1986 y(our)30 b(non-linearity)f(is)g(quite)h(smooth,)f(we)h (cannot)g(hope,)h(in)e(general,)j(to)d(obtain)g(an)i(in)l(v)n(ariant)d (manifold)94 2106 y(whose)c(smoothness)e(is)i(greater)g(than)g Fo(C)1507 2063 y Fz(1)p Fh(+)p Fy(\013)1660 2106 y Fr(,)g(since)g(this)f (smoothness)f(is)i(related)g(to)g(the)g(gap)f(between)h(the)94 2225 y(spectrum)h(of)g Fp(\003)662 2252 y Fy(k)711 2225 y Fr(,)g(and)f(that)h (of)g Fq(Q)1291 2252 y Fy(k)1339 2225 y Fj(L)p Fq(Q)1489 2252 y Fy(k)1538 2225 y Fr(,)g(\(see,)g FA(e)o(.g)o(.)f Fr([L)-7 b(W]\).)94 2535 y Fs(3.)46 b(Applications)94 2724 y Fr(Here,)37 b(we)c(sho)n(w)g(ho)n(w)g(the)g(e)o(xistence)g(of)g(the)h(in)l(v)n(ariant)e (manifold)g(implies)g(Theorem)h(1.1)g(and)h(related)94 2844 y(results.)69 b(T)-8 b(o)36 b(pro)o(v)o(e)f(Theorem)h(1.1,)i(we)e(assume)g (that)f(all)h(terms)f(in)h(the)g(non-linearity)f(are)h(irrele)n(v)n(ant.)94 2963 y(This)c(means)g(that)f(Eq.\(2.3\))h(holds.)57 b(Suppose)32 b(further)g(that)g Fq(k)40 b Fp(=)f Fr(1)32 b(and)g(that)g Fq(`)37 b(>)h Fr(2)p Fq(n)26 b Fo(\000)h Fr(1)f Fp(+)g Fq(d=)p Fr(2)32 b(and)94 3083 y Fq(m)48 b(>)g Fp(\()p Fq(n)30 b Fp(+)h Fr(2)p Fp(\))p Fq(=)p Fp(\()p Fr(2)p Fq(n)p Fp(\))p Fr(.)77 b(These)39 b(hypotheses)e(guarantee)j(that)e(Theorem)h(2.4)g(applies)f(and)h (hence)h(an)o(y)94 3226 y(solution)25 b(near)h(the)g(origin)f(must)f (approach)i(a)h(solution)d(on)h(the)h(in)l(v)n(ariant)e(manifold,)h(at)h(a)g (rate)h Fj(O)3550 3145 y Fl(\000)3596 3226 y Fq(e)3654 3151 y Fg(2)p Fd(\000)p Ff(")p 3654 3168 113 4 v 3676 3207 a Fg(2)p Ff(n)3779 3183 y Fy(\034)3828 3145 y Fl(\001)94 3345 y Fr(in)e Fq(H)280 3372 y Fy(m;`)412 3345 y Fr(.)294 3465 y(The)f(equations)e(on)i(the) g(in)l(v)n(ariant)e(manifold)h(can)h(be)g(written)g(as)g(a)g(system)e(of)i (ordinary)g(dif)n(ferential)94 3585 y(equations:)856 3701 y Fp(_)-46 b Fq(y)887 3728 y Fz(0)971 3701 y Fp(=)1093 3620 y Fl(\012)1140 3701 y Fq(')1205 3658 y Fn(\003)1205 3728 y Fz(0)1251 3701 y Fo(j)p Fq(f)1339 3620 y Fl(\000)1384 3701 y Fq(y)t(;)17 b(h)p Fp(\()p Fq(\021)t(;)g(y)t Fp(\))p Fq(;)g(\021)1866 3620 y Fl(\001)l(\013)1980 3701 y Fq(;)796 3862 y Fp(_)-46 b Fq(y)827 3888 y Fz(1)p Fy(;j)971 3862 y Fp(=)45 b Fo(\000)1207 3822 y Fz(1)p 1183 3839 85 4 v 1183 3896 a(2)p Fy(n)1279 3862 y Fq(y)1328 3888 y Fz(1)p Fy(;j)1450 3862 y Fp(+)1550 3781 y Fl(\012)1597 3862 y Fq(')1662 3819 y Fn(\003)1662 3888 y Fz(1)p Fy(;j)1763 3862 y Fo(j)p Fq(f)1851 3781 y Fl(\000)1895 3862 y Fq(y)t(;)17 b(h)p Fp(\()p Fq(\021)t(;)g(y)t Fp(\))p Fq(;)g(\021)2377 3781 y Fl(\001)m(\013)2492 3862 y Fq(;)116 b(j)33 b Fp(=)28 b Fr(1)p Fq(;)17 b(:)g(:)g(:)d(;)j(d)24 b(;)892 4028 y Fp(_)-46 b Fq(\021)48 b Fp(=)d Fo(\000)1207 3989 y Fz(1)p 1183 4005 V 1183 4062 a(2)p Fy(n)1279 4028 y Fq(\021)29 b(;)3669 3864 y Fp(\()p Fr(3)p Fq(:)p Fr(1)p Fp(\))94 4185 y Fr(where)36 b Fq(')438 4142 y Fn(\003)438 4212 y Fz(0)519 4185 y Fr(and)e Fq(')762 4142 y Fn(\003)762 4212 y Fz(1)p Fy(;j)898 4185 y Fr(are)i(the)e(projections)g(onto)g(the)h(spectral)f(subspace)h(of)g Fq(\025)2973 4212 y Fz(0)3048 4185 y Fr(and)f Fq(\025)3284 4212 y Fz(1)3366 4185 y Fp(=)42 b Fo(\000)p Fr(1)p Fq(=)p Fp(\()p Fr(2)p Fq(n)p Fp(\))p Fr(,)94 4305 y(respecti)n(v)o(ely)-6 b(.)34 b(Note)25 b(that)f Fq(\025)1077 4332 y Fz(1)1142 4305 y Fr(has)h(a)g Fq(d)p Fr(-dimensional)e(spectral)i(subspace.)294 4424 y(The)j(important)f(observ)n(ation)f(to)i(mak)o(e)g(at)g(this)g(point)f 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Fr(for)22 b(some)f Fq(p)28 b Fo(\025)g Fr(1.)35 b(Since)21 b Fq(\021)t Fp(\()p Fq(\034)11 b Fp(\))27 b(=)i Fq(e)1364 4852 y Fn(\000)p Fy(\034)8 b(=)p Fh(\()p Fz(2)p Fy(n)p Fh(\))1663 4895 y Fq(\021)t Fp(\()p Fr(0)p Fp(\))p Fr(,)22 b(this)e(implies)g (immediately)g(that)h(solutions)e(of)j(Eq.\(3.1\))94 5015 y(beha)n(v)o(e)j (as)1424 5121 y Fq(y)1473 5148 y Fz(0)1513 5121 y Fp(\()p Fq(\034)11 b Fp(\))43 b(=)i Fq(B)1892 5078 y Fn(\003)1959 5121 y Fp(+)23 b Fj(O)p Fp(\()p Fq(e)2214 5078 y Fn(\000)p Fy(\034)8 b(=)p Fh(\()p Fz(2)p Fy(n)p Fh(\))2514 5121 y Fp(\))25 b Fq(;)1363 5291 y(y)1412 5318 y Fz(1)p Fy(;j)1513 5291 y Fp(\()p Fq(\034)11 b Fp(\))43 b(=)i Fj(O)p Fp(\()p Fq(e)1966 5248 y Fn(\000)p Fy(\034)8 b(=)p Fh(\()p Fz(2)p Fy(n)p Fh(\))2266 5291 y Fp(\))24 b Fq(:)p eop %%Page: 9 9 bop 94 -45 a Fu(Cahn-Hilliard)39 b(Equa)-7 b(tion)2501 b Ft(9)94 194 y Fr(The)37 b(eigenfunction)d(with)i(eigen)l(v)n(alue)f(0)h(of)g Fj(L)g Fr(is)g Fq(e)2005 151 y Fn(\000)p Fy(p)p Fn(\001)p Fy(p)2178 194 y 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b(abo)o(v)o(e)g(that)g(the)h(non-linearity)f(in)g(the)g(Cahn-Hilliard)h (equation)f(is)g(irrele)n(v)n(ant)g(when)g Fq(d)36 b Fo(\025)g Fr(3,)94 1119 y(we)d(immediately)e(see)i(in)g(this)f(case)h(that)f (Eq.\(1.18\))g(holds)g(for)h(initial)e(conditions)g(which)h(are)i(close)f(to) 94 1239 y Fq(u)28 b Fo(\021)g Fr(3)334 1196 y Fn(\000)p Fz(1)p Fy(=)p Fz(2)512 1239 y Fr(,)d(which)f(yields)g(Corollary)h(1.2.)94 1579 y Fs(4.)46 b(The)e(critical)i(case)94 1768 y Fr(W)-8 b(e)36 b(no)n(w)f(consider)g(the)g(Cahn-Hilliard)g(equation)f(in)h(dimension)f Fq(d)42 b Fp(=)h Fr(2,)37 b(which)e(is)g(the)g(critical)h(case)94 1887 y(in)27 b(terms)g(of)h(the)f(renormalization)f(group)h(terminology)e ([BKL].)j(This)f(means)g(that)g(in)g(some)g(non-linear)94 2007 y(terms)e(the)f(inequality)g(Eq.\(1.15\))g(becomes)h(an)f(equality)-6 b(.)294 2127 y(In)27 b(the)g(Cahn-Hilliard)f(equation,)h(when)g Fq(d)j Fp(=)h Fr(2)c(\(and)g Fq(n)k Fp(=)g Fr(2\),)c(we)h(see)f(that)f(the)h (quadratic)g(term)g(is)94 2246 y(critical,)e(and)f(the)h(cubic)g(term)f(is)g (irrele)n(v)n(ant.)35 b(Note)24 b(that)g(Theorem)h(2.4)f(still)g(implies)f (the)h(e)o(xistence)g(of)h(an)94 2366 y(in)l(v)n(ariant)i(manifold)f(tangent) g(at)h(the)h(origin)e(to)h(the)g(eigenspace)g(of)g Fq(\025)2577 2393 y Fz(0)2617 2366 y Fr(.)44 b(This)26 b(means)h(that)g(when)g(written)94 2485 y(in)g(the)g(form)g(of)g(Eq.\(2.2\),)g(the)g(non-linearity)f(can)h(be)g (written)g(as)g(the)f(sum)h(of)g(2)g(pieces\320one)f(quadratic)94 2605 y(in)g Fq(y)k Fr(and)c Fq(z)32 b Fr(which)25 b(is)h(independent)g(of)g Fq(\021)31 b Fr(\(and)26 b(hence)h(critical\))f(and)g(a)h(cubic)f(piece)h(in) e Fq(y)30 b Fr(and)d Fq(z)k Fr(which)26 b(is)94 2724 y(linear)e(in)f Fq(\021)28 b Fr(\(and)c(hence)g(irrele)n(v)n(ant\).)34 b(This)23 b(implies)f(that)h(the)h(Eqs.\(3.1\),)f(when)h(reduced)g(to)f(the)g(in)l(v)n (ariant)94 2844 y(manifold,)h(tak)o(e)h(the)g(form)247 3054 y Fp(_)-46 b Fq(y)278 3081 y Fz(0)362 3054 y Fp(=)484 2973 y Fl(\012)531 3054 y Fq(')596 3011 y Fn(\003)596 3081 y Fz(0)642 3054 y Fo(j)p Fq(f)730 3011 y Fh(\()p Fz(2)p Fh(\))831 2973 y Fl(\000)876 3054 y Fq(y)t(;)17 b(h)p Fp(\()p Fq(\021)t(;)g(y)t Fp(\))p Fq(;)g(\021)1358 2973 y Fl(\001)m(\013)1470 3054 y Fp(+)1570 2973 y Fl(\012)1617 3054 y Fq(')1682 3011 y Fn(\003)1682 3081 y Fz(0)1728 3054 y Fo(j)p Fq(f)1816 3011 y Fh(\()p Fz(3)p Fh(\))1917 2973 y Fl(\000)1963 3054 y Fq(y)t(;)g(h)p Fp(\()p Fq(\021)t(;)g(y)t Fp(\))p Fq(;)g(\021)2445 2973 y Fl(\001)l(\013)2559 3054 y Fq(;)186 3224 y Fp(_)-46 b Fq(y)217 3251 y Fz(1)p Fy(;j)362 3224 y Fp(=)45 b Fo(\000)573 3184 y Fz(1)p 573 3201 35 4 v 573 3258 a(4)620 3224 y Fq(y)669 3251 y Fz(1)p Fy(;j)791 3224 y Fp(+)891 3143 y Fl(\012)938 3224 y Fq(')1003 3181 y Fn(\003)1003 3251 y Fz(1)p Fy(;j)1104 3224 y Fo(j)p Fq(f)1192 3181 y Fh(\()p Fz(2)p Fh(\))1293 3143 y Fl(\000)1339 3224 y Fq(y)t(;)17 b(h)p Fp(\()p Fq(\021)t(;)g(y)t Fp(\))p Fq(;)g(\021)1821 3143 y Fl(\001)l(\013)1932 3224 y Fp(+)2032 3143 y Fl(\012)2079 3224 y Fq(')2144 3181 y Fn(\003)2144 3251 y Fz(1)p Fy(;j)2244 3224 y Fo(j)p Fq(f)2332 3181 y Fh(\()p Fz(3)p Fh(\))2434 3143 y Fl(\000)2479 3224 y Fq(y)t(;)g(h)p Fp(\()p Fq(\021)t(;)g(y)t Fp(\))p Fq(;)g(\021)2961 3143 y Fl(\001)m(\013)3076 3224 y Fq(;)116 b(j)33 b Fp(=)28 b Fr(1)p Fq(;)17 b Fr(2)24 b Fq(;)283 3390 y Fp(_)-47 b Fq(\021)49 b Fp(=)c Fo(\000)573 3351 y Fz(1)p 573 3367 V 573 3425 a(4)620 3390 y Fq(\021)29 b(:)3669 3217 y Fp(\()p Fr(4)p Fq(:)p Fr(1)p Fp(\))294 3621 y Fr(W)-8 b(e)23 b(no)n(w)e(e)o(xploit)g(the)h(form)h(of)f (the)g(non-linear)g(term)h(in)f(Eq.\(1.17\),)g(namely)g(3)3074 3578 y Fz(1)p Fy(=)p Fz(2)3189 3621 y Fp(\001\()p Fq(w)3385 3578 y Fz(2)3425 3621 y Fp(\))16 b(+)i(\001\()p Fq(w)3771 3578 y Fz(3)3810 3621 y Fp(\))p Fr(,)94 3741 y(plus)24 b(the)h(f)o(act)g(that)g (the)f(eigenfunction)g Fq(')1564 3698 y Fn(\003)1564 3768 y Fz(0)1638 3741 y Fo(\021)k Fr(1.)36 b(Thus)24 b(if)h(we)g(inte)o(grate)f(by)g (parts,)h(we)g(\256nd)g(that)825 3884 y Fl(\012)872 3965 y Fq(')937 3922 y Fn(\003)937 3991 y Fz(0)983 3965 y Fo(j)p Fq(f)1071 3922 y Fh(\()p Fz(2)p Fh(\))1172 3884 y Fl(\000)1218 3965 y Fq(y)t(;)17 b(h)p Fp(\()p Fq(\021)t(;)g(y)t Fp(\))p Fq(;)g(\021)1700 3884 y Fl(\001)l(\013)1811 3965 y Fp(+)1911 3884 y Fl(\012)1958 3965 y Fq(')2023 3922 y Fn(\003)2023 3991 y Fz(0)2069 3965 y Fo(j)p Fq(f)2157 3922 y Fh(\()p Fz(3)p Fh(\))2258 3884 y Fl(\000)2304 3965 y Fq(y)t(;)g(h)p Fp(\()p Fq(\021)t(;)g(y)t Fp(\))p Fq(;)g(\021)2786 3884 y Fl(\001)l(\013)2920 3965 y Fp(=)44 b Fr(0)25 b Fq(;)94 4175 y Fr(so)i(that)f(in)g(Eq.\(4.1\),)44 b Fp(_)-46 b Fq(y)917 4202 y Fz(0)987 4175 y Fo(\021)30 b Fr(0)d(and)f(thus)g Fq(y)1583 4202 y Fz(0)1622 4175 y Fp(\()p Fq(t)p Fp(\))k(=)g Fq(y)1922 4202 y Fz(0)1962 4175 y Fp(\()p Fr(0)p Fp(\))p Fr(.)40 b(This)26 b(re\257ects)h(the)f(f)o(act)h(that)f(the)h(Cahn-Hilliard)94 4295 y(equation)e(conserv)o(es)f(mass.)294 4414 y(Since)40 b(from)g(Eq.\(4.1\))g(we)h(also)f(see)g(that)g Fq(y)1920 4441 y Fz(1)p Fy(;j)2069 4414 y Fp(=)50 b Fj(O)p Fp(\()p Fq(e)2351 4371 y Fn(\000)p Fy(\034)8 b(=)p Fz(4)2539 4414 y Fp(\))p Fr(,)44 b(we)d(\256nd)f(upon)f(re)n(v)o(erting)g(to)h(the)94 4534 y(unscaled)25 b(v)n(ariables)f(the)h(second)f(main)g(result:)94 4725 y Fm(Theorem)29 b(4.1.)k Fc(F)o(or)18 b Fq(d)27 b Fp(=)i Fr(2)p Fc(,)18 b(if)g(the)f(initial) f(conditions)g(of)i(the)f(Cahn-Hilliard)g(equation)g(are)h(suf)n(\256ciently) 94 4845 y(close)25 b(in)g Fq(H)510 4872 y Fz(1)p Fy(;)p Fz(4)633 4845 y Fc(to)f(the)h(stationary)f(state)g Fq(u)k Fo(\021)g Fr(3)1747 4802 y Fn(\000)p Fz(1)p Fy(=)p Fz(2)1925 4845 y Fc(,)c(then)h(the)f (solution)f(beha)n(v)o(es)i(asymptotically)d(as)1030 5129 y Fq(u)p Fp(\()p Fq(x;)17 b(t)p Fp(\))43 b(=)1538 5062 y Fr(1)p 1480 5106 166 4 v 1480 5199 a(3)1530 5170 y Fz(1)p Fy(=)p Fz(2)1680 5129 y Fp(+)1804 5062 y Fq(B)1885 5019 y Fn(\003)p 1791 5106 152 4 v 1791 5199 a Fq(t)1827 5170 y Fz(1)p Fy(=)p Fz(2)1955 5129 y Fq(f)2015 5086 y Fn(\003)2060 5048 y Fl(\000)2165 5062 y Fq(x)p 2117 5106 V 2117 5199 a(t)2153 5170 y Fz(1)p Fy(=)p Fz(4)2281 5048 y Fl(\001)2349 5129 y Fp(+)22 b Fj(O)2518 5048 y Fl(\000)2677 5062 y Fr(1)p 2577 5106 252 4 v 2577 5199 a Fq(t)2613 5170 y Fz(3)p Fy(=)p Fz(4)p Fn(\000)p Fy(")2840 5048 y Fl(\001)2910 5129 y Fq(:)p eop %%Page: 10 10 bop 94 -45 a Fu(Cahn-Hilliard)39 b(Equa)-7 b(tion)2451 b Ft(10)94 194 y Fs(5.)46 b(The)e(relev)-7 b(an)l(t)46 b(case)94 395 y Fr(Here,)41 b(we)c(consider)f(the)h(case)g(of)g Fq(d)44 b Fp(=)g Fr(1)37 b(where)g(one)g(term)f(of)h(the)g(non-linearity)e(is)h(rele)n(v)n (ant.)71 b(This)94 514 y(necessitates)29 b(a)g(change)h(of)f(strate)o(gy)-6 b(,)28 b(because)i(the)f(quadratic)g(term)f(is)h(proportional)f(to)g Fq(\021)3337 472 y Fn(\000)p Fz(1)3469 514 y Fr(and)h(hence)94 634 y(the)22 b(non-linear)g(terms)f(in)h(Eq.\(2.2\))f(are)i(not)e(smooth)f (enough)i(to)f(apply)g(the)h(in)l(v)n(ariant)f(manifold)g(theorem.)94 754 y(In)34 b(order)h(to)e(circumv)o(ent)g(this)f(dif)n(\256culty)-6 b(,)35 b(we)f(choose)g(a)g(scaling)f(dif)n(ferent)g(from)h(Eq.\(1.7\).)62 b(Consider)94 873 y(again)25 b(the)f(Cahn-Hilliard)h(equation,)f (Eq.\(1.17\),)g(with)g Fq(u)j Fp(=)i Fr(3)2321 830 y Fn(\000)p Fz(1)p Fy(=)p Fz(2)2520 873 y Fp(+)23 b Fq(w)s Fr(.)36 b(In)25 b Fq(d)i Fp(=)h Fr(1,)d(we)g(get)1055 1126 y Fq(@)6 b(w)p 1055 1170 133 4 v 1074 1261 a(@)g(t)1243 1193 y Fp(=)45 b Fo(\000)1483 1126 y Fq(@)1542 1083 y Fz(4)p 1455 1170 156 4 v 1455 1261 a Fq(@)6 b(x)1571 1233 y Fz(4)1622 1193 y Fq(w)25 b Fp(+)d Fr(3)1867 1150 y Fz(1)p Fy(=)p Fz(2)2023 1126 y Fq(@)2082 1083 y Fz(2)p 1995 1170 V 1995 1261 a Fq(@)6 b(x)2111 1233 y Fz(2)2162 1112 y Fl(\000)2208 1193 y Fq(w)2282 1150 y Fz(2)2321 1112 y Fl(\001)2389 1193 y Fp(+)2529 1126 y Fq(@)2588 1083 y Fz(2)p 2501 1170 V 2501 1261 a Fq(@)g(x)2617 1233 y Fz(2)2668 1112 y Fl(\000)2714 1193 y Fq(w)2788 1150 y Fz(3)2828 1112 y Fl(\001)2898 1193 y Fq(:)743 b Fp(\()p Fr(5)p Fq(:)p Fr(1)p Fp(\))94 1498 y Fr(No)n(w)25 b(let)f Fq(w)s Fp(\()p Fq(x;)17 b(t)p Fp(\))26 b(=)j Fq(t)893 1455 y Fn(\000)p Fz(1)p Fy(=)p Fz(2)1070 1498 y Fq(W)14 b Fp(\()p Fq(x=t)1360 1455 y Fz(1)p Fy(=)p Fz(4)1475 1498 y Fq(;)j Fr(log)f Fq(t)p Fp(\))p Fr(.)35 b(Then)25 b Fq(W)39 b Fr(satis\256es)579 1733 y Fq(@)6 b(W)p 579 1777 167 4 v 606 1868 a(@)g(\034)802 1800 y Fp(=)45 b Fo(\000)p Fq(@)1060 1757 y Fz(4)1054 1827 y Fy(\030)1099 1800 y Fq(W)36 b Fp(+)1341 1733 y Fr(1)p 1341 1777 50 4 v 1341 1868 a(4)1403 1800 y Fq(\030)26 b Fo(\001)c Fq(@)1576 1827 y Fy(\030)1619 1800 y Fq(W)37 b Fp(+)1861 1733 y Fr(1)p 1861 1777 V 1861 1868 a(2)1923 1800 y Fq(W)f Fp(+)23 b Fr(3)2203 1757 y Fz(1)p Fy(=)p Fz(2)2318 1800 y Fq(@)2377 1757 y Fz(2)2371 1827 y Fy(\030)2416 1719 y Fl(\000)2462 1800 y Fq(W)2570 1757 y Fz(2)2610 1719 y Fl(\001)2677 1800 y Fp(+)g Fq(e)2823 1757 y Fn(\000)p Fy(\034)8 b(=)p Fz(2)3011 1800 y Fq(@)3070 1757 y Fz(2)3064 1827 y Fy(\030)3109 1719 y Fl(\000)3155 1800 y Fq(W)3263 1757 y Fz(3)3303 1719 y Fl(\001)3373 1800 y Fq(:)268 b Fp(\()p Fr(5)p Fq(:)p Fr(2)p Fp(\))94 2099 y Fr(Proceeding)25 b(as)e(in)h(the)f(other)h(cases,)g(we)g(de\256ne)g(the)g (linear)g(operator)g Fq(L)2636 2126 y Fz(1)2703 2099 y Fp(=)k Fo(\000)p Fq(@)2944 2056 y Fz(4)2938 2126 y Fy(\030)3004 2099 y Fp(+)3113 2060 y Fz(1)p 3113 2076 35 4 v 3113 2134 a(4)3160 2099 y Fq(\030)5 b(@)3262 2126 y Fy(\030)3324 2099 y Fp(+)3433 2060 y Fz(1)p 3433 2076 V 3433 2134 a(2)3480 2099 y Fr(,)24 b(which)g(in)94 2219 y(F)o(ourier)h(v)n(ariables)f(becomes)1481 2350 y Fj(L)1552 2377 y Fz(1)1636 2350 y Fp(=)45 b Fo(\000)p Fq(p)1885 2307 y Fz(4)1947 2350 y Fo(\000)2059 2311 y Fz(1)p 2059 2327 V 2059 2384 a(4)2106 2350 y Fq(p@)2209 2377 y Fy(p)2277 2350 y Fp(+)2389 2311 y Fz(1)p 2389 2327 V 2389 2384 a(4)2460 2350 y Fq(;)94 2573 y Fr(so)28 b(that)g(it)g(has)g(eigen)l(v)n(alues)f Fq(\026)1186 2600 y Fy(m)1294 2573 y Fp(=)1416 2528 y Fz(1)p Fn(\000)p Fy(j)p 1416 2550 135 4 v 1465 2607 a Fz(4)1562 2573 y Fr(,)i Fq(j)38 b Fp(=)32 b Fr(0)p Fq(;)17 b Fr(1)p Fq(;)g(:)g(:)g(:)25 b Fr(.)46 b(Thus,)28 b(unlik)o(e)f(the)h(operator)h Fj(L)p Fr(,)g(we)f(ha)n(v)o(e)g(one)94 2712 y(eigen)l(v)n(alue)23 b(lying)g(in)g(the)g(right)g(half-plane.)35 b(Let)c Fp(~)-57 b Fq(\021)31 b Fp(=)e Fq(e)2083 2670 y Fn(\000)p Fy(\034)8 b(=)p Fz(8)2271 2712 y Fr(,)23 b(and)h(let)f Fq(y)2659 2739 y Fz(0)2722 2712 y Fr(and)h Fq(y)2939 2739 y Fz(1)3002 2712 y Fr(denote)f(the)h(amplitudes)94 2832 y(of)h(the)g(eigen)l(v)o(ectors)f (with)g(eigen)l(v)n(alues)g Fq(\026)1623 2859 y Fz(0)1688 2832 y Fr(and)g Fq(\026)1916 2859 y Fz(1)1956 2832 y Fr(.)36 b(Then)25 b(Eq.\(5.2\))f(tak)o(es)h(the)f(form)1306 3092 y Fp(_)-45 b Fq(y)1338 3119 y Fz(0)1422 3092 y Fp(=)1555 3053 y Fz(1)p 1555 3069 35 4 v 1555 3126 a(4)1602 3092 y Fq(y)1651 3119 y Fz(0)1713 3092 y Fp(+)23 b Fq(f)1862 3119 y Fz(0)1901 3092 y Fp(\()p Fq(y)1989 3119 y Fz(0)2029 3092 y Fq(;)17 b(y)2123 3119 y Fz(1)2162 3092 y Fq(;)g(\021)t(;)g(y)2358 3049 y Fn(?)2423 3092 y Fp(\))25 b Fq(;)1306 3258 y Fp(_)-45 b Fq(y)1338 3285 y Fz(1)1422 3258 y Fp(=)44 b Fq(f)1592 3285 y Fz(1)1632 3258 y 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y Fz(1)1201 3949 y Fr(,)g(and)d Fq(f)1501 3906 y Fn(?)1603 3949 y Fr(are)h(the)f(projections)f(of)i(the)f (non-linearity)f(onto)h(the)g(v)n(arious)94 4068 y(subspaces.)294 4188 y(Since)23 b(the)f(spectrum)g(of)h Fq(QL)1323 4215 y Fz(1)1362 4188 y Fq(Q)f Fr(lies)g(in)g(the)h(half-plane)f(Re)c Fq(\026)27 b Fo(\024)2627 4149 y Fz(1)p 2627 4165 V 2627 4222 a(4)2674 4188 y Fr(,)c(we)g(can)g(construct)f(an)g(in)l(v)n(ariant)94 4323 y(manifold)38 b(for)h(Eq.\(5.3\))g(which)f(is)g(the)h(graph)g(of)g(a)g (function)f Fq(h)2461 4281 y Fn(?)2528 4323 y Fp(\()p Fq(y)2616 4350 y Fz(0)2656 4323 y Fq(;)17 b(y)2750 4350 y Fz(1)2789 4323 y Fq(;)g(\021)t Fp(\))p Fr(,)41 b(and)e(e)n(v)o(ery)f(solution)f(of)94 4443 y(Eq.\(5.3\))32 b(which)g(remains)g(in)g(a)h(neighborhood)e(of)h(the)g (origin)g(will)f(approach)i(this)e(manifold)h(at)g(a)h(rate)94 4563 y Fj(O)p Fp(\()p Fq(e)249 4520 y Fn(\000)p Fy(\034)8 b(=)p Fz(4)438 4563 y Fp(\))p Fr(.)47 b(What)29 b(is)f(more,)h(the)g(equations)e 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a(8)2622 5291 y Fp(~)-57 b Fq(\021)29 b(:)p eop %%Page: 11 11 bop 94 -45 a Fu(Cahn-Hilliard)39 b(Equa)-7 b(tion)2451 b Ft(11)94 194 y Fr(Thus,)28 b FA(as)f(long)f(as)h(the)h(solution)d(of)i(Eq.\(5.3\))g(r) l(emains)g(in)g(a)g(neighborhood)e(of)i(the)h(origin)p Fr(,)e(it)h(will)g(be) g(of)94 314 y(the)e(form)1573 420 y Fq(y)1622 446 y Fz(0)1661 420 y Fp(\()p Fq(\034)11 b Fp(\))44 b(=)h Fq(e)2006 377 y Fy(\034)8 b(=)p Fz(4)2132 420 y Fq(y)2181 446 y Fz(0)2220 420 y Fp(\()p Fr(0)p Fp(\))25 b Fq(;)1573 569 y(y)1622 596 y Fz(1)1661 569 y Fp(\()p Fq(\034)11 b Fp(\))44 b(=)h Fq(y)2009 596 y Fz(1)2048 569 y Fp(\()p Fr(0)p Fp(\))25 b Fq(;)1615 739 y Fp(~)-57 b Fq(\021)t Fp(\()p Fq(\034)11 b Fp(\))44 b(=)h Fq(e)2006 696 y Fn(\000)p Fy(\034)8 b(=)p Fz(8)2201 739 y Fp(~)-57 b Fq(\021)t Fp(\()p Fr(0)p Fp(\))24 b Fq(;)1542 909 y(y)1595 866 y Fn(?)1661 909 y Fp(\()p Fq(\034)11 b Fp(\))44 b(=)h Fj(O)p Fp(\()p Fq(e)2115 866 y Fn(\000)p Fy(\034)8 b(=)p Fz(4)2303 909 y Fp(\))25 b Fq(:)3669 659 y Fp(\()p Fr(5)p Fq(:)p Fr(4)p Fp(\))94 1085 y Fr(Thus,)34 b(we)f(see)f(that)g(the)g(solution)f(either)h(lea)n(v)o(es)g (the)g(neighborhood)f(of)h(the)h(origin,)g(or)f(its)g(asymptotic)94 1204 y(beha)n(vior)24 b(can)g(be)g(read)h(of)n(f)e(from)h(Eq.\(5.4\).)35 b(Note)23 b(that)h(the)f(solutions)f(that)i(remain)f(near)i(the)e(origin)g (must)94 1324 y(ha)n(v)o(e)i Fq(y)353 1351 y Fz(0)420 1324 y Fp(=)k Fr(0.)35 b(Thus:)94 1515 y Fm(Theorem)j(5.1.)e Fc(Suppose)24 b(that)g(the)g(initial)g(condition)f(of)h(the)h(Cahn-Hilliard)f(equation)f (is)h(of)h(the)f(form)94 1635 y Fq(u)151 1662 y Fz(0)237 1635 y Fp(=)46 b Fr(3)410 1592 y Fn(\000)p Fz(1)p Fy(=)p Fz(2)617 1635 y Fp(+)30 b Fq(w)795 1662 y Fz(0)873 1635 y Fc(with)37 b Fq(w)1159 1662 y Fz(0)1236 1635 y Fc(small)g(in)g(the)h Fq(H)1848 1662 y Fy(m;`)2018 1635 y Fc(norm)f(for)h(some)f Fq(`)45 b Fo(\025)h Fr(4)38 b Fc(and)g Fq(m)46 b Fo(\025)g Fr(2)p Fc(.)74 b(Assume)94 1771 y(furthermore)25 b(that)770 1691 y Fl(R)836 1715 y Fn(1)817 1806 y(\0001)980 1771 y Fr(d)p Fq(x)17 b(w)1175 1798 y Fz(0)1214 1771 y Fp(\()p Fq(x)p Fp(\))28 b(=)g Fr(0)p Fc(.)35 b(Then)25 b(the)g(solution)e(is)h(of)h(the)g(form)1002 2074 y Fq(u)p Fp(\()p Fq(x;)17 b(t)p Fp(\))43 b(=)1510 2006 y Fr(1)p 1452 2051 166 4 v 1452 2144 a(3)1502 2115 y Fz(1)p Fy(=)p Fz(2)1651 2074 y Fp(+)1763 2006 y Fq(B)1844 1964 y Fn(\003\003)p 1763 2051 168 4 v 1771 2144 a Fq(t)1807 2115 y Fz(1)p Fy(=)p Fz(2)1942 2074 y Fq(f)2002 2031 y Fn(\003\003)2088 1993 y Fl(\000)2193 2006 y Fq(x)p 2146 2051 152 4 v 2146 2144 a(t)2182 2115 y Fz(1)p Fy(=)p Fz(4)2309 1993 y Fl(\001)2377 2074 y Fp(+)23 b Fj(O)2547 1993 y Fl(\000)2706 2006 y Fr(1)p 2605 2051 252 4 v 2605 2144 a Fq(t)2641 2115 y Fz(3)p Fy(=)p Fz(4)p Fn(\000)p Fy(")2868 1993 y Fl(\001)2939 2074 y Fq(;)94 2329 y Fc(where)709 2481 y Fq(B)790 2438 y Fn(\003\003)921 2481 y Fp(=)1043 2345 y Fl(Z)1142 2370 y Fn(1)1098 2572 y(\0001)1261 2481 y Fr(d)p Fq(x)17 b(xw)1513 2508 y Fz(0)1553 2481 y Fp(\()p Fq(x)p Fp(\))24 b Fq(;)116 b(f)1916 2438 y Fn(\003\003)2002 2481 y Fp(\()p Fq(\030)5 b Fp(\))43 b(=)2377 2413 y Fr(1)p 2306 2458 194 4 v 2306 2478 a Fo(p)p 2389 2478 111 4 v 84 x Fr(2)p Fq(\031)2527 2345 y Fl(Z)2627 2370 y Fn(1)2583 2572 y(\0001)2746 2481 y Fr(d)p Fq(p)17 b(e)2909 2438 y Fy(ip\030)3022 2481 y Fq(e)3068 2438 y Fn(\000)p Fy(p)3171 2408 y Fg(4)3231 2481 y Fq(:)94 2786 y Fm(Remark.)55 b Fr(The)43 b(constant)e Fq(B)1231 2743 y Fn(\003)1319 2786 y Fr(in)g(Theorem)h(1.1)g(is)g(not)f(as)i(easy)f(to)g(describe)g (because)h(there,)j(the)94 2906 y(non-linearity)24 b(in)g(the)h(equation)f (for)h Fq(y)1441 2932 y Fz(0)1506 2906 y Fr(did)f FA(not)g Fr(necessarily)h(disappear)-5 b(.)94 3082 y Fm(Pro)s(of.)45 b Fr(The)30 b(proof)g(is)f(an)h(ob)o(vious)e(modi\256cation)h(of)h(the)g(one) g(of)g(Theorem)g(1.1,)h(taking)e(into)g(account)94 3202 y(the)c(special)g (form)f(of)h(the)g(eigenfunctions)e(corresponding)h(to)h(the)f(eigen)l(v)n (alues)g Fq(\026)3020 3228 y Fz(0)3085 3202 y Fr(and)h Fq(\026)3314 3228 y Fz(1)3353 3202 y Fr(.)94 3552 y Fs(App)t(endix.)60 b(Bounds)43 b(on)i(the)g(linear)h(semi-group)94 3741 y Fr(In)39 b(this)e(appendix,)j(we)f (sk)o(etch)e(the)h(proof)g(of)g(Proposition)f(2.1.)75 b(The)38 b(proof)g(is)f(quite)h(similar)f(to)g(the)94 3860 y(estimates)25 b(on)g(the)g(linear)g(semi-group)f(in)h(Appendix)f(B)i(of)g([EWW],)f(\(which) g(w)o(as)h(gi)n(v)o(en)d(for)j(the)f(case)h(of)94 3980 y(a)h(one-dimensional) e(Laplacian,)i(or)f(in)h(the)f(present)g(notation)f Fq(n)30 b Fp(=)h Fq(d)f Fp(=)g Fr(1\))d(so)f(we)h(concentrate)g(only)e(on)94 4100 y(the)g(points)f(where)h(the)g(present)f(ar)n(gument)h(dif)n(fers)f (from)h(the)f(one)h(in)f([EWW].)294 4219 y(W)-8 b(e)25 b(be)o(gin)f(with)g (the)g(representation)1012 4448 y Fl(\000)1058 4529 y Fq(e)1104 4486 y Fy(\034)8 b(L)1208 4448 y Fl(\001)1254 4529 y Fp(\()p Fq(x)p Fp(\))44 b(=)1569 4461 y Fq(e)1627 4392 y Ff(\034)6 b(d)p 1627 4403 77 4 v 1631 4443 a Fg(2)p Ff(n)p 1566 4506 157 4 v 1566 4597 a Fr(2)p Fq(\031)1677 4568 y Fy(d)1752 4393 y 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y(the)k(part)g(f)o(ar)g(from)g(the)f(origin)g(and)g(that)h (close)f(to)g(the)h(origin.)34 b(The)23 b(ne)n(w)f(dif)n(\256culty)g(here)h (is)f(that)g(we)h(do)g(not)p eop %%Page: 12 12 bop 94 -45 a Fu(Cahn-Hilliard)39 b(Equa)-7 b(tion)2451 b Ft(12)94 194 y Fr(ha)n(v)o(e)26 b(an)f(e)o(xplicit)f(representation)h(of)h Fq(g)i Fr(as)e(in)f(the)g(case)h Fq(n)i Fp(=)h Fr(1.)38 b(Ho)n(we)n(v)o(er)l (,)24 b(the)h(technique)g(of)h(estimating)94 314 y(the)36 b(long-time)e(beha) n(vior)h(will)g(remain)g(essentially)g(the)g(same.)68 b(Let)36 b Fq(\037)2703 341 y Fy(R)2804 314 y Fr(be)g(a)g(smooth)e(characteristic)94 433 y(function)23 b(which)h(v)n(anishes)e(for)i Fo(j)p Fq(x)p Fo(j)j Fq(<)h(R)d Fr(and)e(is)h(equal)f(to)h(1)g(for)g Fo(j)p Fq(x)p Fo(j)j Fq(>)h Fr(4)p Fq(R)q(=)p Fr(3.)34 b(W)-8 b(e)24 b(start)g(by)f(studying)f(the)94 553 y(re)o(gion)i(f)o(ar)i(from)f(the)f (origin.)35 b(The)25 b(analog)f(of)h(Proposition)f(B.2)h(of)g([EWW])g(is)94 744 y Fm(Prop)s(osition)35 b(A.1.)g Fc(F)o(or)22 b(e)n(v)o(ery)g Fq(`)27 b Fo(\025)h Fr(1)22 b Fc(and)g(e)n(v)o(ery)g Fq(m)28 b Fo(\025)g Fr(0)p Fc(,)22 b(there)h(e)o(xist)e(a)h Fq(\015)33 b(>)28 b Fr(0)22 b Fc(and)g(a)g Fq(C)7 b Fp(\()p Fq(`;)17 b(m)p Fp(\))27 b Fq(<)h Fo(1)94 864 y Fc(such)d(that)f(for)h(all)g Fq(v)31 b Fo(2)e Fq(H)999 891 y Fy(`;m)1156 864 y Fc(one)c(has)317 1088 y Fl(\015)317 1148 y(\015)372 1173 y Fq(\037)434 1200 y Fy(R)500 1173 y Fq(e)546 1130 y Fy(\034)8 b(L)651 1173 y Fq(v)703 1088 y Fl(\015)703 1148 y(\015)758 1213 y Fy(`;m)935 1173 y Fo(\024)1091 1105 y Fq(C)f Fp(\()p Fq(`;)17 b(m)p Fp(\))p 1069 1150 374 4 v 1069 1196 a Fl(\000)1114 1276 y Fq(a)p Fp(\()p Fq(\034)11 b Fp(\))1300 1196 y Fl(\001)1374 1183 y Ff(q)p 1357 1201 69 4 v 1357 1240 a Fg(2)p Ff(n)1455 1173 y Fq(e)1501 1130 y Fh(\()1559 1103 y Ff(\034)p 1544 1114 V 1544 1154 a Fg(2)p Ff(n)1625 1130 y Fh(\)\()p Fy(d)p Fh(+)p Fy(`)p Fh(\))1876 1062 y Fl(\020)1935 1173 y Fq(e)1981 1130 y Fn(\000)p Fy(\034)d(m=)p Fz(2)2262 1173 y Fp(+)23 b Fq(e)2408 1130 y 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b(A.2.)c Fc(The)25 b(k)o(ernel)g Fq(g)t Fp(\()p Fq(z)t(;)17 b(\034)11 b Fp(\))23 b Fc(decays)i(f)o(aster)g (than)g(an)o(y)f(in)l(v)o(erse)g(po)n(wer)g(of)h Fq(z)30 b Fc(for)25 b Fo(j)p Fq(z)t Fo(j)g Fc(lar)n(ge.)94 2253 y(In)g(f)o(act,)h(one)e (has)h(the)g(estimate)962 2489 y Fo(j)p Fq(g)t Fp(\()p Fq(z)t(;)17 b(\034)11 b Fp(\))p Fo(j)42 b(\024)j Fq(C)7 b(a)p Fp(\()p Fq(\034)k Fp(\))1726 2446 y Fn(\000)1815 2419 y Ff(d)p 1799 2430 V 1799 2470 a Fg(2)p Ff(n)1902 2489 y Fr(e)o(xp)2044 2408 y Fl(\000)2090 2489 y Fo(\000)p Fq(\015)6 b Fp(\()p Fo(j)p Fq(z)t Fo(j)2370 2446 y Fz(2)p Fy(n)2458 2489 y Fq(=a)p Fp(\()p Fq(\034)11 b Fp(\)\))2805 2419 y Fg(1)p 2744 2430 148 4 v 2744 2469 a(2)p Ff(n)p Fd(\000)p Fg(1)2908 2408 y Fl(\001)2979 2489 y Fq(;)640 b Fp(\()p Fr(A)p Fq(:)p Fr(4)p Fp(\))94 2724 y Fc(for)26 b(some)e Fq(\015)33 b Fp(=)28 b Fq(\015)6 b Fp(\()p Fq(n;)17 b(d)p Fp(\))25 b Fq(>)j Fr(0)p Fc(.)94 2962 y Fm(Remark.)39 b Fr(If)25 b Fq(n)i Fp(=)i Fr(1,)24 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Fz(2)p Fy(n)p Fn(\000)p Fz(1)3159 4604 y Fl(\001)3229 4685 y Fq(:)94 4964 y Fr(If)26 b(we)f(rescale)g(the)g(v)n(ariables)f(as)h Fq(p)j Fp(=)g Fq(z)t(t)p Fr(,)d(and)g Fq(q)32 b Fp(=)c Fq(z)t(s)p Fr(,)d(then)g(we)g(ha)n(v)o(e)773 5245 y Fq(X)52 b Fp(=)45 b Fq(z)1080 5203 y Fy(d)1144 5110 y Fl(Z)1260 5245 y Fr(d)p Fq(t)17 b Fr(d)1413 5203 y Fy(d)p Fn(\000)p Fz(1)1556 5245 y Fq(s)g Fr(e)o(xp)1761 5165 y Fl(\000)1807 5245 y Fo(\000)p Fq(a)p Fp(\()p Fq(\034)11 b Fp(\))p Fq(z)2120 5203 y Fz(2)p Fy(n)2209 5165 y Fl(\000)2255 5245 y Fp(\()p Fq(t)2330 5203 y Fz(2)2392 5245 y Fp(+)22 b Fq(s)g Fo(\001)g Fq(s)p Fp(\))2696 5203 y Fy(n)2772 5245 y Fp(+)g Fr(2)p Fq(int)3051 5165 y Fl(\001\001)3167 5245 y Fq(:)p eop %%Page: 13 13 bop 94 -45 a Fu(Cahn-Hilliard)39 b(Equa)-7 b(tion)2451 b Ft(13)94 194 y Fm(Remark.)39 b Fr(Note)24 b(that)h(the)f(polynomial)f Fp(\()p Fq(t)1655 151 y Fz(2)1717 194 y Fp(+)f Fq(s)g Fo(\001)g Fq(s)p Fp(\))2021 151 y Fy(n)2097 194 y Fp(+)h Fr(2)p Fq(int)h Fr(is)h(independent)f(of)h Fq(z)t Fr(.)294 314 y(W)-8 b(e)30 b(will)g(bound)f Fq(X)38 b Fr(by)30 b(taking)g(adv)n(antage)g(of)g(the)g(f)o (act)h(that)f(the)g(inte)o(grand)f(is)h(an)h(entire)f(function)94 433 y(and)25 b(translate)f(the)g(contour)g(of)g(inte)o(gration)f(so)h(that)g (it)g(passes)g(through)f(at)i(least)f(one)g(critical)g(point)g(of)g(the)94 553 y(e)o(xponent.)52 b(These)30 b(critical)g(points)f(occur)i(at)f Fq(s)35 b Fp(=)h Fr(0)30 b(and)g(the)h(roots)e(of)i Fq(t)2720 510 y Fz(2)p Fy(n)p Fn(\000)p Fz(1)2941 553 y Fp(=)36 b Fq(i)31 b Fr(\261)f FA(i.e)o(.)p Fr(,)h(at)f(the)g(points)94 685 y Fq(t)130 712 y Fy(k)207 685 y Fp(=)e Fr(e)o(xp)p Fp(\()p Fq(i)540 637 y Fy(\031)s Fh(\()p Fz(4)p Fy(k)r Fh(+)p Fz(1)p Fh(\))p 540 662 287 4 v 544 719 a Fz(2)p Fh(\()p Fz(2)p Fy(n)p Fn(\000)p Fz(1)p Fh(\))838 685 y Fp(\))p Fr(,)d Fq(k)30 b Fp(=)e Fr(0)p Fq(;)17 b Fr(1)p Fq(;)g Fr(2)p Fq(;)g(:)g(:)g(:)12 b(;)17 b Fr(2)p Fq(n)k Fo(\000)i Fr(2.)294 815 y(Inserting)j(this)f(e)o(xpression)h (into)f(the)i(e)o(xponent)e(of)i(the)g(inte)o(grand)e(of)i Fq(X)8 b Fr(,)26 b(we)h(see)g(that)f(the)h(v)n(alue)f(of)94 934 y(the)f(polynomial)e(at)i(the)f(critical)h(points)e(is)813 1221 y Fo(\000)p Fq(a)p Fp(\()p Fq(\034)11 b Fp(\))p Fq(z)1126 1178 y Fz(2)p Fy(n)1232 1081 y Fl(\022)1305 1221 y Fr(e)o(xp)p Fp(\()p Fq(i)1533 1154 y Fr(2)p Fq(n\031)t Fp(\()p Fr(4)p Fq(k)23 b Fp(+)g Fr(1)p Fp(\))p 1533 1198 524 4 v 1590 1290 a Fr(2)p Fp(\()p Fr(2)p Fq(n)f Fo(\000)g Fr(1)p Fp(\))2069 1221 y(\))f Fo(\000)i Fr(2)p Fq(ni)17 b Fr(e)o(xp)o Fp(\()p Fq(i)2617 1154 y(\031)t Fp(\()p Fr(4)p Fq(k)24 b Fp(+)f Fr(1)p Fp(\))p 2617 1198 415 4 v 2620 1290 a Fr(2)p Fp(\()p Fr(2)p Fq(n)e Fo(\000)i Fr(1)p Fp(\))3043 1221 y(\))3082 1081 y Fl(\023)1103 1498 y Fp(=)45 b(\()p Fr(2)p Fq(n)22 b Fo(\000)g Fr(1)p Fp(\))p Fq(a)p Fp(\()p Fq(\034)11 b Fp(\))p Fq(z)1820 1455 y Fz(2)p Fy(n)1925 1498 y Fr(e)o(xp)p Fp(\()p Fq(i)2153 1430 y(\031)t Fp(\()p Fr(4)p Fq(k)24 b Fp(+)e Fr(2)p Fq(n)p Fp(\))p 2153 1475 475 4 v 2185 1566 a Fr(2)p Fp(\()p Fr(2)p Fq(n)g Fo(\000)g Fr(1)p Fp(\))2639 1498 y(\))i Fq(:)94 1781 y Fr(In)h(particular)l(,)g(if)g(we)g(tak) o(e)g Fq(k)30 b Fp(=)f Fr(0,)24 b(then)h(the)f(real)i(part)e(of)h(the)g (critical)f(v)n(alue)h(is)758 2034 y Fp(\()p Fr(2)p Fq(n)d Fo(\000)g Fr(1)p Fp(\))p Fq(a)p Fp(\()p Fq(\034)11 b Fp(\))p Fq(z)1353 1991 y Fz(2)p Fy(n)1458 2034 y Fr(cos)1591 1953 y Fl(\000)1637 2034 y Fq(\031)t(=)p Fr(2)21 b Fp(+)i Fq(\031)t(=)p Fp(\()p Fr(4)p Fq(n)d Fo(\000)j Fr(2)p Fp(\))2388 1953 y Fl(\001)2477 2034 y Fo(\031)45 b(\000)p Fq(a)p Fp(\()p Fq(\034)11 b Fp(\))p Fq(z)2912 1991 y Fz(2)p Fy(n)3024 2034 y Fo(\001)3085 1966 y Fq(\031)p 3085 2011 61 4 v 3090 2102 a Fr(2)3183 2034 y Fq(;)94 2291 y Fr(when)22 b Fq(n)f Fr(is)g(lar)n(ge)h(\(and)g(is)f(ne)o(gati)n(v)o(e) e(for)j(all)f Fq(n)27 b(>)i Fr(0\).)34 b(Inte)o(grating)21 b(o)o(v)o(er)f(the)i(re)o(gion)e Fm(R)15 b Fp(+)g Fq(t)3254 2318 y Fz(0)3316 2291 y Fr(and)21 b(observing)94 2411 y(that)k(there)g(is)f 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Fr(0)d(for)g(all)f Fq(x)p Fr(\).)94 3904 y Fm(Lemma)41 b(A.3.)i Fc(The)27 b(operator)g Fq(H)39 b Fp(=)32 b Fq(T)1598 3861 y Fn(\000)p Fz(1)1700 3904 y Fq(LT)41 b Fc(is)27 b(self-adjoint)f(on)h (\(a)h(dense)f(domain)f(in\))h Fr(L)3453 3861 y Fz(2)3493 3904 y Fp(\()p Fm(R)3618 3861 y Fy(d)3664 3904 y Fp(\))g Fc(and)94 4024 y(has)e(the)g(same)f(eigen)l(v)n(alues)g(as)h Fq(L)p Fc(.)94 4259 y Fm(Pro)s(of.)46 b Fr(The)32 b(proof)g(is)f(a)h(straightforw)o(ard)f (calculation.)56 b(W)-8 b(e)32 b(note)f(further)h(that)g(if)f Fq(')3229 4286 y Fy(\013)3318 4259 y Fr(are)h(the)g(eigen-)94 4379 y(functions)24 b(of)h Fq(L)g Fr(then)f(the)h(eigenfunctions)e(of)i Fq(H)33 b Fr(are)26 b Fq( )2081 4406 y Fy(\013)2137 4379 y Fp(\()p Fq(x)p Fp(\))h(=)2405 4298 y Fl(\000)2450 4379 y Fq(T)2522 4336 y Fn(\000)p Fz(1)2624 4379 y Fq(')2689 4406 y Fy(\013)2746 4298 y Fl(\001)2792 4379 y Fp(\()p Fq(x)p Fp(\))p Fr(.)294 4498 y(If)g(we)h(tak)o(e)f(the)g(in)l(v)o(erse)f(F)o(ourier)h(transform)g(of) g(the)g(eigenfunctions)f Fq(')2853 4525 y Fy(\013)2910 4498 y Fp(\()p Fq(p)p Fp(\))g Fr(of)h(Eq.\(1.12\),)g(we)h(see)94 4618 y(that)1242 4737 y Fo(j)16 b Fp(~)-66 b Fq(')1335 4764 y Fy(\013)1391 4737 y Fp(\()p Fq(x)p Fp(\))p Fo(j)44 b(\031)h Fq(C)7 b Fo(j)p Fq(x)p Fo(j)1911 4694 y Fn(j)p Fy(\013)p Fn(j)2031 4737 y Fr(e)o(xp)o Fp(\()p Fo(\000)p Fq(\015)f Fo(j)p Fq(x)p Fo(j)2510 4667 y Fg(2)p Ff(n)p 2471 4678 148 4 v 2471 4718 a Fg(2)p Ff(n)p Fd(\000)p Fg(1)2635 4737 y Fp(\))25 b Fq(;)94 4923 y Fr(for)33 b(some)g Fq(\015)43 b(>)d Fr(0,)34 b(using)e(the)g(same)h (sort)f(of)h(estimates)f(as)g(those)h(used)f(to)g(bound)g(the)h(k)o(ernel)g Fq(g)i Fr(of)e(the)94 5042 y(semi-group.)i(Thus,)24 b(for)h Fo(j)p Fq(x)p Fo(j)f Fr(suf)n(\256ciently)g(lar)n(ge,)h(we)g(get)1213 5291 y Fo(j)1261 5265 y Fp(~)1241 5291 y Fq( )1306 5318 y Fy(\013)1362 5291 y Fp(\()p Fq(x)p Fp(\))p Fo(j)43 b(\031)i Fq(C)7 b Fo(j)p Fq(x)p Fo(j)1881 5248 y Fn(j)p Fy(\013)p Fn(j)2001 5291 y Fr(e)o(xp)p Fp(\()p Fo(\000)2272 5252 y Fz(1)p 2272 5268 35 4 v 2272 5326 a(2)2319 5291 y Fq(\015)f Fo(j)p Fq(x)p Fo(j)2539 5222 y Fg(2)p Ff(n)p 2500 5233 148 4 v 2500 5272 a Fg(2)p Ff(n)p Fd(\000)p Fg(1)2665 5291 y Fp(\))24 b Fq(:)p eop %%Page: 14 14 bop 94 -45 a Fu(Cahn-Hilliard)39 b(Equa)-7 b(tion)2451 b Ft(14)294 194 y Fr(The)29 b(usefulness)g(of)g(introducing)f(the)h(operator)g Fq(H)38 b Fr(is)29 b(that)g(it)f(is)h(sectorial,)h(since)f(it)g(is)g (self-adjoint)94 314 y(and)42 b(bounded)g(belo)n(w)-6 b(.)86 b(Therefore,)47 b(the)41 b(associated)h(semi-group)f(can)h(be)g(estimated)f (from)h(spectral)94 433 y(information)20 b(alone.)34 b(In)21 b(particular)l(,)g(if)f Fq(P)1534 460 y Fy(k)1604 433 y Fr(denotes)g(the)h (projection)e(onto)h(the)h(spectral)f(subspace)h(spanned)94 553 y(by)g(the)h(eigenfunctions)d(with)i(eigen)l(v)n(alues)f(0,)i Fo(\000)1861 514 y Fz(1)p 1836 530 85 4 v 1836 587 a(2)p Fy(n)1932 553 y Fr(,)1991 514 y Fn(\000)p Fz(2)p 1991 530 98 4 v 1998 587 a(2)p Fy(n)2100 553 y Fr(,)p Fq(:)17 b(:)g(:)e 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1188 y Fh(\()p Fz(0)p Fh(\))678 1270 y Fy(k)806 1240 y Fr(the)g(projection)f (associated)g(with)g(the)h(spectral)g(subspaces)f(of)h Fq(L)g Fr(\(as)g(we)g(did)f(for)h Fq(H)8 b Fr(\),)94 1360 y(the)25 b(we)g(ha)n(v)o(e)g(the)f(identity:)986 1588 y Fq(e)1032 1546 y Fy(\034)8 b(L)1137 1588 y Fq(Q)1216 1537 y Fh(\()p Fz(0)p Fh(\))1216 1618 y Fy(k)1318 1588 y Fq(v)48 b Fp(=)d Fq(e)1582 1546 y Fy(\034)8 b(L)1687 1588 y Fq(Q)1766 1537 y Fh(\()p Fz(0)p Fh(\))1766 1618 y Fy(k)1868 1588 y Fp(\()p Fr(1)21 b Fo(\000)i Fq(\037)2140 1615 y Fy(R)2206 1588 y Fp(\))p Fq(v)j Fp(+)c Fq(e)2464 1546 y Fy(\034)8 b(L)2569 1588 y Fq(Q)2648 1537 y Fh(\()p Fz(0)p Fh(\))2648 1618 y Fy(k)2750 1588 y Fq(\037)2812 1615 y Fy(R)2878 1588 y Fq(v)28 b(:)94 1801 y Fr(Since)d Fq(\037)402 1828 y Fy(R)467 1801 y Fq(v)j Fr(is)c(localized)f(a)o(w)o(ay)h(from)f(the)h (origin,)f(it)g(can)h(be)g(studied)f(with)g(the)h(help)f(of)h(Proposition)e (A.1,)94 1921 y(so)j(we)g(only)f(focus)h(on)f(the)h(other)g(term.)35 b(And)25 b(there)g(we)g(get)565 2150 y Fo(k)p Fq(e)661 2107 y Fy(\034)8 b(L)766 2150 y Fq(Q)845 2098 y Fh(\()p Fz(0)p Fh(\))845 2180 y Fy(k)947 2150 y Fp(\()p Fr(1)21 b Fo(\000)i Fq(\037)1219 2177 y Fy(R)1285 2150 y Fp(\))p Fq(v)t Fo(k)1426 2177 y Fy(q)s(;r)1576 2150 y Fp(=)45 b Fo(k)p Fq(T)14 b(T)1892 2107 y Fn(\000)p Fz(1)1994 2150 y Fq(e)2040 2107 y Fy(\034)8 b(L)2145 2150 y Fq(T)14 b(T)2289 2107 y Fn(\000)p Fz(1)2391 2150 y Fq(Q)2470 2098 y Fh(\()p Fz(0)p Fh(\))2470 2180 y Fy(k)2572 2150 y Fq(T)g(T)2716 2107 y Fn(\000)p Fz(1)2818 2150 y Fp(\()p Fr(1)22 b Fo(\000)g Fq(\037)3090 2177 y Fy(R)3156 2150 y Fp(\))p Fq(v)t Fo(k)3297 2177 y Fy(q)s(;r)1576 2320 y Fp(=)1698 2236 y Fl(\015)1698 2296 y(\015)1753 2320 y Fq(T)1825 2240 y Fl(\000)1871 2320 y Fq(e)1917 2278 y Fy(\034)8 b(H)2038 2320 y Fq(Q)2117 2347 y Fy(k)2166 2240 y Fl(\001)o(\000)2257 2320 y Fq(T)2329 2278 y Fn(\000)p Fz(1)2431 2320 y Fp(\()p Fr(1)22 b Fo(\000)g Fq(\037)2703 2347 y Fy(R)2769 2320 y Fp(\))p Fq(v)2860 2240 y Fl(\001)2905 2236 y(\015)2905 2296 y(\015)2961 2360 y Fy(q)s(;r)1576 2550 y Fo(\024)45 b Fq(C)24 b Fr(e)o(xp)1935 2470 y Fl(\000)1981 2550 y Fo(\000)p Fq(\034)2125 2483 y(k)h Fp(+)e Fr(1)p 2125 2527 227 4 v 2184 2619 a(2)p Fq(n)2364 2470 y Fl(\001)2409 2466 y(\015)2409 2525 y(\015)2465 2550 y Fq(T)2537 2507 y Fn(\000)p Fz(1)2639 2550 y Fp(\()p Fr(1)f Fo(\000)g Fq(\037)2911 2577 y Fy(R)2977 2550 y Fp(\))p Fq(v)3068 2466 y Fl(\015)3068 2525 y(\015)3123 2590 y Fy(q)s(;r)3255 2550 y Fq(:)94 2825 y Fr(Using)i(no)n(w)g(the)g(information)f(that)h Fo(j)p Fq(T)1466 2782 y Fn(\000)p Fz(1)1567 2825 y Fp(\()p Fq(x)p Fp(\))p Fo(j)j(\024)h Fq(C)c Fr(e)o(xp)p Fp(\()p Fq(\015)6 b Fo(j)p Fq(x)p Fo(j)2359 2755 y Fg(2)p Ff(n)p 2320 2766 148 4 v 2320 2806 a Fg(2)p Ff(n)p Fd(\000)p Fg(1)2484 2825 y Fp(\))25 b Fr(and)f(that)g Fp(\()p Fr(1)d Fo(\000)g Fq(\037)3160 2852 y Fy(R)3226 2825 y Fq(v)t Fp(\)\()p Fq(x)p Fp(\))27 b(=)h Fr(0)c(when)94 2945 y Fo(j)p Fq(x)p Fo(j)j Fq(>)i Fr(4)p Fq(R)q(=)p Fr(3,)23 b(we)i(get)903 3091 y Fl(\015)903 3151 y(\015)958 3176 y Fq(T)1030 3133 y Fn(\000)p Fz(1)1132 3176 y Fp(\()p Fr(1)d Fo(\000)h Fq(\037)1405 3203 y Fy(R)1470 3176 y Fp(\))p Fq(v)1561 3091 y Fl(\015)1561 3151 y(\015)1616 3216 y Fy(q)s(;r)1767 3176 y Fo(\024)45 b Fq(C)24 b Fr(e)o(xp)2126 3095 y Fl(\000)2172 3176 y Fq(\015)6 b Fp(\()p Fr(4)p Fq(R)q(=)p Fr(3)p Fp(\))2584 3106 y Fg(2)p Ff(n)p 2545 3117 V 2545 3157 a Fg(2)p Ff(n)p Fd(\000)p Fg(1)2709 3095 y Fl(\001)2755 3176 y Fo(k)p Fq(v)t Fo(k)2907 3203 y Fy(q)s(;r)3038 3176 y Fq(;)94 3400 y Fr(so)25 b(that)f(\256nally)482 3546 y Fl(\015)482 3606 y(\015)482 3666 y(\015)537 3661 y Fq(e)583 3618 y Fy(\034)8 b(L)688 3661 y Fq(Q)767 3609 y Fh(\()p Fz(0)p Fh(\))767 3691 y Fy(k)869 3661 y Fp(\()p Fr(1)21 b Fo(\000)i Fq(\037)1141 3688 y Fy(R)1207 3661 y Fp(\))p Fq(v)1298 3546 y Fl(\015)1298 3606 y(\015)1298 3666 y(\015)1353 3731 y Fy(`;m)1530 3661 y Fo(\024)44 b Fq(C)24 b Fr(e)o(xp)1889 3580 y Fl(\000)1934 3661 y Fq(\015)6 b Fp(\()p Fr(4)p Fq(R)q(=)p Fr(3)p Fp(\))2346 3591 y Fg(2)p Ff(n)p 2307 3602 V 2307 3642 a Fg(2)p Ff(n)p Fd(\000)p Fg(1)2471 3580 y Fl(\001)2534 3661 y Fr(e)o(xp)2676 3580 y Fl(\000)2722 3661 y Fo(\000)p Fq(\034)2866 3594 y(k)25 b Fp(+)e Fr(1)p 2866 3638 227 4 v 2924 3729 a(2)p Fq(n)3104 3580 y Fl(\001)3150 3661 y Fo(k)p Fq(v)t Fo(k)3302 3688 y Fy(`;m)3459 3661 y Fq(:)294 3914 y Fr(Thus)h(we)h(ha)n(v)o(e)f(pro)o(v)o(en:)94 4106 y Fm(Prop)s(osition)34 b(A.4.)g Fc(Under)21 b(the)f(hypotheses)g(of)37 b(Proposition)19 b(A.1,)i(there)g(e)o(xist)f(constants)g Fq(C)7 b Fp(\()p Fq(`;)17 b(m)p Fp(\))27 b Fq(>)94 4226 y Fr(0)e Fc(and)g Fq(\015)33 b(>)28 b Fr(0)p Fc(,)d(such)f(that)g(for)h(all)g Fq(v)32 b Fo(2)c Fq(H)1532 4252 y Fy(`;m)1664 4226 y Fc(,)d(one)g(has)482 4391 y Fl(\015)482 4451 y(\015)482 4511 y(\015)537 4506 y Fq(e)583 4463 y Fy(\034)8 b(L)688 4506 y Fq(Q)767 4454 y Fh(\()p Fz(0)p Fh(\))767 4536 y Fy(k)869 4506 y Fp(\()p Fr(1)21 b Fo(\000)i Fq(\037)1141 4533 y Fy(R)1207 4506 y Fp(\))p Fq(v)1298 4391 y Fl(\015)1298 4451 y(\015)1298 4511 y(\015)1353 4575 y Fy(`;m)1530 4506 y Fo(\024)44 b Fq(C)24 b Fr(e)o(xp)1889 4425 y Fl(\000)1934 4506 y Fq(\015)6 b Fp(\()p Fr(4)p Fq(R)q(=)p Fr(3)p Fp(\))2346 4436 y Fg(2)p Ff(n)p 2307 4447 148 4 v 2307 4487 a Fg(2)p Ff(n)p Fd(\000)p Fg(1)2471 4425 y Fl(\001)2534 4506 y Fr(e)o(xp)2676 4425 y Fl(\000)2722 4506 y Fo(\000)p Fq(\034)2866 4438 y(k)25 b Fp(+)e Fr(1)p 2866 4483 227 4 v 2924 4574 a(2)p Fq(n)3104 4425 y Fl(\001)3150 4506 y Fo(k)p Fq(v)t Fo(k)3302 4533 y Fy(`;m)3459 4506 y Fq(:)294 4759 y Fr(W)-8 b(e)25 b(no)n(w)f(return)h(to)f(the:)94 4933 y Fm(Pro)s(of)49 b(of)e(Prop)s(osition)g(2.1.)h Fr(As)33 b(in)h([EWW])g(it)g(is)f(only)h(necessary)g(to)g(consider)f(the)h(term)g (with)94 5052 y(highest)19 b(deri)n(v)n(ati)n(v)o(e)f(in)i Fo(k)p Fq(\037)1022 5079 y Fy(R)1087 5052 y Fq(e)1133 5009 y Fy(\034)8 b Fb(L)1237 5052 y Fq(v)t Fo(k)1339 5079 y Fy(`;m)1471 5052 y Fr(.)35 b(All)19 b(other)h(terms)f(are)i(easier)f(to)g(estimate.)33 b(Also,)21 b(as)f(in)f(that)h(paper)l(,)94 5172 y(we)26 b(use)e(the)h(f)o (act)g(that)1456 5291 y(D)1528 5248 y Fy(`)1567 5291 y Fq(e)1613 5248 y Fy(\034)8 b Fb(L)1761 5291 y Fp(=)45 b Fq(e)1929 5248 y Fy(\034)8 b(`=)p Fh(\()p Fz(2)p Fy(n)p Fh(\))2199 5291 y Fq(e)2245 5248 y Fy(\034)g Fb(L)2349 5291 y Fr(D)2421 5248 y Fy(`)2484 5291 y Fq(;)1135 b Fp(\()p Fr(A)p Fq(:)p Fr(7)p Fp(\))p eop %%Page: 15 15 bop 94 -45 a Fu(Cahn-Hilliard)39 b(Equa)-7 b(tion)2451 b Ft(15)94 194 y Fr(where)22 b(D)431 151 y Fy(`)490 194 y Fr(is)e(a)h(shorthand)f (notation)f(for)i(a)g(product)g(of)f(deri)n(v)n(ati)n(v)o(es)e(w)-6 b(.r)h(.t.)20 b(the)g Fq(x)2899 221 y Fy(j)2962 194 y Fr(of)h(total)f(de)o (gree)h Fq(`)p Fr(.)34 b(Thus,)684 415 y Fl(\000)729 495 y Fq(e)775 452 y Fy(\034)8 b Fb(L)879 495 y Fr(D)951 452 y Fy(`)989 495 y Fq(v)1041 415 y Fl(\001)1087 495 y Fp(\()p Fq(x)p Fp(\))44 b(=)1399 428 y Fq(e)1445 385 y Fy(\034)8 b(d=)p Fh(\()p Fz(2)p Fy(n)p Fh(\))p 1399 472 325 4 v 1445 564 a Fp(\()p Fr(2)p Fq(\031)t Fp(\))1634 535 y Fy(d)1753 360 y Fl(Z)1869 495 y Fr(d)1919 452 y Fy(d)1965 495 y Fq(z)22 b(g)t Fp(\()p Fq(z)t(;)17 b(\034)11 b Fp(\))2313 415 y Fl(\000)2357 495 y Fr(D)2429 452 y Fy(`)2467 495 y Fq(v)2519 415 y Fl(\001)2564 495 y Fp(\()p Fq(e)2649 452 y Fy(\034)d(=)p Fh(\()p Fz(2)p Fy(n)p Fh(\))2887 495 y Fp(\()p Fq(x)21 b Fp(+)i Fq(z)t Fp(\)\))i Fq(:)362 b Fp(\()p Fr(A)p Fq(:)p Fr(8)p Fp(\))94 754 y Fr(First)25 b(consider)g(the)f Fq(q)32 b Fp(=)c Fr(0)d(case)g(of)g(\(A.3\).)36 b(Then)331 1034 y Fo(k)p Fq(\037)443 1061 y Fy(R)509 1034 y Fq(e)555 991 y Fy(\034)8 b Fb(L)659 1034 y Fq(v)t Fo(k)761 1061 y Fy(`;m)937 1034 y Fo(\024)1071 967 y Fq(e)1117 924 y Fy(\034)g(d=)p Fh(\()p Fz(2)p Fy(n)p Fh(\))p 1071 1011 V 1116 1102 a Fp(\()p Fr(2)p Fq(\031)t Fp(\))1305 1073 y Fy(d)1424 898 y Fl(Z)1541 1034 y Fr(d)1591 991 y Fy(d)1637 1034 y Fq(z)21 b Fo(j)p Fq(g)t Fp(\()p Fq(z)t(;)c(\034)11 b Fp(\))p Fo(j)2040 949 y Fl(\015)2040 1009 y(\015)2093 1034 y Fq(w)2167 991 y Fy(m)2243 1034 y Fq(\037)2305 1061 y Fy(R)2371 953 y Fl(\000)2416 1034 y Fr(D)2488 991 y Fy(`)2527 1034 y Fq(v)2579 953 y Fl(\001)2624 1034 y Fp(\()p Fq(e)2709 991 y Fy(\034)d(=)p Fh(\()p Fz(2)p Fy(n)p Fh(\))2946 1034 y Fp(\()p Fq(:)22 b Fp(+)g Fq(z)t Fp(\)\))3262 949 y Fl(\015)3262 1009 y(\015)3318 1074 y Fz(2)3383 1034 y Fq(;)236 b Fp(\()p Fr(A)p Fq(:)p Fr(9)p Fp(\))94 1321 y Fr(where)23 b Fq(w)j Fr(is)21 b(the)h(operator)h(of)f(multiplication)d(by)j Fp(\()p Fr(1)16 b Fp(+)g Fq(x)g Fo(\001)g Fq(x)p Fp(\))2263 1278 y Fz(1)p Fy(=)p Fz(2)2380 1321 y Fr(.)35 b(Note)22 b(that)g(the)g(conclusions)e(of)j(Lemma)94 1440 y(B.4)h(of)g([EWW])g(do)f(not)g(depend)h(on)f(the)g(e)o(xact)h(form)f (of)h Fq(g)i Fr(and)e(so)f(it)g(also)g(holds)g(in)g(the)g(present)h (situation)94 1560 y(and)h(we)g(ha)n(v)o(e)94 1751 y Fm(Lemma)39 b(A.5.)c Fc(One)25 b(has)g(the)g(bounds)520 1940 y Fl(\015)520 2000 y(\015)575 2025 y Fq(w)649 1982 y Fy(r)694 2025 y Fq(\037)756 2052 y Fy(R)821 2025 y Fq(v)t Fp(\()p Fq(e)958 1982 y Fy(\034)8 b(=)p Fh(\()p Fz(2)p Fy(n)p Fh(\))1195 2025 y Fp(\()p Fq(:)22 b Fp(+)g Fq(z)t Fp(\)\))1511 1940 y Fl(\015)1511 2000 y(\015)1567 1965 y Fz(2)1567 2065 y(2)1651 2025 y Fo(\024)1773 1855 y Fl(\()1870 1955 y Fq(C)7 b(e)1994 1912 y Fn(\000)p Fy(\034)h(m)2177 1955 y Fo(k)p Fq(v)t Fo(k)2329 1912 y Fz(2)2329 1982 y(0)p Fy(;m)2488 1955 y Fq(;)284 b Fc(if)25 b Fo(j)p Fq(z)t Fo(j)j(\024)g Fr(7)p Fq(R)q(=)p Fr(8)c Fc(,)1870 2095 y Fq(C)7 b Fp(\()p Fr(1)22 b Fp(+)g Fo(j)p Fq(z)t Fo(j)2264 2052 y Fz(2)2304 2095 y Fp(\))2343 2052 y Fy(r)2387 2095 y Fo(k)p Fq(v)t Fo(k)2539 2052 y Fz(2)2539 2122 y(0)p Fy(;m)2673 2095 y Fq(;)99 b Fc(if)25 b Fo(j)p Fq(z)t Fo(j)j Fq(>)g Fr(7)p Fq(R)q(=)p Fr(8)c Fc(.)94 2370 y Fm(Remark.)44 b Fr(Note)31 b(that)f(the)h(proof)f(in)h([EWW])g(is)f(also)h(unaf)n(fected)f (by)h(the)f(dimension)f Fq(d)i Fr(in)g(which)f(we)94 2489 y(w)o(ork.)294 2609 y(No)n(w)41 b(use)g(Lemma)g(A.5)g(to)h(bound)e(the)i(inte)o(gral)e(in)i 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3110 y Fq(;)263 b Fp(\()p Fr(A)p Fq(:)p Fr(10)p Fp(\))94 3397 y Fr(where)29 b(the)f(last)f(step)h(used)f(the)h (estimates)f(of)h(Theorem)f(4.1)h(to)f(sho)n(w)g(that)2800 3316 y Fl(R)2883 3397 y Fr(d)p Fq(z)t Fo(j)p Fq(g)t Fp(\()p Fq(z)t(;)17 b(\034)11 b Fp(\))p Fo(j)30 b(\024)j Fq(C)7 b Fr(,)29 b(with)e Fq(C)94 3516 y Fr(independent)d(of)h Fq(\034)11 b Fr(.)294 3636 y(T)-8 b(o)22 b(estimate)g(the)h(inte)o(gral)e(in)h(the)h (outer)f(re)o(gion,)h(we)f(use)h(the)f(second)h(part)g(of)f(Lemma)g(A.5)h (and)f(then)94 3755 y(bound)i(it)h(by)211 4038 y Fq(C)301 3970 y(e)347 3927 y Fy(\034)8 b(d=)p Fh(\()p Fz(2)p Fy(n)p Fh(\))p 301 4015 V 347 4106 a Fp(\()p Fr(2)p Fq(\031)t Fp(\))536 4077 y Fy(d)655 3902 y Fl(Z)710 4128 y Fn(j)p Fy(z)s Fn(j)p Fy(>)p Fz(7)p Fy(R=)p Fz(8)1053 4038 y Fo(j)p Fq(g)t Fp(\()p Fq(z)t(;)17 b(\034)11 b Fp(\))p Fo(j)p Fq(e)1435 3995 y Fn(\000)p Fy(\034)d(r)r(=)p Fz(2)1660 4038 y Fo(k)p Fq(v)t Fo(k)1812 4065 y Fy(`;m)1989 4038 y Fo(\024)44 b Fq(C)7 b Fp(\()p Fq(n;)17 b(m)p Fp(\))p Fq(e)2504 3995 y Fy(\034)8 b(d=)p Fh(\()p Fz(2)p Fy(n)p Fh(\))2799 4038 y Fr(e)o(xp)2942 3957 y Fl(\000)2987 4038 y Fo(\000)p Fq(\015)e(R)3250 3968 y Fg(2)p Ff(n)p 3210 3979 148 4 v 3210 4019 a Fg(2)p Ff(n)p Fd(\000)p Fg(1)3375 3957 y Fl(\001)3421 4038 y Fo(k)p Fq(v)t Fo(k)3573 4065 y Fy(`;m)3730 4038 y Fq(;)3597 4231 y Fp(\()p Fr(A)p Fq(:)p Fr(11)p Fp(\))94 4350 y Fr(for)40 b(some)g Fq(\015)53 b(>)c Fr(0,)43 b(where,)h(again,)f(we)d(ha)n(v)o(e)f (used)h(the)f(estimates)g(of)h(decay)g(in)f(Theorem)h(4.1)f(both)94 4485 y(to)e(e)o(xtract)f(the)h(f)o(actor)g(of)g(e)o(xp)1204 4404 y Fl(\000)1250 4485 y Fo(\000)p Fq(\015)6 b(R)1513 4415 y Fg(2)p Ff(n)p 1473 4426 V 1473 4466 a Fg(2)p Ff(n)p Fd(\000)p Fg(1)1638 4404 y Fl(\001)1720 4485 y Fr(as)37 b(well)g(as)f(to)h(bound)f(the) g(inte)o(gral)g(o)o(v)o(er)g Fq(z)t Fr(.)73 b(Combining)94 4605 y(Eqs.\(A.1\),)25 b(\(A.10\),)f(and)h(\(A.11\),)g(we)g(get)f(the)h Fq(q)32 b Fp(=)c Fr(0)d(case)g(of)g(Eq.\(A.3\).)294 4724 y(W)-8 b(e)29 b(ne)o(xt)e(indicate)h(ho)n(w)f(to)h(treat)h(the)f Fq(q)37 b(>)c Fr(0)28 b(cases)h(of)f(Eq.\(A.3\).)46 b(Consider)29 b(the)f(case)h Fq(q)36 b Fp(=)d Fr(1.)47 b(W)-8 b(e)94 4844 y(can)34 b(re)n(write)f (Eq.\(A.8\))f(by)h(inte)o(grating)e(by)i(parts)f(once)i(w)-6 b(.r)h(.t.)31 b(one)i(component)f(of)h Fq(z)t Fr(,)j(for)d(e)o(xample)f Fq(z)3809 4871 y Fz(1)3849 4844 y Fr(.)94 4963 y(Then,)214 5157 y Fl(\000)260 5238 y Fq(e)306 5195 y Fy(\034)8 b Fb(L)410 5238 y Fr(D)482 5195 y Fy(`)520 5238 y Fq(v)572 5157 y Fl(\001)617 5238 y Fp(\()p Fq(x)p Fp(\))44 b(=)930 5171 y Fq(e)976 5128 y Fy(\034)8 b(d=)p Fh(\()p Fz(2)p Fy(n)p Fh(\))1255 5171 y Fq(e)1301 5128 y Fn(\000)p Fy(\034)g(=)p Fh(\()p Fz(2)p Fy(n)p Fh(\))p 930 5215 671 4 v 1148 5306 a Fp(\()p Fr(2)p Fq(\031)t Fp(\))1337 5277 y Fy(d)1629 5102 y Fl(Z)1745 5238 y Fr(d)1795 5195 y Fy(d)1841 5238 y Fq(z)1891 5157 y Fl(\000)1938 5238 y Fr(D)2010 5265 y Fy(z)2048 5275 y Fg(1)2083 5238 y Fq(g)2135 5157 y Fl(\001)2179 5238 y Fp(\()p Fq(z)t(;)17 b(\034)11 b Fp(\))2407 5157 y Fl(\000)2452 5238 y Fr(D)2524 5195 y Fy(`)p Fn(\000)p Fz(1)2660 5238 y Fq(v)2712 5157 y Fl(\001)2757 5238 y Fp(\()p Fq(e)2842 5195 y Fy(\034)d(=)p Fh(\()p Fz(2)p Fy(n)p Fh(\))3079 5238 y Fp(\()p Fq(x)22 b Fp(+)h Fq(z)t Fp(\)\))i Fq(:)119 b Fp(\()p Fr(A)p Fq(:)p Fr(12)p Fp(\))p eop %%Page: 16 16 bop 94 -45 a Fu(Cahn-Hilliard)39 b(Equa)-7 b(tion)2451 b Ft(16)94 194 y Fr(Dif)n(ferentiating)24 b(\(A.2\))h(w)-6 b(.r)h(.t.)23 b Fq(z)1213 221 y Fz(1)1278 194 y Fr(gi)n(v)o(es)701 417 y Fp(\()p Fr(D)812 444 y Fy(z)850 454 y Fg(1)885 417 y Fq(g)937 336 y Fl(\001)982 417 y Fp(\()p Fq(z)t(;)17 b(\034)11 b Fp(\))43 b(=)1375 281 y Fl(Z)1491 417 y Fr(d)1541 374 y Fy(d)1587 417 y Fq(k)20 b(ik)1745 444 y Fz(1)1818 417 y Fr(e)o(xp)p Fp(\()p Fq(iq)26 b Fo(\001)c Fq(z)t Fp(\))17 b Fr(e)o(xp)2402 336 y Fl(\000)2448 417 y Fp(\()p Fq(k)25 b Fo(\001)d Fq(k)s Fp(\))2708 374 y Fy(n)2762 417 y Fp(\()p Fr(1)f Fo(\000)i Fq(e)3018 374 y Fn(\000)p Fy(\034)3130 417 y Fp(\))3169 336 y Fl(\001)3240 417 y Fq(:)329 b Fp(\()p Fr(A)p Fq(:)p Fr(13)p Fp(\))94 658 y Fr(T)-8 b(o)33 b(estimate)g(\(A.13\),)h(\256rst)g(replace)f Fq(k)k Fr(by)32 b Fq(p)1698 685 y Fy(j)1779 658 y Fp(=)40 b Fq(a)p Fp(\()p Fq(\034)11 b Fp(\))2082 615 y Fz(1)p Fy(=)p Fh(\()p Fz(2)p Fy(n)p Fh(\))2308 658 y Fq(k)2360 685 y Fy(j)2402 658 y Fr(,)35 b(where,)h(as)d(before)g Fq(a)p Fp(\()p Fq(\034)11 b Fp(\))38 b(=)i Fr(1)26 b Fo(\000)i Fq(e)3737 615 y Fn(\000)p Fy(\034)3849 658 y Fr(.)94 778 y(Then,)144 988 y Fp(\()p Fr(D)255 1014 y Fy(z)293 1024 y Fg(1)328 988 y Fq(g)380 907 y Fl(\001)425 988 y Fp(\()p Fq(z)t(;)17 b(\034)11 b Fp(\))43 b(=)1022 920 y Fq(i)p 830 965 419 4 v 830 1058 a(a)p Fp(\()p Fq(\034)11 b Fp(\))1016 1029 y Fy(d=)p Fh(\()p Fz(2)p Fy(n)p Fh(\))1454 920 y Fr(1)p 1273 965 413 4 v 1273 1058 a Fq(a)p Fp(\()p Fq(\034)g Fp(\))1459 1029 y Fz(1)p Fy(=)p Fh(\()p Fz(2)p Fy(n)p Fh(\))1713 852 y Fl(Z)1829 988 y Fr(d)1879 945 y Fy(d)1926 988 y Fq(p)17 b(p)2043 1014 y Fz(1)2099 988 y Fr(e)o(xp)2241 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1671 325 4 v 1043 1762 a Fp(\()p Fr(2)p Fq(\031)t Fp(\))1232 1733 y Fy(d)1351 1558 y Fl(Z)1467 1694 y Fr(d)1517 1651 y Fy(d)1563 1694 y Fq(z)1647 1609 y Fl(\014)1647 1669 y(\014)1680 1694 y Fr(D)1752 1720 y Fy(z)1790 1730 y Fg(1)1825 1694 y Fq(g)t Fp(\()p Fq(z)t(;)17 b(\034)11 b Fp(\))2105 1609 y Fl(\014)2105 1669 y(\014)2170 1579 y(\015)2170 1639 y(\015)2170 1699 y(\015)2225 1694 y Fq(w)2299 1651 y Fy(m)2375 1694 y Fq(\037)2437 1720 y Fy(R)2503 1613 y Fl(\000)2548 1694 y Fr(D)2620 1651 y Fy(`)p Fn(\000)p Fz(1)2756 1694 y Fq(v)2808 1613 y Fl(\001)2853 1694 y Fp(\()p Fq(e)2938 1651 y Fy(\034)d(=)p Fz(2)3064 1694 y Fp(\()p Fq(:)21 b Fp(+)i Fq(z)t Fp(\)\))3380 1579 y Fl(\015)3380 1639 y(\015)3380 1699 y(\015)3436 1763 y Fz(2)3517 1694 y Fq(;)52 b Fp(\()p Fr(A)p Fq(:)p Fr(15)p Fp(\))94 1914 y Fr(and)32 b(then)f(proceeds)h (as)f(in)g(the)h(case)g(when)f Fq(q)41 b Fp(=)d Fr(0,)32 b(breaking)g(the)f (inte)o(gral)f(o)o(v)o(er)h Fq(z)36 b Fr(into)31 b(the)g(same)h(tw)o(o)94 2034 y(pieces)f(as)g(before.)53 b(These)31 b(tw)o(o)f(pieces)g(are)i(then)e (estimated)f(with)h(the)h(aid)f(of)h(Lemma)f(A.5.)52 b(Note)31 b(that)94 2153 y(while)37 b(the)g(f)o(actor)g Fq(a)p Fp(\()p Fq(\034)11 b Fp(\))966 2110 y Fn(\000)p Fy(d=)p Fh(\()p Fz(2)p Fy(n)p Fh(\))1298 2153 y Fr(of)37 b(Eq.\(A.14\))f(will)g(be)h(absorbed)g (when)f(one)h(inte)o(grates)f(w)-6 b(.r)h(.t)36 b Fq(z)t Fr(,)k(the)94 2283 y(remaining)34 b(f)o(actor)h(of)g Fq(a)p Fp(\()p Fq(\034)11 b Fp(\))1103 2240 y Fn(\000)p Fz(1)p Fy(=)p Fh(\()p Fz(2)p Fy(n)p Fh(\))1425 2283 y Fr(will)34 b(remain)g(in)g(the)h(\256nal)f(bound)g (of)h(Eq.\(A.3\).)64 b(The)34 b(bounds)g(for)94 2403 y Fq(q)e Fp(=)c Fr(2)p Fq(;)17 b Fr(3)p Fq(;)g(:)g(:)g(:)30 b Fr(2)p Fq(n)22 b Fo(\000)g Fr(1)j(follo)n(w)f(in)g(a)h(similar)f(f)o(ashion.)294 2522 y(T)-8 b(o)24 b(complete)g(the)h(proof)g(of)g(Proposition)e(2.1,)h (\256rst)h(re)n(write)333 2692 y Fq(e)379 2649 y Fy(\034)8 b Fb(L)483 2692 y Fq(Q)562 2719 y Fy(k)655 2692 y Fp(=)45 b Fq(e)823 2649 y Fy(\034)8 b Fb(L)p Fy(=)p Fz(2)1003 2692 y Fq(Q)1082 2719 y Fy(k)1130 2692 y Fq(e)1176 2649 y Fy(\034)g Fb(L)p Fy(=)p Fz(2)1400 2692 y Fp(=)45 b Fq(e)1568 2649 y Fy(\034)8 b Fb(L)p Fy(=)p Fz(2)1747 2692 y Fq(Q)1826 2719 y Fy(k)1875 2692 y Fq(\037)1937 2719 y Fy(R)2003 2692 y Fq(e)2049 2649 y Fy(\034)g Fb(L)p Fy(=)p Fz(2)2251 2692 y Fp(+)22 b Fq(e)2396 2649 y Fy(\034)8 b Fb(L)p Fy(=)p Fz(2)2576 2692 y Fq(Q)2655 2719 y Fy(k)2704 2692 y Fp(\()p Fr(1)21 b Fo(\000)i Fq(\037)2976 2719 y Fy(R)3042 2692 y Fp(\))p Fq(e)3127 2649 y Fy(\034)8 b Fb(L)p Fy(=)p Fz(2)3331 2692 y Fq(:)238 b Fp(\()p Fr(A)p Fq(:)p Fr(16)p Fp(\))94 2875 y Fr(The)23 b(second)f(of)h(these)f(terms)g(in)l (v)n(olv)o(es)f(an)i(estimate)f(of)g(the)h(action)f(of)g Fq(e)2640 2832 y Fy(\034)8 b Fb(L)p Fy(=)p Fz(2)2820 2875 y Fq(Q)2899 2902 y Fy(k)2970 2875 y Fr(on)23 b(a)g(function)e(localized)94 2994 y(near)26 b(the)f(origin,)e(so)i(by)f(Proposition)g(A.4,)g(we)h(get)g(a) g(bound)145 3221 y Fo(k)p Fq(e)241 3178 y Fy(\034)8 b Fb(L)p Fy(=)p Fz(2)421 3221 y Fq(Q)500 3248 y Fy(k)549 3221 y Fp(\()p Fr(1)21 b Fo(\000)i Fq(\037)821 3248 y Fy(R)887 3221 y Fp(\))p Fq(e)972 3178 y Fy(\034)8 b Fb(L)p Fy(=)p Fz(2)1151 3221 y Fq(v)t Fo(k)1253 3248 y Fy(`;m)1429 3221 y Fo(\024)45 b Fq(C)1622 3248 y Fy(q)1683 3221 y Fr(e)o(xp)1842 3080 y Fl(\022)1915 3221 y Fq(\015)1973 3140 y Fl(\000)2030 3153 y Fr(4)p Fq(R)p 2030 3198 127 4 v 2068 3289 a Fr(3)2168 3140 y Fl(\001)2214 3161 y Fz(2)p Fy(n=)p Fh(\()p Fz(2)p Fy(n)p Fn(\000)p Fz(1)p Fh(\))2610 3221 y Fo(\000)2721 3181 y Fz(1)p 2721 3198 35 4 v 2721 3255 a(2)2780 3153 y Fq(k)25 b Fp(+)e Fr(1)p 2780 3198 227 4 v 2838 3289 a(2)p Fq(n)3018 3080 y Fl(\023)3108 3221 y Fo(k)p Fq(v)t Fo(k)3260 3248 y Fy(`)p Fn(\000)p Fy(q)s(;m)3519 3221 y Fq(:)50 b Fp(\()p Fr(A)p Fq(:)p Fr(17)p Fp(\))94 3443 y Fr(W)-8 b(e)26 b(use)e(Proposition)g(A.1)g(to)h(bound)f(the)g(\256rst)h (term)g(of)g(\(A.16\):)956 3599 y Fq(C)7 b Fp(\()p Fq(`;)17 b(m)p Fp(\))p 952 3644 337 4 v 952 3759 a Fq(a)p Fp(\()p Fq(\034)11 b Fp(\))1193 3677 y Ff(q)p 1149 3695 123 4 v 1149 3736 a Fe(\()p Fg(2)p Ff(n)p Fe(\))1301 3667 y Fq(e)1347 3624 y Fy(\034)d(`=)p Fz(2)1523 3556 y Fl(\020)1582 3667 y Fq(e)1628 3624 y Fn(\000)1703 3597 y Fg(1)p 1703 3608 25 4 v 1703 3648 a(4)1740 3624 y Fy(\034)g(m)1882 3667 y Fp(+)23 b Fq(e)2028 3624 y Fn(\000)p Fy(\015)t(R)2196 3594 y Fg(2)p Ff(n=)p Fe(\()p Fg(2)p Ff(n)p Fd(\000)p Fg(1)p Fe(\))2514 3556 y Fl(\021)2590 3667 y Fo(k)p Fq(v)t Fo(k)2742 3694 y Fy(`)p Fn(\000)p Fy(q)s(;m)3001 3667 y Fq(:)568 b Fp(\()p Fr(A)p Fq(:)p Fr(18)p Fp(\))94 3913 y Fr(As)25 b(a)g(preliminary)f(step,)g (we)h(note)g(that)f(if)h(we)g(\256rst)g(choose)g Fq(r)i Fr(and)e Fq(R)g Fr(such)g(that)945 4109 y Fq(e)991 4066 y Fy(\034)8 b(q)s(=)p Fz(2)1172 3999 y Fl(\020)1232 4109 y Fq(e)1278 4066 y Fn(\000)1352 4040 y Fg(1)p 1352 4051 V 1352 4090 a(4)1389 4066 y Fy(\034)g(r)1500 4109 y Fp(+)23 b Fq(e)1646 4066 y Fn(\000)p Fy(\015)t(R)1814 4036 y Fg(2)p Ff(n=)p Fe(\()p Fg(2)p Ff(n)p Fd(\000)p Fg(1)p Fe(\))2132 3999 y Fl(\021)2236 4109 y Fo(\024)44 b Fq(e)2403 4066 y Fn(\000)p Fy(\026)p Fh(\(\()p Fy(k)r Fh(+)p Fz(1)p Fh(\))p Fy(=)p Fh(\()p Fz(2)p Fy(n)p Fh(\)\))2996 4109 y Fq(;)94 4341 y Fr(then)26 b(for)g(suf)n(\256ciently)f(small)g Fq(\026)h Fr(\(roughly)f(speaking)g Fq(\026)30 b Fo(\030)2191 4302 y Fz(1)p 2191 4318 35 4 v 2191 4376 a(2)2238 4261 y Fl(\000)2284 4341 y Fr(1)22 b Fp(+)i(\()p Fr(4)p Fq(=)p Fr(3)p Fp(\))2685 4298 y Fz(2)p Fy(n=)p Fh(\()p Fz(2)p Fy(n)p Fn(\000)p Fz(1)p Fh(\))3057 4261 y Fl(\001)3103 4281 y Fn(\000)p Fz(1)3205 4341 y Fr(\),)i(the)g(Eqs.\(A.17\))94 4461 y(and)f(\(A.18\))g(imply)466 4678 y Fo(k)p Fq(e)562 4635 y Fy(\034)8 b Fb(L)666 4678 y Fq(Q)745 4705 y Fy(k)793 4678 y Fo(k)44 b(\024)1190 4611 y Fq(C)p 1021 4655 417 4 v 1021 4748 a(a)p Fp(\()p Fq(\034)11 b Fp(\))1207 4720 y Fy(q)s(=)p Fh(\()p Fz(2)p Fy(n)p Fh(\))1449 4678 y Fq(e)1495 4635 y Fn(\000)p Fy(\026)1618 4604 y Ff(k)q Fe(+)p Fg(1)p 1618 4620 115 4 v 1641 4659 a(2)p Ff(n)1750 4678 y Fo(k)p Fq(v)t Fo(k)1902 4705 y Fy(`)p Fn(\000)p Fy(q)s(;k)2153 4678 y Fp(=)2456 4611 y Fq(C)p 2287 4655 417 4 v 2287 4748 a(a)p Fp(\()p Fq(\034)g Fp(\))2473 4720 y Fy(q)s(=)p Fh(\()p Fz(2)p Fy(n)p Fh(\))2715 4678 y Fq(e)2761 4635 y Fn(\000)p Fy(\026)p Fn(j)p Fy(\025)2942 4646 y Ff(k)q Fe(+)p Fg(1)3063 4635 y Fn(j)3091 4678 y Fo(k)p Fq(v)t Fo(k)3243 4705 y Fy(`)p Fn(\000)p Fy(q)s(;k)3475 4678 y Fq(:)294 4901 y Fr(This)29 b(sho)n(ws)f(that)i(the)g(projection)e(of)i(the) g(semi-group)f(onto)g(the)h(complement)e(of)i(the)g(eigenspace)94 5020 y(spanned)e(by)g(the)g(\256rst)g Fq(k)j Fr(eigen)l(v)n(alues)c(decays)i (with)e(a)i(rate)f(proportional)f(to)h(the)f(eigen)l(v)n(alue)h Fq(\025)3528 5047 y Fy(k)r Fh(+)p Fz(1)3673 5020 y Fr(.)46 b(W)-8 b(e)94 5150 y(can)25 b(sharpen)g(the)g(decay)g(rate)g(so)f(that)g(we)h (obtain)f(a)h(rate)g(lik)o(e)f(e)o(xp)2430 5069 y Fl(\000)2476 5150 y Fo(\000)p Fp(\()p Fr(1)e Fo(\000)g Fq(")p Fp(\))p Fo(j)p Fq(\025)2934 5177 y Fy(k)r Fh(+)p Fz(1)3079 5150 y Fo(j)3107 5069 y Fl(\001)3177 5150 y Fr(by)i(the)h(techniques)94 5270 y(of)g([EWW],)h(\(see)f(Eq.)35 b(B.14)25 b(and)g(follo)n(wing\))e(and)i(this) f(completes)g(the)g(proof)h(of)g(Proposition)e(2.1)p eop %%Page: 17 17 bop 94 -45 a Fu(Cahn-Hilliard)39 b(Equa)-7 b(tion)2451 b Ft(17)94 194 y Fs(References)94 443 y Fw([BKL])285 b(Bricmont,)14 b(J.,)h(A.)e(K)o (upiainen,)h(and)f(G.)g(Lin:)25 b(Renormalization)13 b(group)g(and)g (asymptotics)g(of)g(solutions)g(of)g(nonlinear)593 533 y(parabolic)20 b(equations.)29 b(Comm.)g(Pure)20 b(Appl.)28 b(Math.)20 b Fv(47)p Fw(,)g(893\261922)f(\(1994\).)94 623 y([EG])338 b(Eckmann,)19 b(J.-P)-9 b(.,)19 b(and)g(Th.)g(Gallay:)29 b(Front)19 b(solutions)g(of)f(the) i(Ginzb)n(ur)o(g-Landau)d(equation.)29 b(Comm.)19 b(Math.)g(Phys.)593 712 y Fv(152)p Fw(,)g(221\261248)g(\(1993\).)94 802 y([C])392 b(J.)19 b(Carr:)29 b Fa(The)21 b(Centr)m(e)f(Manifold)h(Theor)m(em)f(and)f (its)i(Applications)p Fw(,)e(Springer)n(-V)-9 b(erlag)20 b(\(1983\).)94 891 y([CH])334 b(Cahn,)19 b(J.)f(W)-7 b(.,)19 b(and)f(H.)g(E.)g(Hilliard:)28 b(Free)20 b(ener)o(gy)e(of)f(a)i(nonuniform)e(system)i(I.)f(Interf)o(acial)h (free)g(ener)o(gy.)27 b(J.)19 b(Chem.)593 981 y(Phys.)g Fv(28)p Fw(,)h(258\261267)e(\(1958\).)94 1071 y([EWW])246 b(Eckmann,)22 b(J.-P)-9 b(.,)22 b(C.E.)g(W)-6 b(ayne,)23 b(and)f(P)-9 b(.)22 b(W)m(ittwer:)33 b(Geometric)23 b(stability)g(analysis)f(for)f(periodic)h (solutions)g(of)g(the)593 1160 y(Swift-Hohenber)o(g)17 b(equation.)30 b(Comm.)20 b(Math.)g(Phys.)f Fv(190)p Fw(,)g(173\261211)g(\(1997\).)94 1250 y([H])387 b(Henry)-5 b(,)19 b(D.:)29 b Fa(Geometric)21 b(Theory)g(of)g(Semilinear)g(P)-6 b(ar)o(abolic)19 b(Equations)p Fw(,)h(Lecture)h(Notes)f(in)g(Mathematics,)j Fv(840)p Fw(,)593 1340 y(Springer)m(,)c(Berlin)h(\(1981\).)94 1429 y([L)-6 b(W])327 b(Lla)n(v)o(e,)28 b(R.,)f(and)f(C.E.)g(W)-6 b(ayne:)41 b(On)26 b(Irwin')l(s)d(proof)i(of)g(the)i(pseudostable)f(manifold)g(theorem.)47 b(Math.)26 b(Z.)g Fv(219)p Fw(,)593 1519 y(301\261321)18 b(\(1995\).)94 1609 y([S])401 b(G.)17 b(Schneider:)29 b(Dif)n(fusi)n(v)o(e)17 b(stability)j(of)d(spatial)j(periodic)e(solutions)h(of)e(the)i (Swift-Hohenber)o(g)d(equation.)29 b(Comm.)593 1698 y(Math.)20 b(Phys.)f Fv(178)p Fw(,)g(679\261702)g(\(1996\).)94 1788 y([W1])330 b(W)-6 b(ayne,)27 b(C.E.:)41 b(In)m(v)n(ariant)26 b(manifolds)f(for)g (parabolic)i(partial)g(dif)n(ferential)f(equations)g(on)g(unbounded)f (domains.)593 1878 y(Arch.)19 b(Rat.)h(Mech.)h(Anal.)e Fv(138)p Fw(,)h(279\261306)f(\(1997\).)94 1967 y([W2])330 b(W)-6 b(ayne,)21 b(C.E.:)31 b(In)m(v)n(ariant)20 b(manifolds)h(and)g(the)g(asymptotics)h(of)e (parabolic)h(equations)h(in)e(c)o(ylindrical)i(domains)f(in)593 2057 y(book)g(Proceedings)i(of)e(the)h(\256rst)g(US/China)h(Conference)f(on)g (Dif)n(ferential)g(Equations,)g(Hangzhou)f(PR.)36 b(Interna-)593 2147 y(tional)20 b(Press)g(\(1997\).)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF