Stability of the blow-up profile for equations of the type $u_t=\Delta u+|u|^{p-1}u$

$Page 1: Stability of the blow-up profile for equations of the type $u_t=\Delta u+|u|^{p-1}u$$
Vol. 86, No. 1 DUKE MATHEMATICAL JOURNAL (C) 1997

STABILITY OF THE BLOW-UP PROFILE FOREQUATIONS OF THE TYPE ut- Au + lulP-lu

FRANK MERLE AND HATEM ZAAG

1. Introduction.equation:

In this paper, we are concerned with the following nonlinear

ut Au + lulP-uu(.,O) uo H,

(1)

where u(t): x IRiv u(x, t) IR, A stands for the Laplacian in IRv. We noteH WI,p+I(IR/v) L(IR/V). We assume in addition the exponent p subcritical:if N > 3, then 1 < p < (N + 2)/(N- 2); otherwise, 1 < p < +. Other types ofequations will be also considered.The local Cauchy problem for equation (2) can be solved in H. Moreover, one

can show that either the solution u(t) exists on [0, +c), or on [0, T) withT < +oo. In this former case, u blows up in finite time in the sense that

Ilu(t)ll/ +o when T.

(Actually, we have both Ilu(t)lln) + and Ilu(t)llw,,+,n) +o whent--. T.)Here we are interested in blow-up phenomena. (For such a case, see, for

example, Ball [1] and Levine [14].) We now consider a blow-up solution u(t)and note T its blow-up time. One can show that there is at least one blow-uppoint a (that is, a IRs such that lu(a, t)l --, + when T). We will considerin this paper the case of a finite number of blow-up points (see [15]). More pre-cisely, we will focus for simplicity on the case where there is only one blow-uppoint. We want to study the profile of the solution near blow-up, and the stabil-ity of such behavior with respect to initial data.

Standard tools, such as center manifold theory, have been proven nonefficientin this situation (cf. [6], [4]). In order to treat this problem, we introduce sim-ilarity variables (as in [10]):

, (2)Y= x/T-

Received 18 January 1996. Revision received April 1996.

143

144 MERLE AND ZAAG

s -log(T t),

WT,a(y,s) (Z t)I/(p-1)U(X, t), (3)

where a is the blow-up point and T the blow-up time of u(t).The study of the profile of u as t - T is then equivalent to the study of the

asymptotic behavior of wr, (or w for simplicity), as s --, o, and each result for uhas an equivalent formulation in terms of w. The equation satisfied by w is thefollowing:

1 wWs Aw--y.Vw- -]-- [w[ p-1

p- 1w. (4)

Giga and Kohn showed first in [10] that for each C > O,

lim sup Iw(y,s)-cl--O,s--*+o [Yl < C

with x (p 1) -1/(p-I), which gives, if stated for u,

lim sup I(T t)l/(P-1)u(a + yv/T- t, t) x O.t-*TlYl<C

This result was specified by Filippas and Kohn [6], who established that indimension N, if w does not approach x exponentially fast, then for each C > 0,

suplylC

x 1

which gives, if stated for u,

suplylC

(T- t)l/(p-1)u(a + yv/T- t, t) Is: +o((-log(T- t))-l).

2p Ilog(T t)N lYl

(5)

Velazquez obtained in [16] a related result, using the maximum principle.Relying on a numerical study, Berger and Kohn [2] conjectured that in the

case of a nonexponential decay, the solution u of (1) would approach an explicituniversal profile f(z), depending only on p and independent from initial data asfollows:

(T- t)V(P-1)u(a + v/(T t)[log(T- t)lz, =f(z) + O((-log(T- t)) -1) (6)

STABILITY OF THE BLOW-UP PROFILE 145

in Llgc with

f(z)= p-l+(p- 1)2 2 (7)

This behavior shows that in the case of one isolated blow-up point, therewould be a free-boundary moving in (x, t) coordinates at the rate

v/(T t)Ilog(T t)].

This flee-boundary roughly separates the space into two regions:(1) the singular one, at the interior of the free-boundary, where Au can be

neglected with respect to lul-u, so equation (1) behaves like an ordinarydifferential equation, and blows up;

(2) the regular one, after the free-boundary, where Au and lul-u are of thesame order.

Herrero and Velazquez in [12] and [13] showed in the case of dimension one(N 1) using the maximum principle that u behaves in three manners, one ofwhich is suggested by Berger and Kohn, and they proved that estimate (6) is trueuniformly on z belonging to compact subsets of IR (without estimating the error).Going further in this direction, Bricmont and Kupiainen constructed a solu-

tion for (1) satisfying (6) in a global sense. For that they used, on the one hand,ideas close to the renormalization theory and, on the other hand, hard analysison equation (4).

In this paper, we shall give a more elementary proof of their result, based on amore geometrical approach and on techniques of a priori estimates.

TI-IEOREM 1 (Existence of a blow-up solution with a free-boundary behavior ofthe type (6)). There exists To > 0 such that for each T (0, To], Vg H withI[gl[ < (log T)-2, one can find do e IR and dl IR such that for each a e IR1,equation (1) with initial data

uo(x) T-1/(p-1) f(z) 1 +

z (x- a)(llog TIT)-/2,

has a unique classical solution u(x, t) on IRN x [0, T) and(i) u has one and only one blow-up point a;

(ii) a free-boundary analogous to (6) moves through u such that

lim (T- t)l/(p-1)u(a + ((T- t) [log(T- t)l)l/2z, t) =f(z)t--.T

(8)

146 MERLE AND ZAAG

uniformly in z lRv, with

f(z)= p-l+(p- 1)2

4-----

Remark. We took do and dl, respectively, in the direction of ho(y) 1 andhx (y)= y, the two first eigenfunctions of ’ (cf. Section 2), but we could havechosen other directions Do(y) and Dx (y) (see Theorem 2). We can notice that wehave a result in H W,p+I(IWv) L(IWV). We can also obtain blow-up resultsin Hi(jRN) L(IRV). If p < 1 / (4/N), then f(z) n1, and we use the samearguments to solve the problem in HI(IRN) L(IRV). If p > 1 + (4IN), theresult in H follows directly from the stability result (see Theorem 2).

Remark. Such behavior is suspected to be generic.

Remark 1.1. One can ask the following questions:(a) Why does the free-boundary move at such a speed?(b) Why is the profile precisely the function f?

As in various physical situations, we suspect that the asymptotic behavior ofw - x is described by self-similar solutions of equation (4).

Since we are dealing with an equation of the heat type (cf. (4)), the naturalscaling is y/x/. Let us hence try to find a solution of the form v(y/x/), with

v(0) x, lim I(z)l =0. (9)

A direct computation shows that v must satisfy the following equation, for eachs > 0 and each z IRN:

1 1Av(z)

1z.Vv(z)

1-2z’Vv(z) - - p- 1 v(z) 4-[o(z)IP-II)(z). (10)

According to Giga and Kohn [8], the only solutions of (10) are the constant ones:0, x,-x, which are ruled out by (9). We can then try to search formally regularsolutions of (4) of the form

and compare elements of order 1/sJ (in one dimension, in the positive case forsimplicity). We obtain, for j 0,

1ZV o(Z 10 --i ,o(z) + ,o(z)


and for j 1 (z - 0)

v (z) + a(z)vl(z) b(z)

with a(z) (2/z)((1/(p- 1) pro(z)p-) and b(z) v(z) + (2/z)v’(z). The sol-ution for v0 is given by

vo(z) (p- 1 + coz2) -1/(p-1)

for an integration constant co > 0. Using this to solve the equation on Vl yields

[ I ]vl (z) vo(z)Pz2 c + (-2vo(()-Pb(() d(

for another integration constant cl. Since we want V to be regular, it is natural torequire that v is analytic at z 0. vx is regular if and only if the coefficient of ( inthe Taylor expansion of vo(()-Pb(() near (= 0 is zero, which turns to be equi-valent to co (p- 1)2/(4p) after simple calculation. Therefore, vo(z) (p- 1 +((p- 1)2/(4p))z2)-1/(p-1). Hence, the first term in the expansion of V is preciselythe profile function f. Carrying on calculus yields

v (z)p- 1 (p- 1)2

2---- f(z)P + 4z2f(z)p log f(z) + cz2f(z)p. (11)

We note that 1)1 (0) x/(2p).Unfortunately, we are not able to calculate every vj. In conclusion, we take

another approach to obtain approximate self-similar solutions (see the proof ofTheorem 1).As in the paper of Bricmont and Kupiainen [4], we will not use the maximum

principle in the proof. The technique used here will allow us, using geometricalinterpretation of quantities of the type of do and dl, to derive stability resultsconcerning this type of behavior for the free-boundary, with respect to perturba-tions of initial data and the equation.

THEOREM 2 (Stability with respect to initial data of the free-boundary be-havior). Let fro be initial data constructed in Theorem 1. Let fa(t) be the solution

of equation (1) with initial data fao, J" its blow-up time, and t its blow-up point.Then there exists a neighborhood of fro in H, which has the following property:For each uo in q/;, u(t) blows up in finite time T T(uo) at only one blow-up pointa a(uo), where u(t) is the solution of equation (1) with initial data uo. Moreover,u(t) behaves near T(uo) and a(uo) in an analogous way as fi(t):

lim (T- t)/(P-X)u(a + ((T- t)[log(T- t)l)/Ez, t) f(z)t---T

uniformly in z IRv.

148 MERLE AND ZAAG

Remark. Theorem 2 yields the fact that the blow-up profile f(z) is stablewith respect to perturbations in initial data.

Remark. From [15], we have T(uo) , a(uo) (t, as u0 t0 in n.Remark. For this theorem, we strongly use a finite-dimension reduction of

the problem in IRl+v, which is the space of liberty degrees of the stability theo-rem: (T, a).Remark 1.2. Theorem 2 is true for a more general rio: It is enough that fi(t)

satisfies the key estimate of the proof of Theorem 1.

Remark. Since we do not use the maximum principle, we suspect that suchanalysis can be carried on for other types of equations, for example,

and

Ut -A2u -k-[u]2u,

ut Au + lulp-lu + ilul’-lu, (12)

where 1 < r < p (p < (N + 2)/(N 2) if N > 3).See also for other applications [18].According to a result of Merle [15], we obtain the following corollary for

Theorem 2.

COROLLARY 1.1. Let D be a convex set in IRN, or D lRN. For an arbitrarilygiven set of k points x1,..., Xk in D, there exist initial data uo such that the sol-ution u of (1) with initial data uo (with Dirichlet boundary conditions in the caseD v lRN) blows up exactly at xl,..., Xk.

Remark. The local behavior at each blow-up point xi (Ix -xil < Pi) is alsogiven by (8).

2. Formulation of the problem. We omit the (T, a) or (do, dl) dependence inwhat follows to simplify the notation.

2.1. Choice of variables. As indicated before, we use similarity variables:

x--a

Y=x/T-t’s -log(T t),

w(y,s) (T- t)l/(p-1)u(x,t).

We want to prove for suitable initial data that

lim II(T- t)l/IP-lu(a / ((T- t)llog(Z- t)l)X/2z, t) -f(z)ll 0t-*T


or stated in terms of w,

limllw(y,s)-f(s) LoO=0,

where

f(z)= (p- l + (p-1)24------ Iz12

We will not study, as usually is done, this limit difference as s +

but we introduce instead

q(y,s) w(y,s) N[ps+ (p 1+(p-l)2 y2

4ps(13)

The added term in (13) can be understood from Remark 1.1, where we tried toobtain for w an expansion of the form _,j+=(1/sJ)vj(y/x/). We got v0 =f andfor vl the expression (11). Hence, it is natural to study the difference w(y,s)-(vo(y/x/) + (1/s)v(y/v/)). Since the expression of Vl is a bit complicated (see(11)), we study instead w(y,s) (vo(y/x/) + (1/s)v(O)), which is (13) for N 1.Now, if we introduce

ps +f --ps + (14)

we have

q(y, s) w(y, s) tp(y, s).

Thus, the problem in Theorem 1 is to construct a function q satisfying

lim IIq(.,s)llz,o o.

From (4) and (13), the equation satisfied by q is the following: for s > 0,-- (y, s) .L’v(q)(y,s)+ B(q(y,s))+ R(y,s), (15)

150 MERLE AND ZAAG

wherethe linear term is

.L’v(q) q(q) + V(y, s)q (16)

with

1,E’(q) Aq--y.Vq + q and V(y, s) p (q9p-1

the nonlinear term (quadratic in q for p large) is

B(q) Iq9 + q[p-l(q9 + q) q)P pq)p-lq, (17)

and the rest term involving q9 is

R(y, s) Aq9 y.Vq9p 1 t + tp

cos (18)

It will be useful to write equation (15) in its integral form: for each so > 0, for eachSl > so,

q(sl) K(Sl,So)q(so)+ dz K(Sl, z)B(q(z))+ dz K(s1, z)R(’), (19)

where K is the fundamental solution of the linear operator ’v defined for eachso > 0 and for each Sl > so by

ts,K(Sl So c’gK(Sl So

K(s0, so) Identity.(20)

2.2. Decomposition of q. Since Lay will play an important role in our analy-sis, let us point out some facts.

(i) The operator is selfadjoint on (La) c L2(IRN, d#) with

e-lyl2/4 dyd#(y)

(4nlV/2(21

Note here that there is a weight decaying at infinity. The spectrum of &a is explicit.More precisely,

mspec() 1- m IN},


and it consists of eigenvalues. The eigenfunctions of .La are derived from Hermitepolynomials:

N=I:All the eigenvalues of are simple. For 1- (m/2) corresponds the eigen-function

[m/2] m!hm(y) Z n!(m- 2n)’n--O

(- 1)"ym-En (22)

hm satisfies

hnhm d/a 2nn!tnm

2(We will note also km hm/llhmllL.)N>2:We write the spectrum of as

1 ml +... + msspec(’)2

ml,...,mN IN}.For (ml,...,mN)]N, the eigenfunction corresponding to 1- (ml +...+mu)/2 is

y hm, (Yl)’" hmN (YN),

where hm is defined in (22). In particular,"1 is an eigenvalue of multiplicity 1, and the corresponding eigenfunction is

Ho(y)- 1. (23)

"1/2 is of multiplicity N, and its eigenspace is generated by the orthogonalbasis {nl,i(y)li 1,... ,N}, with nl,i(y) hl(yi); we note

H1 (y) (H1,1 (Y), H1,v (Y)). (24)

*0 is of multiplicity N(N + 1)/2, and its eigenspace is generated by theorthogonal basis {n2,ij(y)li,j 1,..., N, < j}, with n2,,(y) h2(yi), andfor/< j, n2,ij(y) hi (yi)hl (yj); we note

H2(y) (H2,ij(Y), j) (25)

(ii) The potential V(y,s) has two fundamental properties that will influencestrongly our analysis.

152 MERLE AND ZAAG

(a) We have V(. ,s) -+ 0 in the L2(]RN, d#) when s --+ +oo. In particular, theeffect of V on the bounded sets or in the "blow-up" region (Ixl < C/)inside the free-boundary will be a "perturbation" of the effect of .

(b) Outside the free-boundary, we have the following property: Ve > 0,3C > 0, 3s such that

supss, lyl/x/c

with-p/(p- 1) <-1.Since one is the biggest eigenvalue of ", we can consider that outside the free-boundary, the operator v will behave as one with fully negative spectrum, whichsimplifies greatly the analysis in this region.

Since the behavior of V inside and outside the free boundary is different, let usdecompose q as the following: Let 0 e C([O, +oo)), with supp(7‘0) [0,2] and;to 1 on [0, 1]. We define then

{" lyl ’;(y, s) ;o ,Kos/2j, (26)

where K0 > 0 is chosen large enough so that various technical estimates hold. Wewrite q qb + qe where

qb q7‘ and qe q(1 7,).

Let us remark that

supp qb(s) B(0, 2K0v) and supp qe(S) c IRN \B(0, KoV).

Then we study qb using the structure of 0. Since a has 1 + N expandingdirections (corresponding to eigenvalues 1 and 1/2) and N(N + 1)/2 neutralones, we write qb with respect to the eigenspaces of a as follows:

2

qb(Y, S) Z qm(s).Hm(y) + q-(y, s) (27)m=0

whereqo(s) is the projection of qb on H0;ql,i(s) is the projection of qb on Ha,i, ql(s)= (qa,i(s),...,qa,N(s)), Hi(y) isgiven by (24);q2,ij(S) is the projection of qb on H2,ij, i<j, q2(s)= (q2,ij(s),i <j), H2(y) isgiven by (25);q_(y,s) P-(qb) and P_ the projector on the negative subspace of


In conclusion, we write q into 5 "components" as follows:

2

q(y, s) E qm(s).Hm(y) + q-(y, s) + qe(Y, s) (28)m=0

(Note here that qm are coordinates of qb and not of q.)In particular, if N 1 and rn 0, 1, 2, qm(s) and Hm(y) are scalar functions,

and nm(y) hm(y). We write in this case

2

q(y, s) E qm(s)hm(y) + q-(y, s) + qe(Y, S). (29)m=0

Let us now prove Theorem 1.

3. Existence of a blow-up solution with the given free-boundary profile.section is devoted to the proof of Theorem 1.

This

3.1. Transformation of the problem. As in [4], we give the proof in onedimension (same proof holds in higher dimension). We also assume a to be zero,without loss of generality. Let us consider initial data:

UO,do,dl (X) T-1 f(z) do + dlz1 + + g(z)p-1+1)z2(p-

4p

where

z-- x(llog TIT)-1/2.

We want to prove first that there exists To > 0 such that for each T (0, To], forevery g n with Ilgll, < (log T)-2, we can find (d0, dl) ]R2 such that

lim (T- t)l/(p-1)Udo,d, (((T- t)Ilog(T- t)l)V2z, t) =f(z)t---T

(3O)

uniformly in z IR, where udo,a, is the solution of (1) with initial data uO,do,dl, and

f(z) (P- l + (P-1)2 )4pZ2 (31)

This property will imply that Udo,d blows up at time T at one single point: x 0.Indeed, we have the following proposition.

154 MERLE AND ZAAG

PROPOSITION 3.1 (Single blow-up point properties of solutions).solution of equation (1). Ifu satisfies the property

Let u(t) be a

lim II(T- t)/(P-)u(x/(T- t)llog(T- t)lz, t) -f(z)llL= 0t-*T

(32)

then u(t) blows up at time T at one single point: x O.

Proof. For each b lR, we have from (32)

lim (T-t)l/(p-1)u(b,t)-f .x/(T t)llog(T t)l

Using (31), we obtain limt_T(T- t)l/(p-1)u(O, t) X and for b 0,limt-r(T- t)l/(p-1)u(b,t) 0. A result by Giga and Kohn in [10] shows that bis a blow-up point if and only if limtr(T t)U(P-1)u(b, t) -Z-_ . This concludesthe proof of Proposition 3.1.

Therefore, it remains to find (d0, dl)e ]R2 so that (30) holds to conclude theproof of Theorem 1.

If we use the formulation of the problem in Section 2, the problem reduces tofinding So > 0 such that for each so > So, g e H with IIllLoO 1/S, we can find(d0, dl) IR2 so that the equation (15)

qt- (y’ s) v(q)(Y, s) + B(q(y, s)) + R(y, s),

with initial data at s so

qdo,dl(Y, So) (P l + (P--1)2 )-p/(p-1)

y2 (do + dly/v/o)4pso

+

2pso

(33)

has a solution q(d0, dl) satisfying

lim suPlqao,a (y, s)l 0. (34)s-,o ylR

q will always depend on g, do, and dl, but we will omit these dependences in thenotation (except when it is necessary).The convergence of q to zero in L(IR) follows directly if we construct a q(s)

solution of equation (15) satisfying a geometrical property, that is, q belongs to aset Va C([s0, +), L2(]R, d#)), such that Va shrinks to q 0 when s


More precisely, we have the following definitions.

Definition 3.2. For each A > 0, for each s > 0, we define l(s) as being theset of all functions r in L 2(IR, d) such that

Irm(S)[ As-2, m O, 1,

[r2(s)l < A2(log s)s-2,

Ir-(y,s)l < A(1 + {yl3)s-2,

IIr,(s)ll A2s-1/2,

where r(y) 2m=0 rm(s)hm(y) + r_(y, s) + re(y, s) (cf. decomposition (29)).

Definition 3.3. For each A > 0, we define 1 as being the set of all functions qin C([s0, +oo), L2(IR, d#)) satisfying q(s) Va(s) for each s > so.

Indeed, assume that Vs> so q(s) Va(s). Let us show that Vs> sosupy Iq(Y, s)l < C(A)/vq, which implies (34).We have from the definitions of qb and qe

q(y, s) qb(Y, S) + qe(Y, S)

qb(y,s)’l{lyl < 2Kox/} + qe(y,s)

=(qm(s)hm(Y)+q-(y’s))"l{lyl<2K}(y’s)+qe(y’s)’m=oUsing the definitions of hm (cf. (22)) and 1, the conclusion follows.

3.2. Proof of Theorem 1. Using these geometrical aspects, what we have todo is finally to find A > 0 and So > 0 such that for each so > So, g e H withI111. < 1/s, we can find (do, dl) JR.2 so that s > so,

qdo,dl ($) VA($). (35)

Let us explain briefly the general ideas of the proof.In a first part, we will reduce the problem of controlling all the components of

q in 1 to a problem of controlling (qO, ql)(S). That is, we reduce an infinite-dimensional problem to a finite-dimensional one.

In a second part, we solve the finite-dimensional problem, that is, to find(do, d1) IR2 such that (q0, ql)(s) satisfies certain conditions. We will proceed bycontradiction and use dynamics in dimension 2 of (qo, ql)(S) to reach a topo-logical obstruction (using the index theory).

156 MERLE AND ZAAG

The constant C now denotes a universal one independent of variables, onlydepending upon constants of the problem such as p.

Part I. Reduction to a finite-dimensional problem. In this section, we showthat finding (do, dl) IR2 such that Vs > so, q(s) V(s) is equivalent to finding(do, d1) IR2 such that Iqm(s)l < A/s2 Vs > so, Ym {0, 1). For this purpose, wegive the following definition.

Definition 3.4. For each A > 0, for each s > 0, we define l?A(s) as being theset [-a/s2,a/s2] 2 = IR2.For each A > 0, we define I?A as being the set of all (qo, ql) in C([so, +), IR2)

satisfying (qo, qa)(s) l?a(s) s > so.Step 1. Reduction for initial data. Let us show that for a given A (to be

chosen later), for so > Sl(A), the control of q(so) in l(so) is equivalent to thecontrol of (qo, ql)(so) in l?a(so).LEMMA 3.5. (i) For each A > O, there exists sa(A) > 0 such that for each

so > s(A), 9 H with IIOIl.oo < 1/s, if (do, d1) is chosen so that (qo, q)(so)’a (So), then

Iq2(so)l < (log so)so -2,

[q-(y, so)l C(1 + [y[a)so-2,

Ilq(, ,so)ll S0 -1/2"

(ii) There exists A1 > 0 such that for each A > A, there exists Sl(A)> 0such that for each so > sa(A), 9 H with I]911.oo < 1/s, we have the followin9equivalence:

q(so) Va(so) if and only if (qo, ql)(So) ("4(so).

Proof. We first note that part (ii) of the lemma follows immediately frompart (i) and Definition 3.2. We prove then only part (i). Let A > 0, so > 0, and0 H such that ]I011L= < 1/s. Let (d0, dl) IR2. We write initial data (cf. (33)) as

q(y, so) qO(y, so) + ql(y, so) + qE(y, so) + qa(y, so)

where qO(y, so)=doF(y/x/), qi(y, so)=dl (y/x/)F(y/x/), qE(y, so) -c/(2pso),q3(y, so) 9(Y/x/), and f(y/x/ (p- 1 + ((p- 1)2/(4pso))y2) -p/(p-1). Wedecompose all the qi as suggested by (29).From [Io[l 1/s, we derive that ]q(so)l + Iq(so)[ + ]q(so)l + [[q(so)[[

C/s, and then [q(y, so)[ (C/s)(1 + [y[3).Using simple calculations, we obtain Iq(so)l C/so, q(so)= o, Iq(so)l

Ce-s, [q(y, so)[ CsC2(1 + [y13), and Ilq(so)llL Cs1.


For q0, we have qo(So do d#(z)ZsoF(z/v) doC(p) (so oo), ql (SO 0,q2(so) do f d#(z)7.soF(Z/x/)(z2 2)/8 do(C’(p)/so) (so - ), [qO (y, so)l <do(C/so)(1-t-ly13), and IIq(so)ll < Cdo. All these last bounds are simple toobtain, perhaps except that for q0. Indeed, we write q(y, so) doZsoF(Y/V)-do f d#(z) 7.soF(z/x/) do f d#(z) )(.sof(z/x/) ((z2 2) /8)(y2 2). The last term canbe bounded by (Cdo/so)(1 + ly13). We write the first term as do{Jtso(Y)F(y/x)-Zso(0)F(0) -[d#(z)(Zsot(Z/x/ -;(so(0)r(0))}. Using a Lipschitz property, wehave [Zso(Y)F(y/x/)- ;ts0(0)F(0)[ < Cy2/so, and the conclusion follows.

Similarly, we obtain for ql, q(so)=0, q(so)=(d/x/o)fd#(z)ZsoF(Z/v/’)x(z/2)y dl(C"(p)/x/o) (so-- c), q (so) 0, [q (y, so)[ < dl (C/s/2)(1 -+- [y[3), andIlqe(S0)ll < C(dl/x/ro).Hence, by linearity, we write

qo(so) doao(so) + b0(g, so)(36)

q(so) dial(so) + b(g, so)

with ao(so) C(p), at(so) C"(P)/x/o, Ibo(g, so)l < C/so, and [b(g, so)[ < C/s.Therefore, we see that if (d0,dl) is chosen such that (qo, ql)(So) a(so) and ifso > s (A), we obtain Idm[ < C/so for m {0, 1}. Using linearity and the aboveestimates, we obtain [q2(s0)[ < C/s2o, [q-(y, so)[ < (C/sg)(1 + [y[3)and [[qe(So)[loo <C/so. Taking s (A) larger, we conclude the proof of Lemma 3.5.

Step 2. A priori estimates. This step is the crucial one in the proof of Theo-rem 1. Here we will show through a priori estimates that for s > so, the controlof q in Va(s) reduces to the control of (q0, ql) in l?,i(s). Indeed, this result willimply that if for s, > so, q(s,) 8Va(s,), then (qo(s,),q(s,)) OfZa(s,) (comparewith Definition 3.2).

Remark 3.6. We shall note here that for each initial data q(so), equation (15)has a unique solution on [so, S] with either S +oe or S < +oe and [[q(s)[lt.oo+oe, when s S. Therefore, in the case where S < +oe, there exists s, > so suchthat q(s,) V(s,), and the solution is, in particular, defined up to s,.

PRO’OSITION 3.7 (Control of q by (qo, ql) in V). There exists A2 > 0 suchthat for each A > A2, there exists s2(A)> 0 such that for each so > s2(A), foreach g n with Ilgl[L(R) < 1/s, we have the following properties: if (do, d1) ischosen so that (qo(so),ql(sO)) lea(so), and if for Sl > so, we have Vs [so, s],q(s) Va(s), then Vs [so, s],

[q2(s)[ A2s-2 log s- s-3

A 3) -2Iq_(y,s)l < -(1 + [y[ s

A2

[[qe(S)[[LOO < 2x/"

158 MERLE AND ZAAG

Proof. See the subsection "Proof of Proposition 3.7" below.

Step 3. Transversality. Using now the fact that (q0, ql) controls the evolutionof q in 1, we show a transversality condition of (q0, ql) on lT’a(s,).LEMMA 3.8. There exists A3 > 0 such that for each A > A3, there exists s3(A)

such that for each so s3(A), we have the following properties:(i) Assume there exists s, so such that q(s,) Va(s,) and (q0, ql)(s,)^

then there exists rio > 0 such that Vdi (0, di0), (q0, ql)(S, + fi) Va(s, + fi).(ii) If q(so) V(so), q(s) Va(s) Vs [s0, s,] and q(s,)dVa(s,), then there

exists 60 > 0 such that V (0, 6o), q(s, + 6) q Va(s, + 6).

Proof. Part (ii) follows from Step 2 and part (i). To prove part (i), we willshow that for each m (0, 1}, for each e {-1, 1}, if qm(S,)= e(A/s,), then(dqm/ds)(s,) has the opposite sign of (d/ds)(eA/s)(s,) so that (q0, ql) actuallyleaves l)’a at s, for s, > so where so will be large. Now let us compute (dqo/ds)(s,)and (dql/ds)(s,) for q(s,) Va(s,) and (qo(s,),ql(s,))dP,(s,). First, we notethat in this case, IIq(s,)ll < CA2/v/, and Iqb(y,s,)l < CA2(log s,/s2,)(1 4-]y[3)(provided A > 1). Below, the classical notation O(l) stands for a quantity whoseabsolute value is bounded precisely by and not C1.For m {0, 1 }, we derive from equation (15) and (22):

I d#z(s,)(t3q/c3s)km= I d#z(s,)&’qkm+ I d#z(s,)Vqkm+ J d#(s,)B(q)km

+ J d#7.(s,)R(s,)km.

We now estimate each term of this identity:(a) I d# Z(s,)(t3q/3s)km dqm/ds I d#(dz/ds)qkml I d#(dz/ds)qkmld#](dz/ds)lCA2/x/,lkml < Ce-s* if so > s3(A).(b) Since is selfadjoint on L 2(IR, d#), we write

d# Z(s,)&’qkm J d# -(Z(s,)km)q.

Using (Z(s,)km) (1 (m/2))jt(s,)km + (O2Z/OsE)km + (OZ/Oy)(2(Okm/Oy)-(y/2)km), we obtain f d# .(s,)qkm (1 (m/2))qm(S,)2+ O(CAe-S*).

(c) We then have from (16): gy, IV(y,s)l < (C/s)(1 / lyl ). Therefore,

< t|d#C(1 + ly15)CA2 logs,Ikml <

S, S,2J

CA2 log s,

(d) A standard Taylor expansion combined with the definition of 1 showsthat IZ(y,s,)B(q(y,s,)) < Clql 2 < C(Iqb124- lqe[ 2) (CAa(log S,)22/S4,)(1 4- lyl3)2 4-l(lyl>K,)(y)(h2/x/,). Thus, If d# Z(s,)n(q)kml < (Ch4(log s,) /s4,) 4- Ce-s*.


(e) A direct calculus yields If d# ;t(s,)R(s,)kml < C(p)/s2, (actually, it is equalto 0 if rn 1). Indeed, in the case m 0, we start from (18) and (14) and expandeach term up to the second order when s . Since qg(y,s)=f(y/x/)+(x/2ps), we derive

(1) d#z(s)p- 1 p- 1 x--ps+-ps +O(cs-2

(2) d# X(s)qp d# fP +ps dl pfp-1 + O(Cs-2)

r t pp-1 2(p-1)s 2psp-1

+p-1

O(Cs-2)

(3) q)s(Y, s)p 1 y2fp4ps2 2ps2 and then I d# Z(s)(-qg) O(Cs-2);

(4) qy(y, s) 2ps l ( 1 ) xyfP and then d# ;t(s) yoy + O(Cs-2)__

J t -2).(5) qgyy(y,s)--p 1

fp+ (p-1)2y2f2p_1 then d# Z(S)q)yy --FO(Cs2ps 4ps2 2ps

Adding all these expansions, we obtain f d# 7.s,R(s,)= O(C(p)s22). Concludingsteps (a) to (e), we obtain

dqmds ( )eAsz (C(p)) logs,)(s,)= 1- _-5-+0 +0 CA4,S, S,

whenever qm(S,) 13A/$2,. Let us now fix A > 2C(p), and then we take s3(A) largerso that for so > s3(A), Vs > so, (C(p)/s2) +O(CA4(log s/s3)) < 3C(p)/2s2. Hence,if e -1, (dqm/ds)(s,) < 0, if e 1, (dqm/ds)(s,) > 0. This concludes the proof ofLemma 3.8.

Now, let us fix A > sup(A2, A3).Part II. Topological argument. Now we reduce the problem to studying a

two-dimensional one. We give this problem’s initialization in the followinglemma.

160 MERLE AND ZAAG

LEMMA 3.9 (Initialization of the finite-dimensional problem). There exists

s4(A) > 0 such that for each so > s4(A), for each 9 n with I111, < 1/s, thereexists a set g,so c IR2 topologically equivalent to a square with the followin9property:

q(d0, dl, so) 1 (so) /f and only if (do, dl) g,s0.

Proof. As stated by Lemma 3.5 (ii), if we take so > sl(A) and g H withI1 11, 1/s , then it is enough to prove that there exists a set ,s0 topologicallyequivalent to a square satisfying

(qo, ql)(So) a(So) if and only if (d0, dl) v,s0.

If we refer to the calculus of qm(So) (cf. (36) and what follows), and takes4(A) > so (A) and s4(A) large enough, then this concludes the proof of Lemma 3.9.

NOW we fix So sup(sl(A),sE(A),s3(A),s4(A)) and take so > So. We start theproof of Theorem 1 for A and so(A) and a given g H with I1#11(R) < 1/s.We argue by contradiction: According to Lemma 3.9, for each (do, dl) v,so

q(do, dl,so) Va(so). We suppose then that for each (do, d1) ,o, there existss > so such that q(do, dl,s) Va(s). Let s,(do, dl) be the infimum of all these s’s.(Note here that s,(d0, dl) exists because of Remark 3.6.)

Applying Proposition 3.7, we see that q(do, dl,s,(do, dl)) can leavere’(s,(do, dl)) only by its first two components; hence,

(qo, ql)(do, dl,s,(do, dl)) . c’A(s,(do, dl)).

Therefore, we can define the following function:

S, (do, dl)2 (qo, ql)(do, dl, s, (do, dl)),(do, dl) -- A

where is the unit square of IR2.Now, we claim the following proposition.

PRO’OSlTION 3.10. (i) Og is a continuous mappingfrom a,so to(ii) d(,,0, *, 0) 0.

From that, a contradiction follows (the index theory). This means that thereexists (do(g),dl(g)) such that Vs>so, q(do, di,s) VA(S); that is, q VA. Inparticular,

C(A)IIq(s)ll <

Using Proposition 3.1, this concludes the proof of Theorem 1.


Proof of Proposition 3.10. Step 1: (i). We have (qo, ql)(s) is a continuousfunction of (w(so),s) e H x [so, /) where w(so) is initial data for equation (4).Since w(so) (-q(y, so)+ qg(y, s0); cf. (33) and (14)) is continuous in (do, d1) (it islinear), we have that (qo, q)(s) is continuous with respect to (do, dx,s). Now,using the transversality property of (q0,q) on t317’,4 (Lemma 3.8), we claim thats.(do, d) is continuous. Therefore, Do is continuous.

Step 2: (ii). If (d0,dl)e 0o,so, then, according to the proof of Lemma 3.9,(qo, q)(so)O’,(so). Therefore, using q(so) Va(so) (Lemma 3.5), we haveq(so) OVa(so). Applying (ii) of Lemma 3.8 with so and s. so yields 60 > 0 suchthat ’6 (0, dio), q(so + 6) VA(SO + 6). Hence,

s,(do, dl =SO,

and 0(d0,dl) (s/A)(qo, ql)(So). Formulas (36) show then that there exists anaffine function A" IR2 IR2 such that * o AIo, Id[o,, which yields (ii). Thisconcludes the proof of Proposition 3.10.

Let us now prove Proposition 3.7.

3.3. Proof of Proposition 3.7. For further purpose, we are going to prove amore general proposition which implies Proposition 3.7.

PROPOSITION 3.11. For each . > O, there exists A-2(I) > 0 such that for eachA > A-2(), there exists 2(,A)> 0 such that for each so > 2(,A), for eachsolution q ofequation (15), we have the followin9 property:f

Iq.,(s0)[ < Asff2, rn 0, 1

Iq2(s0)l </s-2 log so,

[q-(y, so)l < s62(1 + lyl3),(37)

Ilqe(S)ll

iffor Sl > So, we have Vs [s0, s], q(s) Va(s), then Vs e [so, s],

[qE(s)[ < a2s-2 log s- s-3,

A(1 + lyla)s-2Iq-(y,s)[ <

a2

Proposition 3.11 implies Proposition 3.7. Indeed, referring to Lemma 3.5, weapply Proposition 3.11 with max(l, C). This gives A-2 > 0, and for each

162 MERLE AND ZAAG

A > A’2, 2(A, A). If we take s2(A) max(2(max(1, C), A), sl (A)) (cf. Lemma 3.5),then, applying Proposition 3.11 and Lemma 3.5, one easily checks that Proposi-tion 3.7 is valid for these values.

Proof of Proposition 3.11. The proof is divided in two parts.In the first part, we_give a priori estimates on q(s) in l(s)’ assume~ that for a

given A > 0 large, A > 0, p > 0, and initial time so > ss(A, A, p), we haveq(s) V(s) for each s [a, a + p], where a s0. Using the equation satisfied by q,we then derive new bounds on q2, q-, and q in [a, a + p] (involving A, A and p).

In a second part, we will use these new bounds to conclude the proof of Prop-osition 3.11.

Step 1. A priori estimates of q. Let us recall the integral equation satisfied byq (of. (19))"

q(s) K(s, a)q(a) + dz K(s, z)B(q(z)) + dz K(s, z)R(z), (38)

where

B(q) Igo + ql p-l(g + q) goP pgop-lq,

1 1 cOgoR(y, s) Ago - y.Vgo

P igo / gop cgs

and K is the fundamental solution of .’ (cf. (16)).We now assume that for each s [a, a + p], q(s) VA(S). Using (38), we derive

new bounds on the three terms in the right-hand side of (38), and then on q.In the case a so, from initial data properties, it turns out that we obtain

better estimates for s [so, so + p]. More precisely, we have the following lemma.

LEMMA 3.12. There exists A5 > 0 such that for each A > As, fl > O, p* > O,there exists ss(A, fl, p*) >0 with the following property: Vso>ss(A,.,p*),Vp < p*, assume Vs [o’, tr + p], q(s) V(s) with tr > so.

I. Case a > so. We have Vs [a, tr + p],(i) (linear term)

I :(s)llogS2

/ (S-- iv)GAs-3,

[_(y, s)[ < C(e-(1/EI(-IA + e-(-/A2)(l + lyi3)s-2,

where

IIoe(S)llzoo < C(A2e-(s-a)/p / Ae(S-a))s-1/2,

2

K(s, a)q(a) a(y, s) E am(s)hm(y) + a_(y, s) + ae(Y, S)m=O


(ii) (nonlinear term)

S3+1/2

Ifl-(y, s)l < (s-a)(1 + lyla)s-2-,

where

(p) > o,

and

2

dz r(s, z)B(q(z)) fl(y,s) flm(s)hm(y) + fl_(y,s) + fle(Y,S).m=O

(iii) (corrective term)

le=(s)l (s- )Cs-,ly-(y, s)l (s-tr)C(1 + lyl3)s-2,

IIw(s)llzo (s- O’)S-3/4,

where

2

dz K(s, z)R(., z) (y, s) /m(s)hm(y) + _(y, s) + Te(Y, S).m=0

II. Case tr so. Assume in addition that q(so) satisfies (37). Then, Vs[so, so + p], (linear term)

l2(s)log so

s2 + C max(A, A)(s- so)s-3

I-(y, s)l C(1 + lylS)s-2,

II(s)ll c/(1 -t-e(S-s))s-1/2

We will give the proof of this lemma later.

Step 2. Lemma 3.12 implies Proposition 3.11. Let A be an arbitrary positivenumber. Let A >/2(), where 2(/), will be defined later. Let so > 0 be chosen

164 MERLE AND ZAAG

larger than 2(A), where 2(A) will be defined later. Let q be a solution of equation(15) satisfying (37), and sl > so. Assume in addition that Vs e [so, Sl], q(s) Va(s).We want to prove that Ys [so, sl]

[q2(s)[ A2 log s 1 A A282 S3, [q-(y,s)l < s2( 1 + lY[ ), []qe(s)l[ioo < -. (39)

Let p > P2 two positive numbers (to be fixed in terms of A later). It is then enoughto prove (39), on the one hand for s-so < px, and on the other hand fors-so > P2. In both cases, we use Lemma 3.12. Hence, we suppose A > As,so > max(ss(A, ,, p), ss(A, , P2)).

Case 1: s- so < Pl. Since we have Vz [s0,s], q(z) l(z), we apply Lemma3.12 (II (i), I (ii), (iii)) with A, p* p, and p s- so.From (38), we obtain

log so[q2(s)[ z,S2 -- Cl(max(A, A) + 1)(s- s0)s-3 + (s- So)S-3-1/2

Iq-(y,s)[ < (C1/ nt- Cl(S- so))(1 + lyl3)s-2 + (s- so)(1 + lyl3)s-2-e (40)

Ilqe(S)llcoo (C+ Cles-S)s-1/2 + (s- so)s-3/4 d-- (s- so)s-(1/2)-e

To have (39), it is enough to satisfy

log so A2 log s

As_2C1As-2 q’- C1 (s so)s-2 - (41)

A2s-1/2C1s-1/2 -- Cl"eS-Ss-1/2 "-4-on the one hand and

Cl(max(A, d) + 1)(s- s0)s-3 + (s- so)s-3-/2 < A2 log s

2 s2

A s-2(s- so)s (42)

A2

(S- So)S-3/4 "+" (S- So)S-(1/2)-e "-S-1/2

on the other hand.


If we restrict Pl to satisfy ClPl < A/8, Cl.epl < A2/8 (which is possible if wefix Pl (3/2) log A for A large), and A to satisfy < A, .i < A2/2, C1.’ < A/8,and C1. < A2/8 (that is, A > A6()), then, since s so < Pl, (41) is satisfied. Withthis value of Pl, (42) will be satisfied if the following is true:

3CI(A + 1)-log As-3

3 As_3_12 A2 log s s-+ log < 2 S2

3log As-2- < A $-2

3 -(1/2)-e A2 -1/23log As-3/4 + log As < s

which is possible, if so > s6(A).This concludes Case 1.

Case 2: s- so >/92. Since we have Vz e [a, s], q(z) Fa (z), we apply Part I ofLemma 3.12 with A, p p* P2, a s- P2. From (38), we derive

IqE(s)! < A2 1og(s- P2)s2

+ C2AP2s-3 + C2P2s-3 + P2s-3-1/2

Iq-(y,s)l < C2(e-(1/2)p2A + e-p22A2 + P2)( 1 + lyl3)s-2 + P2( 1 + lyl3)s-2-" (43)

IIq,(s)ll < c2(A2e-p2/p + AeP2)s-1/2 + P2s-3/4 + P2s-(1/2)-.

To obtain (39), it is enough to have

o

C2(e-(1/2)p2A + e-P22A2 + P2) < A4 (44)

a2

C2(A2e-p/p + Aep:) < ---with

fA,m (S) A2 log ss2

$-3 [A2 log(s-/92)$2 + C2(A + 1)p2s-3 + p2s-3-1/2]

166 MERLE AND ZAAG

on the one hand and

As-2p2$-2-e < -_1_ A2 1/2p2s-3/4 + pEs < S-

(45)

on the other hand.Now, it is convenient to fix the value of/92 such that C2Aep2 A2/8; that is,

P2 log (A/8C2). The conclusion follows from this choice, for A large. Indeed,for arbitrary A, we write

-3fA, Iog(A/8C2) (S) S ( AA2 log 2- 1 C2(A + 1)log 2

CA2( A)2

S3+1/2 log

Then we take A > A7 such that

( AA210g2-1-C2(A+l)log2 />1

((A),-1/2 A) AC2 2 A + e-(lg(A/8C2))2A2 + log2 < "

We introduce s7(a) > 0 such that for s/> so/> s7(A), we haves-3-1/2CA2(log(A/8C2))2 < (1/2)s-3, and (45) is satisfied. This way, (44) and (45)are satisfied, for A > A7 and so > ST(A), which concludes Case 2.

We remark that for A > As, we have pl (3/2) log A > P2 log (A/8C2). Ifnow we take A2 sup(As, A6(), A7, As), and then s2 max(ss(A, , pl(A)),ss(A, , p2(A)), s6(A), s7(A)), then this concludes the proof of Proposition 3.7.

Proof of Lemma 3.12. Let A > A5 with A5 > 0 to be fixed later. Let A > 0;p* > 0. We take p < p* and so > ss(A,,p*). We consider tr > so such that/s [tr, tr + p], q(s)

_VA(S). For each part I (i), (ii), (iii) and II of the lemm_a, we

want to find ss(A, A, Po) such that the concerned part holds for so > ss(A, A, p*).The proof is given in two steps. First, we give various estimates on different

terms appearing in equation (19). Second, we use these estimates to conclude theproof.


Step 1. Estimatesfor equation (38)

(i) Estimates on K

LEIIA 3.13 (Bricmont-Kupiainen). We have

(a) Vs > z > l with s < 2z, Vy, x lR,

IK(s, , y, x)l < Ce(s-)’ (y, x), with

e(y,x) eo

X/’4n(1 e-)exp

4(1

(b) For each A’> O, A"> O, A’> O, p*> O, there exists s9(A’,A",A"’,p*)with the followino property: Vso > s9; assume that for tr > so,

Iqm(a)l < Ata-2, m O, 1,

IqE(a)l < A"(log a)a-2,

Iq-(y, r) A"(1 + lYl 3)a-E,(46)

then Vs [tr, tr + p*]

12(s)l A"1%s2 + (s- a)C max(A’, Am)s-a,

Io-(y, s)l < C(e-(/(-’IA" + e-(-’>A")(1 + lyl3)s-,II(s)ll,. C(A"e-(s-)/p + A"e(’-))s-1/2,

where

2

K(s, tr)q(tr) o(y, s) Z Om(s)hm(y) -I- _(y, s) -t- Oe(y,s).m=O

(47)

(c) Vp* > 0, 3slo(p*) such that Va > Sl0(p*), Vs [tr, tr + p*],

Ir:(s)l < (s- )Cs-,ly-(y, s)l < (s-a)C(1 + lyla)s-2,

168 MERLE AND ZAAG

where

2

dzK(s, z)R(z) V(y,s) E Vm(s)hm(Y) + 7_(y, s) + ye(y,s).tr m-’0

Proof. See Appendix A.

Using the above lemma and simple calculation, we derive the followingcorollary.

COROLLARY 3.14. We have Vs z 1 with s 2z,

I K(s, z, y, x)(1 + ]xl m) dx < Cle(’-)e(y,x)(1 +Ixl ’n) dx < e’-(1 (48)

(ii) Estimates on B

LEMMA 3.15. We have VA > 0, Sll(A) such thatimplies

IZ(Y, z)B(q(y, z)) < Clql 2 (49)

and

In(q)l Clql: (50)

with/ min(p, 2).

Proof. Let A > 0. If q(z) l(z), then [[q(v)ll= < C(A)"c-1/2 (1/2)f(2Ko),if z > Sll (A) (Gf. Definition 3.3, (7) for f, and (26) for K0). Then (49) and (50) areequivalent to (1), (2), and (3), with

(1) p/> 2 and IB(q)l < Clql 2,(2) p < 2 and IX(Y, z)B(q(y, z)) < CIq] 2,(3) p < 2 and IB(q)l < CIq[ p.We prove (1), (2), and (3).For part (1), we Taylor-expand B(q), and use the boundedness of Irpl and

Part (2) holds if Jr(Y, z) 0. Otherwise, we have lYl < 2KoVq Again, we Taylor-expand B(q): Z(Y,’r)IB(q)[ < CZ(y, z)lq[ 2 [(1 0)Itp + Oq[ p- dO, and concludewriting X(Y, s)[o + Oql p-2 < 7.(Y, s)(Iql- Iql)p-2 < (f(2K0)- (X/2)f(2Ko))P-2= C,

For (3), we write

B(q___) I1 4- [P-I(1 -- ) 1 pby setting q.

Iql p I1 p o

We easily check that this expression is bounded for --o 0 and .


(iii) Estimate on R

LEMMA 3.16. We have 3s12 > 0 7 s12,

CJR(y, z)[ < (51)

Proof. From (18) and (14), we compute

p- lfp + (p- 1)2

q)YY 2p---- 4Pz----T- y2f2p-1

tPsp- 1 x

and4Pz2

y2fp -k 2pz2

2 ytpy f + 2p(p 1)z p 1p-14pz

y2fp

-2p(p- 1)z + f + fP;

using a Lipschitz property and simple calculations, the conclusion follows.

(iv) Estimates on q in VaFrom Definition 3.3, we simply derive the following lemma.

LEMA 3.17. We have 3s13 > 0 /A > O, /z > s13; if q(z) Va(z), then

Iq(Y, z)l < CA2"c-2 log z(1 + [yl 3) (52)

and

Iq(Y, )] CA2z-1/2" (53)

Step 2. Conclusion of the proofofLemma 3.12. We choose so > p* in all casesso that if so < tr < z < tr + p and p < p*, we have tr-1 < 2s-1 and z-1 < 2s-.

I (i) linear term in I. We apply (b) of Lemma 3.13 with A’= A, A"= A2,and A"’ A. Take ss(A, p*) s9(A, A2, A, p*).

II linear term in II. We apply (b) of Lemma 3.13 with A’= A, A"= ., and

I (ii) nonlinear term

170 MERLE AND ZAAG

" fl2(S)" By definition,

fl2(s) J d#(y)k2(y)z(y, s)fl(y, s)

dlz(y)k2(y);t(y, s) d’c K(s, , y, x)B(q(x, )) dx I + II, where

I d/.t(y)k2(y))(y, s) dz K(s, z, y, x)jt(x, z)B(q(x, z)) dx, and

II d#(y)kE(y)x(y, s) dz K(s, z, y, x)(1 ;t(x, "c))B(q(x, z)) dx.

For I we write

II[ < dl(y)[k2(y)[ d IK(s, ,, y, x)lx(x, z)lB(q(x, ))[dx

< C dl(y)lk2(y)l dx IK(s, ,, y, x)llq(x, x)l 2 dx (of. (49))

< C dl(Yllk2(y)l d’c Ig(s, x, y,x)laar.-4(log 12(1 + Ix161 dx (of. (52))

J< CA4 dlz(y)lk2(y)l dr, -4(log z)2e’-(1 + lyl 6) (cf. Corollary 3.14)

< CA4 1 d/(Y)lkE(y)l(1 + lyl6)(s tr)tr-4(lg s)Ee’-

< CA4(s- r)eS-a(s/2)-4(log s)2 (we take so > p* so that

s < a + p* < a + so < a + a 2tr).

For II, we use (50) and (53) to have

J]III < C e-y2/4 dy;t(y, sllkE(y)l dr, dx(1- X(x,z))

ff’4n(1 e-(S-’O)(ye_(S_)/2 X)2] A2.C-/2exp4(1 e-(’-))

Now we have

e(1/2)[-y2/4-((ye-t/2-x)2/4(1-e-’))] e-C(ro) < e-c


for [y] < 2K0x/ and Ix[ > K0x/ (if so > p*). Hence, we derive

fill C e-x’/8 dylk2(y)I d’c dx(1- x(x, ))

V/4zr(1 e-(S-)l(y__e(s-’)/ -_x)] -csA2O,r,-O/2exp2 4(1 e-(S-)) ]

e

Using a variable change in x, and carrying all calculation, we bound [II[ by(s-tr)e-cs, for s > s14(A,p*). Adding the bounds for I and I1, and takingtr > sis(A, p*), we obtain the estimate for fl2(s).

fl_(y,s): Using (50), (52), and (48), and computing as before yields[fl(y,s)[ < CA2(s a)e(S-’)(1 + [yj3lt(log s/s2). If we multiply this term byand bound in it [y[3p-3 by (x/)op-o, we obtain [flb(Y,S)[ < CA2(s a)e(s-) x(1 + [y[ 3) (v/)3t-3 (log s/s2), hence ]flb(Y,S)[ < CA2t(s- a)e(S-)(1 + [y]3) x(log s)/s(+3)/2, which implies simply the estimate for

_(for a > Sl6(P*) and

some 81(p)).e(y,s): Using (50), (53), and (48), and computing as before yields

Ifl(Y, s)l < CA2(s- a)e(-’)s-(1/2). From this, we derive directly the estimate fore (for a > SiT(p*) and some e2(p)).

Finally, we take tr > max(s5, s16, s17)) ss(A,p*) and e min(e, e2) to havethe conclusion.

I (iii) corrective term. For ?2 and ?_, we use (c) of Lemma 3.13. For ?e, westart from (51) and write ?e(Y,S) (1 Z(y, s) )?(y, s) (1 X) r d’c dxK(s, r,, y,x)R(x, r,), and then as in I (ii), [Ye(y,s)[ < C [s dr, dx e(-(y,x)(C/z)

tr 3/4C (dz/t)e < (C/s)(s- a)e < (s a)s- if a > Sl0(p ).

4. Stability. In this section, we give the proof of Theorem 2. As in Section 3,we consider N 1 for simplicity, but the same proof holds in higher dimension.We will mention at the end of the section how to adapt the proof to the caseN>2.

4.1. Case N 1. Let us consider fi0 an initial data in H, constructed inTheorem 1. Let fi(t) be the solution of equation (1):

Ut Au-4- lulp-lu, u(O) ,o.

Let " be its blow-up time and a be its blow-up point.We know from (35) that there exists A > 0, go > log such that Vs > go,

?,a(s) e (s), where ?,a is defined in (13) by

l?,a(Y, s)= e-S/(P-x)fi(gt + ye-/2, ?- e-s) + 1 + (P 4ps-1)2 yE)’-l/(p-1)]

172 MERLE AND ZAAG

Remark. Following Remark 1.2, we can consider a more general fi0, that is,t0 with the following property: 3(’, a), 3,’[, g0 such that Ys > 0, 7,a(s) e l(s).From Definition 3.3, the definition of 7,a(s), and Proposition 3.1, t(t) blows

up at time " at one single point ti, and behaves as the conclusion of Theorem 1.We want to prove that there exists a neighborhood , of t0 in H with the fol-

lowing property: Vuo q/f,, u(t) blows up in finite time T at only one blow-uppoint a, where u(t) is the solution of equation (1) with initial data u(0)= u0.Moreover, u(t) satisfies

lim (T- t)l/(P-1)u(a + ((T- t)Ilog(T- t)l)I/2z, t) =f(z)t--,T

(54)

uniformly in z IR, with

f(z) (P-- l q-(P--1)2 )-1Z24pThe proof relies strongly on the same ideas as the proof of Theorem 1" use of

finite-dimensional parameters, reduction to a finite-dimensional problem, andcontinuity. For Theorem 2, we introduce a one-parameter group, defined by

T, a) - q1",a

where qT,a is defined by (13), for a given solution u(t) of equation (1) with initialdata u0. This one-parameter group has an important property: V(T, a), qT,a is asolution of equation (15). Therefore, our purpose is to fine-tune the parameter(T, a) to get (T(uo),a(uo)) such that qT(uo),a(uo)(S) VAo(S), for s > So, and A0 andso are to be fixed later. Hence, through the reduction to a finite-dimensionalproblem, we give a geometrical interpretation of our problem, since we deal withfinite-dimensional functions depending on finite-dimensional parameters through aone-parameter group.As indicated in the formulation of the problem in Section 2 and used in Sec-

tion 3 (Definitions 3.2 and 3.3), it is enough to prove the following.

PROPOSITION 4.1 (Reduction). There exist Ao > O, so > O, Do neighborhood of(,ti) in IR2, and o neighborhood of fo in H with the following property:Vuo o, (T, a) Do such that Vs > so, qT,a(S) VAo(S), where qT,a is defined by(13), and u(t) is the solution ofequation (1) with initial data u(0) uo. (We keephere the T, a) dependencefor clearness).

Indeed, once this proposition is proved, (54) follows directly from (3), (13), andDefinitions 3.2 and 3.3. Proposition 3.1 applied to u(x- a, t) then shows directlythat u(t) blows up at time T at one single point: x a.The proof relies strongly on the same ideas as those developed in Section 3,

and geometrical interpretation of T and a. Let us explain briefly its main ideas.


In a first part, as before, we reduce the control of all the components of q toa problem of controlling (qo, ql)(S), uniformly for u0 e and (T,a) D1 (where/i and D1 are, respectively, neighborhoods of t0 and (J", )).

In a second part, we focus on the finite-dimensional variable (q0, ql)(S), andtry to control it. We study the behavior of r,a under perturbations in (T, a) near

(, ) (and some topological structure related to these). We then extend the prop-erties of to q, for u0 near fi0. We conclude the proof, proceeding by contradictionto reach a topological obstruction (using the index theory).The constant C again denotes a universal one independent of variables, only

depending upon constants of the problem such as p.For each initial data uo, u(t) denotes the solution of (1) satisfying u(0) uo,

and for each (T, a) IR2, Wr,a and qT,a denote the auxiliary functions derivedfrom u by transformations (3) and (13).

Part I. Initialization and reduction to afinite-dimensional problem. In this sec-tion, we first use continuity arguments to show that for A, so large enough (to befixed later), for (u0, T, a) close to (t0, , a), qr,a is defined at s so and satisfiesqr,a(So) Va(so) (Step 1). After, we aim at finding (T, a) such that qr,a(S) in Va(s)for s > so. For this purpose, we reduce through a priori estimates the control ofqr,a(S) in Va(s) to the control of (qo,r,a, ql,r,a)(S) in (s) for s > So (Step 2).

Step 1. Initialization. We use here the fact that ,a(s) l(s) for any s > go,and the continuity of qT,a with respect to initial data u0 and (T, a), to ensure thatfor fixed so > 0, qT,a(SO) . V2,(s0), for (u0, T, a) close to (fi0, , a). Hence, if A islarge enough, we have qT,a(SO) VA(So) and qT,a(SO) is "small" in a way.

LEMMA 4.2 (Initialization). For each so > o, there exist 1 neighborhood ofgoin H and Dl(so) neighborhood of (,t) in lR2, such that for each uo 1,(T,a) D(so), q(T,a,s) is defined (at least) for s (-log T, s0], and qr,a(So)v2(o).Proof of Lemma 4.2. VT > O, Va . JR, qT,a(S) is defined on (-log T, +), if

T < J’, or (-log T,-log(T-’)), if T > . Therefore, qT,a(S) is defined on(-log T, so] for T near .

(i) Reduction to the continuity of qT,a(SO) . L(IR). Let so > go. It is enoughto prove that Ve > 0, there exist / and D such that /uo , (T, a) D,

(55)

Indeed, if it is the case, then

Vm (0, 1, 2}, Iqm,T,a(SO) m,,a(So)l Ce,

Iq-,T,a(y, SO)- _,,a(Y, So)l Ce(1 4-ly12),

(56)

(57)

Ilqe, T,a(So) e,La(so)]lLOO(n) < Ce. (58)

174 MERLE AND ZAAG

(56) and (58) follow directly from (55). For (57), write q_(y,s)= ;t(y,s)q(y,s)-2’m=0 qm(s)hm(y), and use (55) and (56). Using ,a(so) l(so) and taking e > 0

small enough yields the conclusion of Lemma 4.2.(ii) Continuity of qT,a(SO) L(IR). We have

qT,a(Y, SO) 7,a(y, SO)

WT,a(y, SO) ff,a(y, SO)

e-S/(P-){u(e-S/y + a, T- e-s) (e-S/2y + , "- e-S)}

e-s/(p-) {u(e-so/2y + a, T e-s) fi(e-S/2y + a, T- e-s)}

+ e-S/(P-){(e-so/2y + a, T- e-s) (e-S/2y + , T- e-S)}

+ e-S/(P-1){f(e-so/2y + , T- e-s) (e-S/2y + fi, - e-S)}.

Since

uou(t) e c (["’-e-2is defined and continuous (for uo near fi0), we have the conclusion.

Step 2. Uniform finite-dimensional reduction. This step is similar to Step 2 ofPart 1 in the proof of Theorem 1. Here we show that for ,4 and so to be fixedlater, if qr,a(So) is "small" in VA(S0), then the control of qr,a(S) in Va(s) for s > soreduces to the control of (q0,7",a, q,r,a)(S) in (s).LEMMA 4.3 (Control of q by (q0, q) in ). There exists A2 > 2.2. such thatfor

each A >/A2, there exists SE(A) > 0 such thatfor each so >/SE(A), we have the fol-lowin# properties:

(i) For any q, solution of equation (15), satisfyin#* q(so) V2i(s0 and,for sl > so, Vs e [so, sl], q(s) e VA(S),we have Ys e [so, si],

Iq2(s)l < A2s-2 log s- s-3

A 3)s_2Iq-(y, s) < (1 / lyl

A2

IIq,(s)ll, < -.


Moreover,(ii) For any q, solution of equation (15), satisfying

q(so) V2i(so)( VA(SO)).For s, > so. q(s) e VA(S) Vs e [So, S,]. andq(s,) 3VA(s,).we have (qo, qi)(s,) 3#a(s,). and there exists (5o > 0 such that V(5 (0, dio)(q0, ql)(S, + b) I,)A(s, + ) (hence, q(s, + ) q Va(s, + b)).

Proof. (i) We apply Proposition 3.11 with max(2, (2)2), and takeA2 max(A-2, 2[), and s2(A) max(0 + 1, 2(, A)) to have the conclusion.

(ii) We apply (i) with sl s,, and use Definition 3.2. Then we apply Lemma3.8.

Part II. Topological argument. Below we use the notation qT,a(S)=q(T, a, s), qT,a(Y, S) q(T, a, y, s), qm,T,a(S) qm(T, a, s).

In Part I, we reduced the problem to a finite-dimensional one: for each u0close to fi0, we have to find a parameter (T,a) (T(uo),a(uo)) near (#,a) suchthat (qo, q)(T,a,s) Va(s) for s > so. We first study the behavior of (l(T,a) for(T, a) close to (, a). Then we show a stability result on this behavior for u0 neart0. Therefore, for a given u0, we proceed by contradiction to prove Proposition4.1, which implies Theorem 2.

Step 1. Study of (t(T, a). We study the behavior of (T, a) for (T, a) close to(]b, d)in IR2.PROPOSITION 4.4 (Behavior of (T, a) near (7, )). There exists A4 > 0 such

that for each A > A4, there exists s4(A)> 0 with the following property. Foreach so > s4(A), there exists D4(so) neighborhood of (,) such that for each(T,a) D4(so)\{(,gt)}:

(i) O( T, a, s) is defined for s e (-log T, so] and O( T, a, so) e VA (So),(ii)s.(T,a)>so such that Vse[so, s,(T,a)], (T,a,s) eVa(s), and

gl(T.a.s,(T.a)) e OVA(s.(T.A)). and if we define

tIJo D4(s0) \{(’, a)} -- ]R2

g,(T. a)2 (59)(T,a) -- A (O’01)(T’a’*(T’a))’

then Im(Wao) . where c is the unit square of JR2.Moreover.

(iii) tPa is continuous;(iv) Ve > 0. there exists a curve F, D4(so) such that d(F,,Wao,0)=-1, and

V(T, a) e F,, I(T,a) (,gt)l < e.

Proof. To prove (i), (ii), and (iii), we take A > A5 with A5 max(2[, A2, A3),so > ss(A) max(0 + 1,s2(A),s3(A)), Ds(s0) Dl(So) (with the notation ofLemma 4.2). For such A and so, we can apply Lemma 4.2 and Lemma 4.3.

176 MERLE AND ZAAG

Proofof (i). By Lemma 4.2, V(T, a) e Ds(so), t(T, a, s) is defined (at least) fors (-log T, s0] and gl(T,a, so) V2,i(s0 c (s0), which proves (i).

Proof of (ii). We claim that V(T,a) eD5(so)\{(’,a)}, s(T,a) > so suchthat (T, a, s) (s). Indeed:

Case 1: T > T. Since (T, a, y, s) e-S/(P-1)f(a + ye-s/2, T- e-s) qg(y, s),(T, a, s) is defined on [so, -log(T )) and not after. Suppose that (T, a, s)does not leave l(s) for s[so,-log(T- )); then, VyelR, /s[so,-log(T-I(T, a, y, s)[ < C(A)/x/q (of. Definition 3.3).

Since t(x, t) (T- t)-V(p-) (t(T, a, (x- a)/(v/T t),-log(T- t)) + tp((x- a)/(x/’T- t),-log(T- t))), lim suPt7 Ila(t)ll,m < c,,a < /. This contradictsthe fact that t(t) blows up at time .

Case 2: T < and (T,a)# (,). gl(T,a,s) is defined on [s0,+o). Sup-pose that (T,a,s) does not leave l(s) for st [s0,+o). Then, VyIR,Ys[s0,+oo), I( T, a, y, s)l < (C(A)/v) (cf. Definition 3.3). Hence, by (13),lim,r II(Z t)l/<P-)u(a / x/(T t)llog(T t)lz, t) -f(z)ll. 0, and fromProposition 3.1, u(t) blows up at time T at one single point; x a. Since(T, a) (, ), we have a contradiction. Therefore, (T, a, s) leaves Va(s) fors > so.

In conclusion, we derive V(T, a) e D \{(, ti) },[so, s.(T,a)], (T,a,s) Va(s) and (T,a,s.(T,a)) tgVa(s.(T,A)). (.(T,a) > sosince (T, a, s) is in V2,i(so), which is strictly included in (s0).) If now we define

Wao by (63), then we see from Lemma 4.3 that Im(Fa0)

Proof of (iii). Let (T,a) Ds(s0)\(,d). We have explicitly for m 0, 1:

glm(T,a,s) I dla km(y)g.(y,s)gl(T,a, y,s)

I dl km(y)z(y, s)e-S/(P-)f(a + ye-s/2, T- e-s)

I d/a km(y)z(y, s)cp(y, s).

From the continuity of u(x, t) with respect to (x, t), and (ii) of Lemma 4.3,g.(T,a) and (g.(T,a)2/A)(qo, q)(T,a,.(T,a)) are continuous with respect to(T,a).

Proof of (iv). Let > 0. We now construct r’ satisfying d(F,Wo, 0)=-1and V(T,a) r’,s,, [(T,a)- (,)1 < . This will be implied by the followinglCmma.

Lh 4.5. There exists A6 > 0 such that A > A6, 3s(A) > 0 satisfyina the


followin9 property: Vso > s6(A), 3D6(so) neiohborhood of (’, t) such that Ve > O,3sl(A, e, so) > so, F, a 1-manifold in D6(so) satisfyino:

V(T, a) F, I(T, a) (, fi)l < e, Vs [so, s1], (T,a,s) Va(s),

(0, I)(T, a, Sl)

d (l"e, (o, 1)(.,., Sl), O) -1.(60)

(a) Proof of Lemma 4.5. The proof is not difficult, but it is a bit technical.See Appendix B for more details.

(b) Lemma 4.5 implies (iv). Let A4 max(As, A6), and A > A4. Let s4(A)max(ss(A),s6(A)), and so > s4(A). Let D4(so) Ds(so) c D6(so), and > 0.

Then, according to the beginning of the proof of Proposition 4.4, (i), (ii), and(iii) hold. We take now Sl Sl(A,e.,so) and F. By Lemma 4.5, we see that(T,a) F,, s.(T,a) Sl, and tPao(T,a (s/A)(o,a)(T,a,s). From (60), wederive d(Y’, Wag, 0) -1, which concludes the proof of Proposition 4.4.

Step 2. Behavior of q(T,a) for uo near rio. Now, we fix Ao= 1+sup(2A, A2, As, A4), and then so so(Ao) sup(go, s2(Ao), s3(Ao), s4(Ao)). Apply-ing Lemma 4.2 gives us U and D1 (so). We then fix Do Dl(So) c D4(so). Apply-ing Proposition 4.4 with so and eo > 0 small enough gives us the curve Fo Fo,included in Do. We consider now Fo as fixed.Our purpose is to show that for uo near rio, the behavior of q(T, a) on the

curve Fo F(fio) is the same as (T,a). More precisely, we have the nextproposition.

PROPOSITION 4.6 (Stability result on the behavior on Fo, for uo near rio).We have Ve > O, / c/i, neiohborhood of fig such that Vuo "//, V(T, a) Fo:

(i) q(T, a, s) is defined for s (-log T, so] and q(T, a, so) VAo (so);(ii) 3s.(T,a)>so such that Vs[so, s.(T,a)], q(T,a,s)VAo(S), and

(qo, ql)(T,a,s.(T,a)) Oo(s.(T,a)). Then we can define

tPuo

(T,a) s,(T,a) 2

Ao(qo, ql)(T,a,s,(T,a))

(61)

where cg is the unit square of IR2.Moreover,

(iii) tPu is a continuous mapping from Fo to Ocg;(iv) II’I’u01r0 q’aolr0 [IL(R)(ro) < e"

Proof. We first show a local result; then, by compactness arguments, weconclude the proof. We claim the following.

178 MERLE AND ZAAG

LEMMA 4.7 (Punctual stability on Fo). We have Ve > O, V(T,a) Fo, D,,aneighborhood of (T,a) in Do, /,T,a neighborhood of fro in r such that:V( T’, a’) D,,T,a, VUo t/,T,a:

(i) q(T’,a’,s) is defined (at least) for s e(-log T, so] and q(T’,a’,so) 0(s0),

(ii) 3s,(T’,a’)>so such that Vs[so, s,(T’,a’)], q(T’,a’,s)Vao(S), and(qo, q)(T’,a’,s,(T’,a’)) tf"ao(s,(T’,a’)).

Moreover,

s,(T’, a’) 2

(qo, ql)(T’, a’, s,(T’, a’)) s,(T’, a’) 2

Ao Ao (o,)(T’,a’,,(T’,a’))

(62)

We remark that Proposition 4.6 follows from Lemma 4.7. Indeed, for e > 0,from the lemma we write

Io U D,T,a(T,a) eFo

and using the compactness of F0, we have the conclusion.

ProofofLemma 4.7. We have explicitly for u0 H, s (-log T,-log(Tif T > ; otherwise s (-log T, +), and m 0, 1, qm,r,a(S) f dl kin(y) xZ(Y, s)q( T, a, y, s) f d# km(y)z(y, s)e-S/(P-1)u(a + ye-s/2, T e-s) d# kin(y) xZ(y,s)q(y,s). Therefore, using the continuity of u(x,t) with respect to (u0,x, t),(qo, ql)(T,a,s) is a continuous function of (u0, T,a,s). Using this fact and thetransversality of (0, I)(T, a, L(T, a)) on 0 (s,(T, a))(Lemma 4.3 (ii)), (i)and (ii)then follow easily.

This concludes the proof of Proposition 4.6.

Step 3. The conclusion of the proof. From continuity properties of thetopological degree, there exists el > 0 such that VW C(F0,1R2) satisfyingtI’ Va0 II(r0) < 81, we have d(F0, W, 0) -1.Applying Proposition 4.6, with e el, we have Vu0 //1, d(F0, Wuo, 0) -1.We claim that the conclusion of Proposition 4.1 follows with Ao, so, Do, and. Indeed, by contradiction as in Section 3: suppose that for uo , we

have V(T,a) Do, there exists s > so, q(T,a,s) q Vao(S). Let s,(T,a) be theinfimum of all these s’s. We now remark that Wu0 is defined on Do (Lemma 4.2and Lemma 4.3). Wuo is continuous from Do to t3c (see the proof of Proposition4.4 (iii)), and d(F0, W0,0) 0, which is a contradiction. Hence, Proposition 4.1 isproved, which concludes the proof of Theorem 2.

4.2. Case N > 2. Let us consider fi0 an initial data in H, constructed in


Theorem 1. Let fi(t) be the solution of equation (1):

u Au + lu]’-u, u(O)

Let J" be its blow-up time and d be its blow-up point.Although the proof of Theorem 1 was given in one dimension, we know that

there exists > 0, 0 > log J" such that ’s > 0, 7,a(s) (s), where:,a is defined in (13) by

qLa(Y, s) e-S/(P-1)fi(d + ye-s/2, - e-s)

Nx (p- 1)2 lyl 2+ p-l+ 4p------

andDefinitions 3.2 and 3.3 are still good to define V2(s), if we understand qm(S)to be a vector-valued function, as defined in Section 2 (see (27) and (28)), andIqm(S)l to be the supremum of all coordinates of qm(S). (By the same way, thedefinition of (s) given in 3.4 is good here.)

With these adaptations, our purpose is summarized in the following proposi-tion, analogous to Proposition 4.1.

PROPOSITION 4.8 (Reduction). There exist Ao > O, so > O, Do neiohborhood of(J’,d) in IRl+n, and o neiohborhood of fi0 in H with the followin9 property:Vuo o, 3(T, a) Do such that /s > so, qT,a(S) Vao(S), where qT,a is defined by(13), and u(t) is the solution of equation (1) with initial data u(O) uo.Indeed, once this proposition is proved, from (3), (13), and Definitions 3.2 and

3.3, we have

lim (T- t)I/(P-1)u(a + ((T- t)Ilog(T- t)l)a/2z, t) =f(z)t---T

uniformly in z IRn, with

f(z)= (p_ l + (P_ l)2 )-U(p-1)4"-’-- [Z[2

Proposition 3.1 (which is true in N dimensions) applied to u(x- a, t) then showsdirectly that u(t) blows up at time T at one single point: x a.

Formally, the proof in the case N > 2 and in the case N 1 have exactly thesame steps with the same statements of propositions and lemmas, under the fol-lowing obvious changes:

180 MERLE AND ZAAG

(, ), (T, a) and (T’, a’) are in IRa+N, and every neighborhood of such apoint is a neighborhood in IRa+U;in Part 2, c denotes the unit (1 + N)-cube of IRa+N, F (and F, F0,...) is aLipschitz N-submanifold of IRa+N, forming the boundary of a bounded con-nected Lipschitz open set of IRa+N, and all introduced topological degreesdifferent from zero are equal to (-1)N.

Moreover, the proofs can be adapted without difficulty to the case N > 2,even:

the proof of Proposition 4.4, which relies on results of Section 3 (Subsection3.3 and Lemma 3.8) that are true in N dimensions (in particular, the Lemma3.13 of Bricmont and Kupiainen, with the adaptation IR IRN);the construction of F given in Appendix B can be simply adapted to thecase N > 2.

APPENDICES

Appendix A. Proof of Lemma 3.5. In this appendix, we prove Lemma 3.13.Equation (15) has been studied in [4]; hence, our analysis will be very close to [4](the proof is essentially the same as in [4]). Lemma 3.13 relies mainly on theunderstanding of the behavior of the kernel K(s, a, y, x) (see (20)). This behaviorfollows from a perturbation method around e(S-)’(y, x).

Step 1. Perturbation formula for K(s,a, y,x). Since is conjugated to thex2/8 xharmonic oscillator e- -We /8 2 (x2/16) + (1/4) + 1, we use definition

(20) of K and give a Feynman-Kac representation for K:

K(s, a, y, x) e(S-*)e(y, x) dyx (og)e0 v(o(,),+,)d, (63)

where d# is the oscillator measure on the continuous paths a)" [0, s- a] IRwith 09(0)= x, o)(s- a)= y, i.e., the Gaussian probability measure with cova-riance kernel F(z,

090(z)ogO(Z’) -4- 2(e-1/21"-’’1 e -1/21"+’’1 -I- e -1/212(s-r)-’c’+’cl e -1/212(s-’r)-’r’’-’cl

(64)which yields d#-%)(z)= co0(z)with

too(z)= sinhS-a z s-a-z

2 ysinh+xsinh

We have in addition

e0e’(y, x) V/4n(. 1 e-)(ye_O/2_ X)2]exp 4(1_-_] -].


Now, we derive from (63) a simplified expression for K(s, tr, y, x) considered asa perturbation of e(S-)e(y, x). To simplify the notation, we write from now on(, tp)for J" d#(y)p(y)tp(y).LEMMA A.1 (Bricmont-Kupiainen).

kernel K(s, tr, y, x) satisfiesWe have Vs > tr > 1 with s < 2tr, and the

(1 )K(s, tr, y,x) e(S-’r)’Z(y,x) 1 +-Pl(s, tr, y,x) + PE(s, tr, y,x)$

where P is a polynomial

Px (s, tr, y, x) Em,n>O,m+n<2

Pm,n(S, tr)ymxn

with IPm,n(S, tr)l < C(s- tr) and

IP2(s, tr, y, x)l < C(s a)(1 + s tr)s-2(1 + lYl + Ix[)4.

Moreover, I(k2, (K(s, tr) (trs-1)2)h2)] < C(s- tr)(1 + s- a)s-2.

Proof. See Lemma 5 in [4].

Step 2. Conclusion of the proofofLemma 3.13

Proofof (a). From (16), it follows easily that V(y, s) < Cs-. Using this esti-mate and (63), we write

IK(s, , y, x)l < e-le(y, x) I d#-;(9)eff’c+O-’ a< e(-)’Z(Y’X) I dla-(()(sx-)c < Ce(-)a"(Y’X)

since s < 2z and d/zx is a probability.

Proofof (c). See Lemma 2 in [4].

Proof of (b). We consider A’ > 0, A" > 0, A" > 0, and p* > 0. Let so > p*,tr > so, and q(tr) satisfying (46). We want to estimate some components of(y, s) K(s, a)q(tr) (see (47)) for each s e [tr, tr + p*].

Since tr > so > p*, we have Vz e [tr, s], < s < 2z. Therefore, up to a multi-plying constant, any power of any z e [tr, s] will be bounded systematically by thesame power of s during the proof.

182 MERLE AND ZAAG

(i) Estimate of 9.(s).

2(s) (k2, Z(., s)K(s, tr)q(tr))

aEs-EqE(a) + (k2, (Jr(., s) ;t(., a))aEs-Eq(a))

+ (k2, Z(. ,s)(K(s, tr) tr2s-2)q(tr)).

From (46), (21), and (26), we have la2s-2q2(a)l Arts-2 log tr and

I(k2, (z(. s) X(. a))r2s-2q(a))l < Ce-C’ra-3/2(s a)a2s-2 max(A",A"’)

CAt(s O’)S-3 for a > so > sl(A’,A",A’,p*).

We write (k2,jt(.,s)(K(s,a)- a2s-2)q(a)) as r2__o br + b_ + be, where br(k2, X(. ,s)(K(s, tr) a2s-2)h,.)qr(a), b_ (k2, X(. ,s)(K(s, tr) tr2s-2)q_(a)), andbe (k2, Z(. s)(K(s, a) tr2s-2)qe(tr)).For r 0 or 1, we use Lemma A.1, Corollary 3.1, (21), (46), the fact that

e(S-r)’hr e(1-r/E)(s-r)hr and (kE, hr) 0, and derive Ib,I- I(k2, z(. ,s)(K(s, a)-e(S-’O’)hr)qr(a) + (k2, Z(., s)(e(s-*)’- trEs-E)hr)qr(tr)l < CA’(s a)s-3 + Ce-cs

(s ca’(s-We have by Lemma A.1 and the same arguments [b21--[(k2,(K(s,a)-

r2s-2)h2)q2(r) / (k2,(-1 / X(.,s))(K(s,a) a2s-2)h2)q2(a)[ < C(s- a)(1 + s-a)s-2A"s-2 log s+ Ce-Cs(s a) < CA’(s a)s-3 if tr > so > s2(A’, A", p*).We can treat b_ exactly as bo; it is bounded by C(s a)Ams-3.Since K(s, a) a2s-2 K(s, a) e(s-)’+ (e(s-)’’- 1) + (1 a2s-2), we

write be be,1 -+" be,2 -1- be,3 with be,1 (k2, X(., s)(K(s, tr) e(S-’O’)qe(Cr)), be,2(k2, z(. ,s) f- dz .e*’qe(tr)), be,3 (k2, z(. ,s)(1 -tr2s-2)qe(tr)).From (46), we bound be,3 by C(s-tr)s-lAt’o’-l/Ee-Ca< C(s-tr)A’s-3 if

tr > so > sa(A’, A", p*). Since is selfadjoint,

Ie-y2/4 Is-rl e(S-’OIbe,2l< dy.W(k2z(.,s))(y) dr. dx

0 V/4(1 e-1)

x expl-(Ye-/2_)2.]A"tr-1/24(1

Now, we have e(1/2)[-(y2/4)-((ye-*/2-x)2/4(1-e-’))] < e-c(tc)s < e-2s, for lYl <


2Kox/ and ]x] > Ko/ (if Ko is big enough and so > p*). Hence,

Ibe,2[ CA"s-1/2 e-y2/8 dy dz dx0

exp4(1

CAtts-1/2(s t)e-s CA’(s r)s-3 if a > so > s4(A’,A", p*).

Using these techniques and Lemma A.1, we bound be,1 in the same way.Adding all these bounds yields the bound for l2(s)l.(ii) Estimate of_ (y, s). By definition,

a_(y,s) P-(I.(. ,s)K(s, a)q(a)) P-(I.(. ,s)K(s, a)q_(a))

2

+ Z qr(tr)P_(z(. ,s)K(s, tr)hr) + P-(Jt(. ,s)K(s, tr)qe(tr)) (65)r=0

where P_ is the L 2 (JR, d#) projector on the negative subspace of (see Subsection2.2). To bound the first term, we proceed as in [4]"

K(s, a)q_(a) J dx eX2/4K(s, tr,., x)f(x) (66)

where f(x) e-X2/4q_(x, a).N(y, x)E(y, x) with

From Step 1, we have eX2/4K(s, a, y, x)

-1/2 a x 4 (s )/2x2 41 (s )N(y,x) [4zr(1 e-(s-")] e e e-(y--- (67)

and E(y,x) d#-q(co)ef-‘V(()’+)d. Let f0 =f and for m > 1, f(-m-1)(y)fY-oo dx f(-m)(X). From (46) and the following lemma, we can bound f(-m).LEMMA g.2. We have If(-)(y)l < CA’s-2(1 / lYla)3-e-Y’/4.Proof. See Lemma 6 in [4].

By integrating by parts, we rewrite (66) as

2

(K(s,a)q_(a))(y) OrxN(Y,X),OxE(y,x)f(-r-1)(x) dxr--O

J C93xN(Y, x)E(y, x)f(-3) (x) dx. (68)

From (67), we get, for s-tr > 1 and re {0,1,2,3}, IOxN(y,x)l < Ce-r(s-’r)/2X(1 + lYl + IXl) ex2/4e(s-’)’z(y, X).

184 MERLE AND ZAAG

Using the integration by parts formula for Gaussian measures (see [11]), wehave

1 s-,r

dr. d’r’OxF(’c, z’) d/;*(eo)V’(og(v), tr + z)o(y,x) = 0 0

I$--0" I+01 d xF(z, z) d#(o)V"(og(’c), tr + z)ef0 a"v(o("),,+")

(69)

By (16), we have V(y,s) < Cs-1 and [dnV/dyn] Cs-n/2 for n 0, 1,2. Combin-ing this with (64) and using s < 2tr, we have

d#S_,(og)ef d"v(o(e’),+") Cyx and

}xE(y,x)l < Cs-(s a)(1 + s r)(ly + [xl).

Using (46), (68), and all these bounds, we get I(K(s, tr)q_(a))(y)l <CA,,,s_2e_(_,)/2(1 + [y[3) if tr > so > ss(p*) and s- a > 1. This yieldsI(P-z(. ,s)K(s, tr)q_(tr))(y)] < CA’s-2e-(S-’)/2(1 + lyl 3) if s- tr > 1. For s- tr <1, we use directly Lemma A.1, Corollary 3.1, (46), and C < e-(s-’)/2 to get thesame estimate.Now, we consider the second term in (65) (r 0, 1,2). From Corollary 3.1,

Lemma A.1, and the fact that lyl < 2Kosl/2, we obtain

[q(r)(Z(., s)g(s, tr)hr) (y) qr(tr)e(S-a)(1-r/2)(Z(., s)h,.)(y)l

< Cmax(A’,A")s-3+/2 log s.(s- ix)(1%- s-tr)eS-’(1 + lyl3). (70)

Hence, P_ {qr(tr)(it(. s)K(s, tr)hr)(y) qr(tr)e(s-’r)(1-r/2) (Z(., s)hr)(y) } satisfies thesame bound. Since P_hr=O and [(1-Z(.,s))h,[ < Cs-/2(l/ly[3), we canbound qr(tr)e(S-’)(1-"/2)P_(it(.,s)h,.) by (70). Hence, the second term of (65) isbounded by CA"s-2e-(S-’r)/2(1 + lyl 3)if tr > so > s6(A’,A",A’,p*).For the last term in (65), we use (46) and (a) of Lemma 3.5 to get

[[(1 4-lyl3)-lz(.,s)K(s, tr)qe(tr)l[LoO < CA"eS-as-1/2sup(1 -b [yl3)-y,x

4(1 e-(S-)) ]x it(Y, tr + (s a))(1 it(x, tr))

STABILITY OF THE BLOW-UP PROFILE

f CAt’s-2 s- tr < toe-s s-a > to

185

for a suitable constant to. This yields a bound on the last term in (65), which can bewritten as CA"e-(S-’)2s-2(1 + ly[ 3) for tr > so large enough.

Hence, combining all bounds for terms in (65), we have

I-(y, s)l < Cs-2(A’"e-(s-)/2 + A"e-(S-)2)(1 + [y[3).

Estimate of e(y,s). We write e(y,s) (1 Z(y,s))K(s, tr)q(tr)(1--7.(y,s))K(s,a)(qb(a) +qe(r)). From (46) and Corollary 3.1, we haveIqb(y,s)l < CA"-1/2 and 11(1- X(Y,s))K(s,)qb()llc < A"eS-s-1/2 if a >so > s7(A’, A", A").LEMMA A.3 [4]. We have IIK(s,)(1 x())ll, < Ce-(S-)/P.

Using (46) and the above lemma from [4], we have

I1(1 X(y,s))K(s, a)qe(a)llL A"e-(S-*)/Ps-1/2,

which yields the conclusion.This concludes the proof of Lemma 3.13.

Appendix B. Proof of Lemma 4.5. Let us recall Lemma 4.5.

LEMMA B.1. There exists A6 > 0 such that VA > a6, 3s6(A)> 0, satisfyin#the followin# property: Vso > s6(A), :lD6(s0) neiohborhood of (,) such thatVe > O, sl(A,e, so) > so, F, a 1-manifold in D6(s0) satisfyin#, V(T,a)](T, a)- (, a)l < e, Vs [s0, s], (T,a, s) (s),

(to, ll)(T, a, sl) cODA(s1), (71)

d(l-’t, (o, 1)(. s1), O) -1. (72)

In this lemma, we want to control the evolution of (T, a, s) in 1 (s), for (T, a)close to (J’, a). Hence, in a first step, we use ,a(s) e l(s) s > 0, to give esti-mates on different components of T,a(S), for (T, a) near (, ). From these esti-mates, we introduce a function (o,?11)(T,a,s) close to (flo,l)(T,a,s), but muchmore simple, and show that (0,1) satisfies properties analogous to (71) and(72). Therefore, we extend this result to (0, 1), by continuity, and then finish theproof of Lemma 4.5.

186 MERLE AND ZAAG

Step 1. Asymptotic development of (T, a) for (T, a) near (J’, d).(13) and (3), one time to (, i) and one time to (T,a), we write

Applying

( Y+l(T, a, y, s) (1 z)-l/(p-1)t ,t,x/’l z,s-log(l-

+ ( (1--’c)-l/(p-1) (p-- l-+(p 1)2 (y + z)2

4p(1 z)(s log(1 z))

+ { (l z)_i/(p_) tc x.__ ) (73)2p(s- log(1- z)) 2ps

with z (T- #)e and (a-d)es/2. Now we use (J, , s) (s) for s > 0,to give a development of (IT,a(Y, S), when I*1 < 1/2, and I1 < 1/2.LEMMA B.2 (Development of (T, a) near (, d)). There exists s7 > 0 such

that Vs > ST, V(T, a) IR2 satisfying [(T- )el < 1/2 and [(a- i)e/2l < 1/2, wehave

(logs z"c2 02 _ls)lo( T, a, s) Oo( T, a, s) + 0

\ s5/2 + -- + +

/ loggll(T,a,s) {ll(T,a,s) + O{

s

s2\

tX2 "C log s++-+ )s s s3

Oglo(T,a,s) 0 (T,a,s) + eS(O(’c + s-t/2)),-

(1)1 (Z, a, s) 11 s) + eSO -O-- -g- r, a,

(ll r, a, s 1 s) + eS/2 ([z__[s 1 [_)Oa -a T’ a’ 0 + +

(74)

(75)

(76)

(77)

(78)


with

glo( T, a, s5r

p-1

ll(T,a,s)s 2p

(79)

and T- )e and (a- )es/2. Moreover,

12(T,a,s)l < Clg ss2

1I_(T,a,y,s)l < C(1 + lYl ) $3/2

CI(T,a, y,s)l < -.

Proof of Lemma B.2. The idea is simple: for s > go, we try to express eachcomponent of (T, a) in terms of the corresponding component of (, a), andbound the residual terms using (, ti, s) e l(s) and other estimates that followfrom it. Hence, we first give various estimates following from t(,a,s)e l(s),and then we prove only some of the estimates in Lemma B.2, since the otherestimates can be obtained in the same way.

(i) We write the estimatesfollowingfrom 1( ’, , s) VA(s).LIMMA B.3 (Consequences of (,,s) l(s)). We have 3s16 > O, Vs > s16,

CI(, , y, s) < , (80)Vs

Ib(, fi, Y, s)l < C log s

s2 (1 + lyl (81)

0(,,s) 5x ()8--+ 0 (82)

11(’, , s)l < C lg---s (83)$3

< C1 +lY_._____[. (84)

188 MERLE AND ZAAG

Proof of Lemma B.3. Estimates (80) and (81) follow directly from Definition3.3.

After some simple calculations, we show that f d# )(y,s)R(y,s) (5x/8ps2) +O(s-3). As in the proof of Lemma 3.8, we write the equation satisfied by q0(s):

5x (log)d,od (’’) 0(,,) +8-+ o\ 3

which implies (82).By the same way, we write

dO (, , s)1 flog s),/--7 gO(,a,s) + o

k s3

which yields (83).From (80), we derive that r c3q/c3s satisfies

dr d2r 1t3 t3y2 2Yy + A(y, s)r + D(y, s),

with IA(y,s)l c and, if p > (3/2)lD(y,s)l < C/s;C/sp-(1/2) By parabolic regularity, (84) follows.

otherwise, IO(y, s)l <

(ii) Proof of some estimates in Lemma B.2: (74) and (75). (The other esti-mates follow from similar techniques.) From (73), we have glo(T,a,s)= 11 +I2 + 13, with

11 (1 z)-1/(v-*) d#(Y)Z(y, s)t , t, xi z,s log(1 z

I2 (1- z)-l/(p-1) I d#(y)it(y s)(p- 1 + (p 1)2 (y + 002 -l/(p-1)4p(l z)(s- log(1 z))]

I d#(Y)Z(Y,s) (P -1+I3 I d#(Y)X(Y,S){ (1- z)-I/(P-1) 2p(s-log(1-z)) 2ps

13: We have easily 1131 < CJls-l,


12: Since all quantities appearing in 12 are bounded, we can write

I2 O(e-S) + I d#(Y) {(P- l + (p 1)2(y -4- 002 -1/(p-1)4p(1 Z)(S- log(1

+ dg(y) p- 1 +p-1

l { (P- 1):(y + ):O(e-s) + o(z2) + d#(y)

4p(1 z)(s log(1 z))

}(p 1)2(y A- 002 -l/(p-1)

4p(1 "c)(s- log(1

(p- 1)2y2 / -14ps p- 1

(p 1)2y2-1-1/(p-1))

(P- 1)2y2}2)4ps

(p 1)2 (y -4- z)2 -l/(p-4p(1 z)(s- log(1

([ { (P- 1)2(Y + )2-4- 0 d(y)

4p(1 "c)(s- log(1 z))

+ +p-1 p-1

Hence,

I2 p-- 1 < Ce- + C’c2 -[- Cll s-1 ._ C02s-1 _[_ CT,2s-2 ..[_ C02s-2 "4- C04s-2

Therefore, [I2 T,x/(p 1)l < Ce-s + Cz2 + Clzls-a -4- C02S-1.11" Using (80), we write I1 O(zs-1/2)+ fd#(y)7,(y,s)(’,d, (y + )/v/1 z,

s log(1 z)). If we introduce a new integration variable z (y + )/v/1 z, weobtain I1 O(zs-1/2) + L1 + L2 with

exp(-(zv/1 z)2)4fL1 dz,jZ(z, s- log(1 log(1 z)) 4n

190 MERLE AND ZAAG

and

L2 J{X(zv/1 z a,s) X(z,s log(1 z)))l(J’,a,z,s log(1 z))

exp(-(zv/1 z02)4x dz.

4rr

L I Z(z, s log(1 z))O(, , z, s log(1 z)) 47gexp (z1/42)

2zx/1 z t2.) dzx exp4

O(zs-/2) + J;t(z,s-log(1 zlll(’,d,z,s-log(1 z))

2zx/’l z 02 1 20zx/’i " 02.+ 4

Using (81), we obtain L1 O(’cS-1/2) "+"/0(’,a,$-- log(1 z)) + tl(,tis-log(1-z))+O(o2s-2 logs). By (82) and (83), we have L1 =-(5x/8ps2)/ o(s-/2) / o(s-) / o(2s-2 og s), IL21 < C l(zV’l ) /vzlx/(- log(1 ))1 (lOb(, 8, z, s- log(1 ))1 + I,(, a, z, s- log(1 ))1) xexp(-Cz2) dz. Using (81) for qb, (80) for qe, and the fact that qe =- 0 for Izl < KoVqyields L2 --< C(l’ls-1/2 q-IOlS-1/E}(s-Elog S + e-S). In conclusion, I1 --(5tcl8ps2) -FO(zs-1/2) + O(s-5/2) + O(02s-Elog s). Adding 11, I2, and I3 yields (74).

We compute t3flo/t3z instead of tlo/COT, and then we use tlo/t3T eS(t3lo/COz)to conclude. With the previous notation, we write (t1o/t3z)(T,a,s) (c311/tz)+(tI2/t’c) + (tI3/tz).

I3Oz"

tI3 1t3--- p 1

(1 z) 2p(s- log(1 z))1


and

1p 1

(1-z)-1-(1/(p-1)) d#(y)z(y, s) 1+(P 1)2(y + 002

4p(1 z)(s log(1 z))

+ (1 z)-I/(p-1) I d#(y)7.(y,s)1 (p 1)2(y + cz)2(1 (s- log(1 z)))

4p(1 z)2 (s log(1 z))2

(P 1)2(y + r)2 )-1-1/(p-1)p- 1 + 4p(1 Z)(S- 1og(l Z))

Computing as for 12, we obtain OI2/z O(z) + (x/(p- 1)) + O(s-1).I1&"

I1 M + ME + M3

with

1 l ( Y+m p- 1 (1 Z) -1-(1/(p-1)) d#(y).(y, s) , a,,sx/1-,log(1 z

M21 l y+ tO

p 1 (1 ,)-1/(p-1) dl(y)z(y, s)2(1 ,)3/2 0

y+e ),s-log(1-z)x ,a,v,i_

M31 l 1 30(,a, Y+e )),s-log(1-z

p- 1 (1 z) -U(p-1) d#(y)z(y, s) 1 z Os v/1 r,

From (80), (84), and integration by parts we derive IOI1/&l < IMll-+-]MaI-t-[M3[ < Cs-1/2.

This concludes the proof of Lemma B.2.

Step 2. Behavior of (0,l) near blow-up. We use the _explicit asymptoticdevelopment given in Lemma B.2 to construct a 1-manifold F that is mapped by(0, 1) into 8(s).LEMMA B.4 (Behavior of (0, 1)). We have C0 Co(p), qA9 > 0 VA > A9,

192 MERLE AND ZAAG

:ls9(A) > 0 ’s/> s9(A), 3r’A,s rectangle in

DA,s (’, dl) + (-CoAe-Ss-2, CoAe-Ss-2) x (-CoAe-S/2s-1, CoAe-S/2s-1)

such that V(T,a) FA,, (o,i)(T,a,s) a(s), and d(Fa,, (o,l)(.,.,s),0)ml.

Proof. Since (t0, tl given in (79) is almost the linear part of (t0,l) (seeLemma B.2), we can first show for (0, 1) an analogous version of Lemma B.4,then use Lemma B.2 to conclude. We use scaling arguments to get uniform esti-mates in s. Indeed, let us introduce

{) ---({)0, 1)" (-CoA, CoA)2-- IR2

(85)

and

s (0, l)s" (-CoA, CoA)2--+ IR2

({, ) "-(0, tl) " + e-s2 + es/2si.., s(86)

wiaere Co Co(p). Note that Q is independent of s, and that

A((lo,l)(T,a, s) (00, 1)((T- )ess2, (a t)eS/2s),

A((lo, (ll)(T,a, s) (00, 01)s((T- )ess2, (a a)eS/2s).

The conclusion of Lemma B.4 follows if we show that there exists a 1-manifold "in (-CoA, CoA)2 such that V(, ) e ’, 0s(, ) e dog, and d(’, 0, 0) -1. FromLemma B.2, we compute for s > SiT(A): lit) t)ll((-c,c=/< c log s/(hv) 0when s - +o.

It is easy to see that_ [0,1), 3’n rectangle such that /(,)’n,Q(, ) e (1 + /)Yg, and d(F, Q, 0) -1.From the continuity of topolo_gical degree, we know that there exist /0 > 0,

e0 > 0 such that for each curve F (indexed by dog) satisfying Ill"-r/ox/ (’0 itself is indexed by t3cg), for each continuous function Q" (-CoA, CoA)2

]R2 satisfying II0 Qlli.,7((_Coa,CoA)2) < co, we have d(’, Q, 0) -1.

Since we have lit) Q IILo (_CoA,Coal= < c log s/(av), and from (98) Jac 0


-tc2/(4p(p 1)A2) < 0, we can take s large enough (s > S11(A, /30, /’]0)) SO that

-V(, )s f’no, )s(, ) e ext (1 +) eft, (87)

-V(, &) (-CoA, CoA)2, Jac O.s(, &) < O;

-Vco e Im (}s c Im O, if co (}s(), then

(88)

0-1(2)1 r/o, (89)

(90)

By_(90)and (87), we have d(’no 0.s, 0) -1. Therefore, by (87), Vco (1 + (0/4_))cg,d(F,o Q, o) =-1. (The degree is the same in the same component of IR2\Qs(F0).)Combining this with (88) and the definition of topological degree for rg functionsyields Va e (1 + (r/o/4))rg, there exists a unique(, ) IR2 such that 0.(, ) 09.

Hence, is a diffeomorphism from (Os)-((1 + (0/4))cg) onto (1 + (r/0/4))c’.Thus there exists a piecewise cgo 1-manifold interior to ’,0, such that Os mapsonto dC ( is diffeomorphic to 0rg). By (103), I 0,a[ < r/0. Therefore, we derived(, 0.s, 0) -1. This concludes the proof of Lemma B.4.

Step 3. Conclusion of the proof of Lemma 4.5. We take A > A9, so >max(g0 + 1,s7, s9(A)), and e > 0. For each Sl > so, we consider Da,s and Fa,sgiven by Lemma B.4. Ifsx > Sl2(A, e,, so), then (r, a) Fa,sx, I(r, a) (, a)] < e,and (T, a) D(so) (with the notation of Lemma 4.2). Therefore, for such Sl, we

s 2have I(r- ’)e I< CA/s and I(a- a)eS/21 < CA/sI. This implies Vs [so, s],I(r- )el < CA/s2 and I(a- a)e/21 < CA/s. What we want to do now is toshow that Vs e [so, Sl], (r,a,s) Va(s). By Lemma B.2, we have: For so > s13(A),V(T, a) e FA,s,, Vs e [so, s]"

CAIo(T,a,s)l < -- (91)

CAIl(T, a,s)l < --fi- (92)

[2(T,a,s) < clog s

s2 (93)

IO- (T, a, y, s) C(1 + lYl 3)1

(94)

CIOe(T,a,y,s)[ < ---. (95)

194 MERLE AND ZAAG

Therefore, if A > A14,

102(T,a,s)l < A2lOg s

s2Io_(r,a,y,s)l A(1 + lyl 3) 1 a2

IOe(T,a, y,s)l -(96)It remains for us to show that Iglm(T,a,s)l A/s2, for m 0, 1. Following theproof of Lemma 3.8, we easily prove the next lemma.

LEMMA B.5 (Transversality property). We have 3A15 > 0, VA > A15, 3sis(A)such that Vs0 > sis(A), VSl > so, for any solution q of (15), satisfying:

properties (91) to (95),for s [so, s1];s e (so, Sl] such that (qo, ql)(S) O!2A(s);we have the following property: >0 such that Vs_e(s-6,s),(qo, ql)(S-) s int((s_)).

If A > A15 and so > sis(A), then by Lemma B.5, V(T, a) e FA,sl

Vs e [so, Sl), (glO, Ol)(T,a,s) int((s)). (97)

Indeed, this follows if we apply Lemma B.5 to S1 ((t0, Ol)(S1) 6 t(S1) by LemmaB.4) and to s e (so, s], and use I {s e [so, sl)lVs’ e [s,s), (Oo, O)(r,a,s’) eint((s’))}.The conclusion of Lemma 4.5 follows for A > A6 max(A9, A14, A15), so >

max(o + 1, s7, s9 (A), s3(A), ss(A)), D6(s0) D (so), and for e > 0, s s2(A, e, so)and F F,.

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prepdnt.

MERLE: DIPARTEMENT DE MATHIMATIQUES, UNIVERSIT DE CERGY-PONTOISE, 8 LE CAMPUS, 95 033CERGY-PONTOISE, FRANCEZAAG: DIPARTEMENT DE MATHIMATIQUES ET INFORMATIQUE, ]COLE NORMALE SUPIRIEURE, 45 RUE

D’ULM, 75 230 PARIS CEDEX 05, FRANCE

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