Unique Continuation Property for a Class ofFifth-order Korteweg-de-Vries Equations?

GAO X iao-hong,ZHENG Xiao-cui

(1-School of mathematics,Northwest University,Xi’an 710127;2-Center for Non linear Studies,Northwest University,Xi’an 710069)

1 In troduction

In this paper,we consider the initial value problem associated with the follow ing fifth-order Korteweg-de-Vries equations

where x∈R,t>0,u=u(x,t)is a real-valued function andα,c1,c2,c3are real constantsw ithα?=0 and c1=c2.The equations in(1)were proposed by Benney[1]as a model for the interaction of short and long waves.For local well-posedness issue of the problem(1),Kenig and Pilod[2]showed that the problem(1)is locally well-posed in H2(R)for s≥2.At the same time,they demonstrated that when c1=c2,the problem(1)adm itted L2(R)-norm conservation law.

In this paper,app lying Bourgain’smethod[3],we focus on investigating the unique continuation property for the problem(1).The unique continuation property says,if a solution u to a certain evolution equation or a vector?u to a system vanishes on some nonem pty open subset O of?,then it vanishes in the horizontal com ponent of O,where?is the domain of the evolution operator under consideration.

A pioneer work in this direction is due to Carleman[4]and hismethod was based on theweighted estimates for the associated solutions.Saut and Scheurer[5]firstly used the Carleman type estimates and obtained the UCP for a general class of dispersive equations in one space dimension.In particular,the KdV equation is one of the class.Later,Zhang[6]used inverse scattering transform and some results from Hardy function theory to prove that the solution to the KdV equation can not be supported in the horizontal half lines at two diff erent moments un less it vanishes identically.He also proved the same result for themodified KdV equation using miura’s transformation.Recently,Bourgain[3]introduced a different approach and proved that,ifa solution u to a dispersive equation has com pact support in a nontrivial time interval I=[t1,t2],then u vanishes identically.Using of com plex variab les technique along with Paley-W iener theorem are the main ingredients in Bourgain’s approach.More recently,Kenig et al[7]introduced a new method and proved that,if a suffi ciently smooth solution u to a generalized KdV equation is supported in a half line at two diff erentmoments of time,then u vanishes identically.To get this result,they derived a Carleman type estimate exp loiting the structure of the generalized KdV equation.Carvajal and Panthee[8]proved the UCP for am ixed equation of the type KdV and Schr¨odinger.A lso,there is a recent work due to Kenig et al[7]dealing with the UCP for the Schr¨odinger equation.

In the follow ing,we elaborate themain result.

Theorem 1 Let u=u(x,t)∈C([0,T],Hs(R)),s≥4,be a solution of the problem(1).If there exists a nontrivial time interval I=[0,T]such that supp u(t)?[?B,B],?t∈ I,where B>0 is a constant,then u(t)≡ 0 for all t∈ I.

2 preliminary

Firstly,we give some notations which w ill be used throughout this work.We use F to denote Fourier transform

We write A.B if there exists a constant c>0 such that A≤cB.A lso,we use ρ?nto denote the convolution productand supp f to denote the support of f.By the Paley-Wiener theorem and the assump tion of Theorem 1,for the solution u=u(x,t)to the prob lem(1)andλ∈R,we define

We proceed to establish the follow ing results that w ill be useful in our analysis.Lemm a 1 Let u=u(x,t)be a smooth solution to the problem(1),if for some B>0,supp u(t)? [?B,B],then for allλ,θ∈ R,we have

proo f Using the Cauchy-Schwarz inequality and L2(R)-norm conservation law,wemay prove this lemma like Carvajal and Panthee in[8].

Considering u0=u(x,0)suffi ciently smooth and taking into account the wellposedness theory for the prob lem(1),we have the follow ing resu lts.

Lemm a 2 Let u=u(x,t)∈C([0,T],Hs(R)),s≥4,bea solution to the problem(1),B as in the Theorem 1,then we have

proo f App lying the Cauchy-Schwarz inequality and L2(R)-norm conservation law,wemay verify this lemma like Carvajal and Panthee in[8].

Rem ark 1 If there exists t such that u(t)?=0 and u(t)has com pact support,then ρ(λ)>0 for allλ ∈ R.

Then from Lemma 2,we derive the follow ing lemma whose proof is sim ilar to the proof of the lemma in[3].

Lemm a 3[3]Let u=u(x,t)be a solution of the problem(1)with the same hypothesis of Theorem 1.Suppose that there exists t∈ I such that u(t) ?=0,then there are numbers c>0 and Q0>0 such that,for any Q>Q0,there are arbitrarily large values ofλ (i.e.for allλ′>0 there existsλ with λ > λ′)such that

Rem ark 2 Ifλ>0 satisfies(6)and(7),then so does?λ.

Coro llary 1[3]Suppose the hypothesis of Lemma 3 are valid,then there exists arbitrarily large λ and Q and tλ:=t1∈ I,such that when t=t1,themaximum ρ(λ)is achieved,i.e.

In what follows,we need some derivative estimates for the entire function.These results and their proofs appear in[3].So we directly exp loit them w ithout proofs.

Lemm a 4[3]LetΦ(z)be an entire function bounded and integrable on the real line such that|Φ(λ +iθ)|.e|θ|B,λ,θ∈ R,B>0,then we have

Coro llary 2[3]Under the hypothesis of Lemma 4,ifθ∈ R such that

then we have

Coro llary 3[3]Let t∈I,Φ(z)=F(u(t))(z),θas in Corollary 1 andρas in(2),then for|θ′|≤ |θ|fixed,we have

During the process of the proof of ourmain result,we need the follow ing estimate.

Lemm a 5 If ρ(λ)is a function defined in(2),then there exist constants a0>0,a1>0,a2>0 such that

proof Since g is an even function,we supposeλ>0.By Lemma 2,it follows that

In the first integral lettingξ?λ=x,we obtain

By usingξ?λ=x in the second integral,it follows that

A ltogether,we obtain

This comp letes the proof of this lemma.

W ithout loss of generality,we assume thatα>0.For the caseα<0,the proof is sim ilar.We considerλ,t1as in Corollary 1,t2∈ I such that|?t|=|t2?t1|=T andθsu ffi ciently small such thatθ?t<0.A fter choosing sign forθ,let us chooseλ suffi ciently large such that|λ|3|θ|&1 and|λ|2|θ|3.1.

3 The proof of Theorem 1

In this section,we firstly rew rite the equations in(1).Notice that

Therefore,the equations in(1)become

Using the Duhamel formula,for t1,t2∈I,we deduce

where

Taking Fourier transform F in the space variable in(9),we obtain

Since u(t)hascom pact support,by Paley-W iener theorem,F(u),F(u2),F(u3),F(u?2xu)have analytic continuation in C.It follows that

Taking absolute value of?t=t2?t1,we deduce that

i.e.,

Next we consider?t>0.For?t<0,the analysis is sim ilar.So we obtain

For arbitrarily small values ofθand|λ|2|θ|3.1,we have

that is

where

In the follow ing,we are going to carry out the proof of Theorem 1.The key point in the proof is the estimate(10).We prove it by contradiction.

proo f Suppose u(t)?=0,for some t∈ I.Using Corollary 3 with

we obtain

Now we proceed to estimate I1,I3,I4and I5.

Estimate for I1.Using Corollary 1,we obtain

where

Using the definition ofρ(λ),(6)and(7),we deduce that

Sim ilarly

On the other hand,we have

App lying(8)in Lemma 5,we have

and(15)yields

Hence from(13),(14)and(16),we obtain the follow ing estimate for(12),

We set

Then using Lemma 3,Lemma 4,Corollary 2,Corollary 3 and taking

we deduce that

From(11),(17)–(20),it yields

Since|λ|3|θ|&1,from(21)it follows thatwith m0>0,m1>0 constants.But this inequality is false for large values of Q andλif we choose Q>2/m1,|λ|>2|log m0|/m1.Hence this concludes the proof of Theorem 1.

Rem ark 3 Wehave chosen|θ|.1/B(1+|logρ(λ)|)which isatmost(B?1Q)/|λ|.Hence

where c4,c5,c6>0 are constants,|λ|,Q are su ffi ciently large.So we can choose θ such that|θ|lies between the second and the third values.

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