修正Kawahara方程空间解析半径的下界.pdf

应用数学MATHEMATICA APPLICATA2024,37(2):530-539Lower Bounds on the Radius of SpatialAnalyticity for the Modified KawaharaEquationGUO Yantao(郭延涛),WANG Huichao(王会超)(School of Science,Xuchang University,Xuchang 461000,China)Abstract:We consider the Cauchy problem for the modified Kawahara equation with acubic nonlinear term in an analytic Gevrey space.Utilizing linear and trilinear estimates inanalytic Bourgain-Gevrey space,we establish the local well-posedness in Gevrey space G,sand show that the radius of spatial analyticity persists during the lifespan.Finally,usingan approximate conservation law,we extend this to a global result in such a way that theradius of analyticity of solutions is uniformly bounded,that the uniform radius of spatialanalyticity of solutions at later time t can decay no faster than 1/|t|as t .Key words:The modified Kawahara equation;Radius of spatial analyticity;GevreyspaceCLC Number:O175.2AMS(2010)Subject Classification:35A20;35Q53Document code:AArticle ID:1001-9847(2024)02-0530-101.IntroductionIn this paper we consider the Cauchy problem for the modified Kawahara equation onreal linetu+5xu+3xu+xu+x(u3)=0,x,t R,u(0,x)=u0(x),(1.1)where =0.This equation was also called fifth-order KdV type equation,and it can modela one-dimensional propagation of small amplitude long waves in various problems of plasmaphysics.13Recently,there is a growing interest in the well posedness of these types of partial differ-ential equations in analytic(or Gevrey)spaces G,s(R).For any 0,G,s(R)denotes theBanach space endowed with the norm f G,s=e|sf()L2(R),here =1+|andf denotes the spatial Fourier transform of f.For =0,the Gevrey space coincides with the standard Sobolev space Hs(R).TheCauchy problem for the Kawahara equation with the quadratic nonlinear term x(u2)wasReceived date:2023-04-25Foundation item:Supported by the Program for the Young Yeachers of Henan Province(2020G-GJS210),Key Scientific Research Projects of Henan Province(22B110014,23A110021)and the Programfor Innovative Research Team of Henan Province(23IRTSTHN018)Biography:GUO Yantao,male,Han,Henan,associate professor,major in partial differential equation.No.2GUO Yantao,et al.：Lower Bounds on the Radius of Spatial Analyticity for the MKE531studied in many works.CUI,DENG and TAO4showed that the equation has a global solu-tion when s=0.Later,many authors extended the results to the negative index spaces.56It should be noted that under the weaker regularity condition on initial data,CHEN andGUO7proved the global well-posedness for s 7/4 by the I-method.Kato89obtainedthe locally well-posedness for s 2 and the global solution for s 38/21.By the Fourierrestriction norm,JIA and HUO10considered the Cauchy problem for the modified Kawa-hara equation with cubic nonlinear term x(u3),and obtained the local well-posedness withs 1/4.In this paper,we study the well-posedness of(1.1)in Gevrey space G,s(R)motivated bythe earlier works on this issue.The key property of the Gevrey space is that every function inspace G,swith 0 has an analytic extension to the strip S=x+iy:x,y R,|y|0,s R.Then the following are equivalent:(i)f G,s;(ii)f is the restriction to the real line of a function F which is holomorphic in the stripS=x+iy:x,y R,|y|and satisfiessup|y|F(x+iy)Hsx 0,and establish an almost conservation law in G,s(R),then to use an iterative process ontime 0,T,with T arbitrarily large.This method was widely discussed for the Korteweg-deVries equation.Bona and Gruji c13proved that the radius(t)for the KdV equation decays exponential-ly et2as t .This was improved greatly by Bona,Gruji c and Kalisch14to a polynomialdecay rate of|t|12.Selberg and da Silva16established a key inequality and an iterativeprocess to extend the solution up to any time and obtained(t)c|t|4/3+,Tesfahun17removed the exponent.Later,HUANG and WANG18proved a higher order almost conser-vation law in Gevrey spaces by the modified I-method.Finally they improved the decay rateup to|t|1/4.For other related works in the radius of analyticity for other partial differentialequations,we refer to 19-21 and references therein.The Kawahara equation is a fifth-order KdV type equation.Ahn,Kim and Seo20dis-cussed lower bounds on the radius of spatial analyticity of solutions to the Kawahara equation532MATHEMATICA APPLICATA2024with the quadratic nonlinear term,and the uniform radius of spatial analyticity of solutionsat later time t can decay no faster than|t|1.In spite of these many works,we discuss themodified Kawahara equation with the cubic nonlinear term.Now we state our first mainresult about the local well-posedness in analytic Gevrey spaces for the modified Kawaharaequation.Theorem 1.1If 0 and s 1/4,then for any u0 G,s(R),there exists a=(u0G,s)0 and a unique solution u(t)of the Cauchy problem(1.1)on the timeinterval,such that u(t)X,s,b C(,G,s(R)with=C(1+u02G,s)afor some constants C 0 and a 5 depending only on s.Moreover the local solution u(t)satisfies the boundsu(t)X,s,b Cu0G,sand sup|t|u(t)G,s Cu0G,s.The next main result deals with the evolution of the radius of analyticity in time,andan algebraic lower bound on the radius of analyticity(t)as t tends to infinity.Theorem 1.2If 0 0 and s 1/4 and u0 G0,s(R),then the solution u(t)obtained in Theorem 1.1 extends globally in time,and for any T 0,and we haveu(t)C(T,T,G(T),s(R),where(T)=min0,CT1,C is a constant depending on u0,0,s.The paper is structured as follows.In Section 2,we introduce some function spaces suchas Bourgain and Bourgain-Gevrey spaces,and some basic properties that will be used in latersections.In Section 3,we present trilinear estimates in Bourgain-Gevrey spaces,solve theCauchy problem with data in G,s,and derive a formula for the lifespan of the local solutiondepending on the norm of the initial data.In Section 4,we prove an almost conservation lawby transforming the equation for u(t)to a similar equation for v(t).=e|Dx|u(t)with an errorterm in the right-hand side,and estimate the error terms via trilinear estimate yields thedesired the almost conservation law.Finally,in Section 5,using this almost conservation law,we derive the lower bound radius of spatial analyticity and complete the proof of Theorem1.2.2.PreliminariesIn this section,we introduce some function spaces and their basic properties which willbe used in later section for the proof of theorems.Throughout this paper,the letter C standsfor a positive constant which may be different at each occurrence,and we denote A.B andA B to mean A CB and B.A.B,respectively.Recall that for all f S,the Fourier transformf is defined by(Ff)()=f=Reixf(x)dx.The inverse Fourier transform of any g is given by(F1g)(x)=g=12Reixg()d.From the definition of the Gevrey space,we can get that the Gevrey space satisfies thefollowing important embedding propertyG,s,G,sfor all 0 1/2.Then X,s,b,C(R,G,s)andsuptRf(t)G,s Cf(t)X,s,b,where C is a constant depending only on b.Lemma 2.210Let 0,s R,1/2 b b 0.ThenfX,s,b Cb,bbbfX,s,b,where Cb,bis a constant depending only on b and b.Lemma 2.310Let 0,s R,1/2 b 0.Then for any time intervalI 0,IfX,s,b CfX,s,b,where I(t)is the characteristic function of I,and C is a constant depending only on b.3.Local Well-posedness in Bourgain-Gevrey SpaceIn this section,we establish the local well-posedness based on the contraction mappingprinciple in space X,s,b,we obtain Theroem 1.1 which is the local well-posedness in G,swith a lifespan 0,implies the radius of analyticity remains strictly positive in a short timeinterval 0 t .Let the linear operator associated with the corresponding linear equation of(1.1)beW(t)=F1eit(53+)F,then,by Duhamels principle,the solution of(1.1)can be written asu(t)=W(t)u0+t0W(t t)F(t,x)dt,(3.1)where F(t,x)=x(u3).Then the following linear X,s,benergy estimate follows directlyfrom 10,20 or also 21.534MATHEMATICA APPLICATA2024Lemma 3.1Let s R,1/2 b 1 and 0 1.ThenW(t)u0X,s,b Cu0G,s,t0W(t t)F(t,x)dtX,s,b CFX,s,b1.We present a trilinear estimate in Bourgain spaces proved in 10 and use the transformv(t).=e|Dx|u(t)to obtain a trilinear estimate in Bourgain-Gevrey spaces which plays a rolein obtaining the local well-posedness.With the aid of this trilinear estimate we will deducean estimate which is another useful tool particularly in obtaining the almost conservation lawin the next section.Lemma 3.2If s 1/4,1/2 b 1/2.Thenx(u1u2u3)Xs,b1 Cu1Xs,bu2Xs,bu3Xs,b.(3.2)Lemma 3.3Let s,b,bbe as in Lemma 3.2.Then for all 0,we havex(u1u2u3)X,s,b1 Cu1X,s,bu1X,s,bu1X,s,b.(3.3)ProofLet b vj=e|b uj(,)then(3.3)is reduced toIL2,.3j=1vjXs,b,(3.4)whereI=is p()b1e(|3j=1|j|)3j=1b vj(j,j)dd,here we used the notationwdd=3j=1j=,3j=1j=w3j=1djdjfor the function w=w(j,j).By the triangle inequality,we have|3j=1j,which implies e(|3j=1|j|)1,andhenceIL2,.x3j=1vjXs,b.(3.5)Thus(3.4)can be reduced to showingx(v1v2v3)Xs,b1 Cv1Xs,bv2Xs,bv3Xs,b.(3.6)This is the conclusion in Lemma 3.2.Proof of Theorem 1.1Consider the setB=u:u(t)X,s,b 2Cu0G,s,define an operator on B as followsu(t)=W(t)u0 t0W(t t)x(u3(t)dt.(3.7)On the one hand,if u(t)B,then using Lemmas 2.2,3.1 and 3.3,we findu(t)X,s,bW(t)u0X,s,b+t0W(t t)x(u3(t)dtX,s,bCu0G,s+Cx(u3(t)X,s,b1No.2GUO Yantao,et al.：Lower Bounds on the Radius of Spatial Analyticity for the MKE535Cu0G,s+Cbbx(u3(t)X,s,b1Cu0G,s+Cbbu(t)3X,s,b,(3.8)with 1/2 b b 5 since 1/2 b b 7/10.Now assume that u and v are solutions of the Cauchy problem(1.1)for initial data u0and v0respectively.Then similarly as above,with the same choice of and for any suchthat 0 ,we haveu vX,s,b u0 v0G,s+12u vX,s,b,which proves the continuous dependence of the solution on the initial data.4.Almost Conservation LawWe know that the solution of the modified Kawahara equation has a conservation law inL2space,i.e.,u(t)L2=u0L2for all t 0,but in Gevrey space G,0,given u0 G,0,by Theorem 1.1 we only have a solution u(t)G,0for 0 t satisfying the boundsup|t|u(t)G,0 Cu0G,0.This local estimate is not sufficient to derive our conclusion,so we shall improve this localestimate to the following almost conservation law for any large time T0,that is,sup|t|T0u(t)G,0 u0G,0+E(0),536MATHEMATICA APPLICATA2024where E(0)satisfies the bound E(0)Cu04G,0,the quantity E(0)can be consideredan error term since in the limit as 0,we have E(0)0.Lemma 4.12223Let min,med,maxdenote the minimum,medium,maximum of|1|,|2|,|3|.Then for 0,1 we have the estimatee3j=1|j|e|3j=1j|12mede3j=1|j|.Lemma 4.2Let f=x(e|Dx|u)3 e|Dx|u3.Given 0 1,there exists 1/2 b 1/2 and 0 1,and be as in Theorem 1.1.Then thereexists a C 0 such that for any 0 and any solution u X,0,bof the Cauchy problem(1.1)on the time interval 0,we have the estimatesupt0,u(t)2G,0 u02G,0+Cu(t)4X,0,b.(4.2)Moreover,we havesupt0,u(t)2G,0 u02G,0+Cu04G,0.(4.3)ProofLet v(t,x)=e|Dx|u(t,x)which is real-valued since the multiplier e|Dx|is evenand u is real-valued.Applying e|Dx|to(1.1),we obtaintv+5xv+3xv+xv+x(v3)=f,(4.4)where f=x(e|Dx|u)3 e|Dx|u3.Multiplying(4.4)by v and integrating in space,we have12Rv2dt+2R(2xv)2x2R(xv)2x+2R(v2)xdx+2Rv4dx=Rfvdx,(4.5)No.2GUO Yantao,et al.：Lower Bounds on the Radius of Spatial Analyticity for the MKE537which implies12Rv2dt=Rfvdx.(4.6)Now integrating in time over the interval 0,we obtainu()2G,0=Rv2(,x)dx=u02G,0+2RR0,vfdxdt.(4.7)We now use Plancherel Theorem and Lemma 2.3,Lemma 4.2 to estimate the integral on theright-hand side of(4.7)as|RR0,vfdxdt|0,vX0,1b0,fX0,b1 vX,0,1bfX,0,b1CuX,0,1bfX,0,b Cu4X,0,b.(4.8)Inequality(4.3)follows from Theorem 1.1 and(4.2)directly.The proof is complete.5.Proof of Theorem 1.2ProofBy invariance of the Kawahara equation under the reflection(t,x)(t,x),we may restrict to positive times.By the embedding property of the Gevrey spaceG,s,G,sholds for all 0 0 depending on u0G0,0and 0.Now fix T arbitrarily large,it suffices to showsupt0,Tu(t)2G,0 2u02G,0,(5.3)for satisfying(5.2)which in turn implies u(t)G(t),0as desire.First,choose n Z+such that n T (n+1),using induction,we shall show thatfor any k 1,2,n+1 thatsupt0,ku(t)2G,0 u02G,0+kC22u04G,0,(5.4)andsupt0,ku(t)2G,0 2u02G,0,(5.5)provided satisfying 0,and2TC22u02G0,0 1.(5.6)Indeed,for k=1,from(5.1),we havesupt0,u(t)2G,0 u02G,0+Cu04G,0 2u02G,0,(5.7)538MATHEMATICA APPLICATA2024where we use the fact that u02G,0 u02G0,0and Cu04G0,0 1 which follows from(5.6).Now assume(5.4)-(5.5)hold for some k 1,2,n.Applying(5.1),(5.4)-(5.5),wehavesuptk,(k+1)u(t)2G,0u(k)2G,0+Cu(k)4G,0u(k)2G,0+C22u04G0,0.(5.8)Combining this with the induction hypothesis(5.4)for k,we getsupt0,(k+1)u(t)2G,0 u02G,0+C(k+1)22u04G0,0,(5.9)which proves(5.4)for k+1.Since k+1 n+1 T/+1 2T/,from(5.6)we also getC(k+1)22u02G0,02TC22u02G0,0 1,which along with(5.9)we prove for k+1.Finally,the condition(5.6)is satisfied for(t)=(C22u02G0,0)1/(1T)1/.Particularly choosing =1,we get(t)CT.The condition C in(t)CTmay be given as C=C22u0G0,0which depends only onu02G0,0.The general case s R.Recall the embedding property of the Gevrey space G,s,G,sfor all 0 and s,s R.For any s R,we use this embedding to getu0 G0,s,G0/2,0.From the local well-posedness result,there is a =(u0G0/2,0)such thatu(t)G0/2,0,for 0 t .Similarly as in the case s=0,for T fixed greater than,we have u(t)G,0for t 0,Tand C/T,where C depends on u0G0/2,0and 0.Applying the embedding again,we concludeu(t)G,s,for t 0,T,where =/2.The proof is complete.References:1 KAWAHARA T.Oscillatory solitary waves in dispersive mediaJ.J.Phys.Soc.Japan.,1972,33:260-264.2 BONA J L,SMITH R S.A model for the two-ways propagation of water waves in a channelJ.Math.Proc.Cambridge Philos.Soc.,1976,79:167-182.3 KICHENASSAMY S,OLVER P J.Existence and nonexistence of solitary wave solutions to higher-order model evolution equationsJ.SIAM J.Math.Anal.,1992,23:1141-1166.4 CUI S B,DENG D G,TAO S P.Global existence of solutions for the Cauchy problem of the Kawaharaequation with L2initial dataJ.Acta Math.Sin.,2006,22:1457-1466.5 WANG H,CUI S B,DENG D G.Global existence of solutions for the Kawahara equation in Sobolevspaces of negative 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