Exponen tial Decay of Energy for a Logarithm icWave Equation

ZHANG Hongw ei,LIU Gongw eiand HU Qingying

D epartm ent ofM athem atics,H enan University ofTechnology,

Zhengzhou 450001,China.

Received 6 June 2015;Accep ted 31 Ju ly 2015

Exponen tial Decay of Energy for a Logarithm icWave Equation

ZHANG Hongw ei?,LIU Gongw eiand HU Qingying

D epartm ent ofM athem atics,H enan University ofTechnology,

Zhengzhou 450001,China.

Received 6 June 2015;Accep ted 31 Ju ly 2015

.In this paperw e consider the initial boundary value p roblem for a class of logarithm ic w ave equation.By constructing an app rop riate Lyapunov function,w e ob tain the decay estim ates of energy for the logarithm ic w ave equation w ith linear dam p ing and som e su itab le initial data.The resu lts extend the early resu lts.

Logarithm icw ave equation;initialboundary value p roblem;decay estim ate.

1 In troduction

In this paper,w e shalldealw ith decay estim atesofenergy for the initialboundary value p roblem of the logarithm icw ave equation

w here ? ? Rn,n ≥ 1,is a bounded dom ain w ith sm ooth boundary ??,k≥ 1.

This type of p roblem s havem any app lications in m any branches of physics such as nuclear physics,op tics and geophysics(see[1–3]and their references).It has been also introduced in the quantum fi eld theory,such kinds non linearity appear natu rally in infl ation cosm ology and in supersymm etric field theories([3]).W hen ? is a finite interval[a,b]in p roblem(1.1)-(1.3)w ithou t dam p ing term ut,it is a relativistic version of logarithm ic quantum m echanics in troduced by Bialynicki-Biru la and M ycielski(see[1,4]). H iram atsu et.al[5]introduced also the follow ing equation

for studying the dynam ics o f Q-ball in theoretical physics.

In[6],Cazenave and Haraux established the existence and uniqueness of a solu tion for the Cauchy p roblem for the fo llow ing equation

in R3.By using com pactnessm ethod Gorka[3]obtained the g lobal existence of w eak solu tions for all u0∈ H10,u1∈ L2to the initial boundary value p roblem of Eq.(1.5)in one-d im ensional case.In[2],Bartkow skiand Gorka show ed the existence of classical solu tionsand investigated w eak solu tions for the correspond ing Cauchy p roblem for Eq. (1.5)in one-d im ensional case.In[7],Han got the global existence of w eak solu tions for all u0∈ H10,u1∈ L2to the initial boundary value p rob lem(1.4)in R3.How ever,no resu lt is given concerning the decay p roperty of solu tions and it is desirable to establish the uniform stabilization of solu tions to(1.1)-(1.3).

We shou ld also point ou t that there is extensive literatu re on the question o f existence,asym ptotic behavior and nonexistenceof solu tion for the follow ing initialboundary value p roblem

Here,w e m ention on ly som e results abou t the interaction betw een the dam ping term and the sou rce term.It w as fi rst considered by Levine[8]in the linear dam p ing case h(ut)=autand a po lynom ialsou rce term of the form f(u)=b|u|p?2u.Then Georgiev and Todorova in[9]extended Levine’s resu ltto thenonlinear dam p ing case h(ut)=a|ut|m?2ut. For theexponen tialdecay for the initialboundary value p roblem for the follow ing equation

has been stud ied for ω ≥ 0 or ν ≥ 0,am ong others,by Nakao[10],Zuazua[11,12]Benaissa[13]and Gebri[14].How ever,them ethod in abovem en tioned w orks cannot be app lied d irectly to the case that the equations have the logarithm ic non linearities term. In troducing the logarithm ic nonlinearity term m akes the p roblem d ifferen t from the one considered in above allm entioned paper.The pu rposeof this paper is to obtain a decay estim ate of so lu tions to the p rob lem(1.1)-(1.3).M ore p recisely w e show thatw e can findinitialdata in the stable set forw hich the solu tion of problem(1.1)-(1.3)decays exponentially.Thekey toolin thep roof isan ideaofHaraux and Zuazua[11]and Zuazua[12],and ithasbeen recen tly used by Benaissaand M essaoud i[13]and Gerbiand Said-Houari[14]. Th ism ethod is based on the construction ofa su itab le Lyapunov function and thiskind o f Lyapunov function isa sm allperturbation of the energy.We find that for the logarithm ic nonlinearity case it is d iffi cu lt and the stable set is also d ifferent from the one considered in[14].

Let us fi nally m ention that a sem ilinear heat equation w ith logarithm ic non linearity w as stud ied by Chen[15].

This article is organized as follow s.Section 2 is concerned w ith som e notation and som e lemm as.In Section 3,w e p rove the decay of the solu tion for the p roblem(1.1)-(1.3).

2 Prelim inariesand som e lemm as

In this paper,w e denoteby||·||pthe Lp(?)norm,||?·||the Dirichletnorm in H10(?).In particu lar,w e denote||·||=||·||2.We also use C to denote a universal positive constant thatm ay take d ifferentvalues in d ifferentp laces.

By a w eak so lu tion u(x,t)o f p roblem(1.1)-(1.3)on ?×[0,T)w em ean

such that u(x,0)=u0(x)in H10,ut(x,0)=u1(x)in L2,and

for any w∈H10(?),a.e.t∈[0,T),and the follow ing energy inequality of thew eak solu tions u holds,i.e.

w here

In order to give ou r resu lts,now,w e give the follow ing lemm a.

Lemm a 2.1.([3,15,16])(Logarithm ic Sobolev inequality)Let u beany function in H10(?)

and a＞0 beany number.Then

By using Galerkin m ethod com bined w ith logarithm ic Sobolev inequality and compact theorem,sim ilar to the p roof in[3,7],w ehave the follow ing resu lt:

Theorem 2.1.Suppose that u0∈ H10(?)and u1∈ L2(?),then there exist globalweak solution to theproblem(1.1)-(1.3).

Now,w e in troduce tw o functionals J(u)and I(u):

Then,it is obvious that

Accord ing to the logarithm ic Sobolev inequality(Lemm a 2.1),J(u)and I(u)are w ell defined on H10(? ).Fu rtherm ore,E(u)=E(t)isw elldefined on H10and E(t)is decreasing for t≥ 0.As in[14],the potentialw elldep th is defi ned as:

We also define thew ell-know n Neharim anifold

Sim ilar to the p roofof[15,Lemm a 2.3],w e have

Lemm a 2.2.For any u ∈ H10(?){0},w ehave

where

Now w e defi ne the subsets of H10(?)related to p rob lem(1.1)-(1.3).Set

Then it is read ily seen that the potentialdep th d is also characterized by

Lemm a 2.3.Letu ∈ H10(?)and l=(2π/k)n2en,if0 ＜ ||u||2＜ l,then I(u)＞ 0;if I(u)=0 and ||u||/=0,i.e.u ∈ N,then||u||2＞ l.

Proof.By the logarithm ic Sobolev inequality,for any a＞ 0,w ehave

Taking any a satisfying 0 ＜ a2≤ 2π/k in(2.13),w e gain

If0 ＜ ||u||2＜ l,then I(u)＞ 0 from(2.14).If I(u)=0 and||u||/=0,then by(2.14),w e have

that is||u||2＞ l.

Lemm a 2.4.Wehave

Proof.If I(u)=0 and||u||/=0,then by Lemm a 2.3w e have||u||2≥ l.Togetherw ith(2.5), w e get

then,w e have d≥k4l.

Lemm a 2.5.Ifu0∈ H10(?),u1∈ L2and 0＜E(0)＜ d,u isaweak solution ofproblem(1.1)-(1.3), then u ∈ W if I(u0)＞ 0.

Proof.Let T bem axim alexistence tim eofw eak solu tion of u.From thedefinition ofw eak solu tion and(2.2),w e have

Then w e claim that u(t)∈ W for all t∈ [0,T).If it is false,then there is a t0∈ (0,T)such that u(t0)∈ ?W,so w ehave

(a)I(u(t0))=0 and||u(t0)||/=0,or(b)J(u(t0))=d.

By(2.15),(b)is im possible,thusw e have I(u(t0))=0 and||u(t0)||/=0.How ever,from the definition of d,one has J(u(t0))≥ d,w hich is contractive w ith(2.15).Then w e have u(t)∈W for all t∈ [0,T).

3 Decay estim ates of the solu tion

Now,w e state the decay estim ates o f the p roblem(1.1)-(1.3).

Theorem 3.1.Letu0∈W,u1∈ L2.Assumefurther that0＜ E(0)＜αk4l＜ d,where α isa positive constantsatisfying 0 ＜ k4nα4n2 πke2＜1,then thereexist tw o positive constants K and ξindependent oftsuch that:

Proof.Let u(t,x)isaw eak solu tion ofp roblem(1.1)-(1.3).Since u0∈W,u1∈L2,by Lemm a 2.5,w e have u ∈W for all t∈ [0,+∞)and then 0 ＜ E(t)＜ d and I(u)＞ 0.

Now,w e construct the Lyapunov functionalby perform ing a suitablem od ifi cation of the energy

w here ∈ ＞ 0 w illbe determ ined in the later.Since

and the defi nition o f E(t)by(2.6),then w e know that L(t)and E(t)are equ ivalent in the sense that thereexist tw o positive constants β1and β2depend ing on ∈such that for t≥0

By taking the tim e derivativeof the function L(t),using Eq.(1.1),and perform ing several integration by parts,w e get

Now,w eestim ate the fi fth term in the righthand sideof(3.3).By using Young inequality, w e obtain,for any δ＞ 0

Consequen tly,inserting(3.4)in to(3.3),w e have

Using the defi nition of E(t),for any positive constant M,w e have

For 0 ＜ M ＜ 1,and by logarithm ic Sobolev inequality(Lemm a 2.1),w ehave

Noting 1? M/2＞ 0 since0 ＜ M ＜ 1,and ln||u||2＜ ln(4J(u)),w egetSince 0 ＜ M ＜ 1,J(u) ＜ E(0) ＜ kαl/4,by taking a satisfying k4nα n4(2π/k)2e2＜ a2＜ 2π/k, w here a can be assured by theassum p tion about α in Theorem 3.1,and taking δ＞0 sm all su ffi cien tly such that

then w e have

Now,choosing ∈ ＞ 0 sm allsu ffi ciently such that

inequality(3.9)becom e

By(3.2),w e have

Setting K=M∈β2＞ 0 and integrating the inequality(3.10)betw een 0 and t gives the follow ing estim ate for the function L Consequently,by using(3.2)onceagain,w e conclude the result.This com p letes the p roof of the theorem.

Acknow ledgem en t

Thisw ork is supported by NationalNatu ral Science Foundation of China(No.11171311) and Basic Research Foundation of Henan University of Technology(171164).Theau thors are gratefu l to the referee for his suggestions.

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?Correspond ing au thor.Emailaddresses:whz661@163.com(H.W.Zhang),gongwei l iu@126.com(G.W.Liu), s l xhqy@163.com(Q.Y.H u)

AMSSub ject Classifi cations:35L20,35L70,35B40,35Q 40

Chinese Library Classifi cations:O175.27,O175.29