Young measure solutions for a fourth-order wave equation with variable growth

R E S E A R C H

Open Access

Young measure solutions for a

fourth-order wave equation with variable

growth

Mingqi Xiang

*_{Correspondence:}

[email protected] College of Science, Civil Aviation University of China, Tianjin, 300300, P.R. China

Abstract

In this paper, we study the existence of Young measure solutions to a fourth-order wave equation with variable exponent nonlinearity on a bounded domain. The asymptotic behavior of the Young measure solutions is also investigated by applying a lemma developed by Nakao.

MSC: 35L35; 35B40; 35D99

Keywords: nonlinear wave equation; Young measure solution: variable exponent space; nonlinear elastodynamics; Galerkin’s approximation; asymptotic stability

1 Introduction

In this paper, we consider the initial boundary value problem of the following model:

∂_u

∂t +

|u|p(x)–u+|u|q(x)–u+a

|u|q(x)dx

∂u

∂t =f(x,t), (x,t)∈QT,

u=u= , (x,t)∈∂×(,T), (.)

u(x, ) =u(x),

∂u(x, )

∂t =u(x), x∈,

where⊂RN (N≥) is a bounded domain with smooth boundary∂,  <T<∞is a

given constant, andQT=×(,T). The coefficienta: [,∞)→(,∞) and the exponents

p,q:→(,∞) are given continuous functions andf:QT→R. PDEs with variable

expo-nent growth conditions are usually called equations with nonstandard growth conditions.

After Kováčik and Rákosník first discussed the variable exponent Lebesgue spaceLp(x)

and Sobolev spaceWk,p(x)_{in [], a lot of research has been done concerning these kinds}

of variable exponent spaces; see for example [, ] for the properties of such spaces and [–] for the applications of variable exponent spaces on partial differential equations. In [] Růžička presented the mathematical theory for the application of variable exponent spaces in electro-rheological fluids. Problems with variable exponent growth conditions also appear in the mathematical modeling of stationary thermo-rheological viscous flows of non-Newtonian fluids [] and nonlinear elastics [–]. Another field of application of equations with variable exponent growth conditions is image restoration [].

We claim that the Young measure solutions of problem (.) can be approximated by the following problem with a viscosity termε∂u

∂t (ε> ):

∂_u

∂t +

|u|p(x)–u+|u|q(x)–u

+a

|u|q(x)dx

∂u ∂t +ε

∂u

∂t =f(x,t), (x,t)∈QT,

u=u= , (x,t)∈∂×(,T), (.)

u(x, ) =u(x),

∂u(x, )

∂t =u(x), x∈.

Whenp(x)≡ and the space dimensionN= , problems of the type (.) are a class of

essential fourth-order wave equations appearing in elastoplastic-microstructure models. They govern the longitudinal motion of an elastoplastic bar and antiplane shearing

de-formation; see []. Forp(x)≡ and the multidimensional case, Chen and Yang []

dis-cussed the global existence, asymptotic behavior and blow-up of solutions to the initial

boundary problem of the equation with weak damping term ∂u

∂t; see also Messaoudi []

for wave equations with nonlinear damping. For the analysis of nonlinear second-order hy-perbolic equations with damping, we refer to the seminal work of Lions and Strauss []; see also Friedman and Ne˘cas [], and Emmrich and Thalhammer [, ]. In recent years, hyperbolic equations with variable exponent growth conditions were studied by Antont-sev in [], Haehnle and Prohl in [], Pinasco in []; see also Autuoriet al.in [, ] for the Kirchhoff equations withp(x)-growth. It is to be noted here that the viscosity term ∂u

∂t plays a key role in the proof of the global existence. The global existence results of

weak solutions for second-order wave equations (even ifp(x)≡constant= ) without the

viscosity term∂_∂u_t have been found only in one space dimension; see DiPerna [] and Shearer []. To the best of our knowledge, the equations without the viscosity term are studied only in [–]. In that work, the concept of Young measure solutions has been introduced and applied to dynamic problems and wave equations.

Thus motivated, in the present paper, we prove the global existence of Young measure solutions of problem (.), we first construct Young measure solutions as the limit of the sequence of solutions of problem (.). Then we give a decay estimate to the Young mea-sure solutions of problem (.).

Our work is organized as follows. In Section , we give some necessary definitions and properties of variable exponent Lebesgue spaces and Sobolev spaces. In Section , we ob-tain the existence of weak solutions of problem (.) by Galerkin’s approximation method. In Section , under some conditions, from the sequence of solutions of problem (.) and somea prioriestimates, we get the existence of Young measure solutions by lettingε→. In Section , we investigate the decay property of Young measure solutions and get a decay rate estimate by using Nakao’s lemma.

2 Preliminaries

In this section, we first recall some necessary properties of variable exponent Lebesgue spaces and Sobolev spaces; see [–] for the details.

Let⊂RN be a domain. A measurable functionp:→[,∞) is called a variable

exponent and we definep–_{=ess inf}

exponentpis said to be bounded. The variable exponent Lebesgue space is

Lp(x)() =

u:→Ris a measurable function;ρp(x)(u) =

u(x)p(x)dx<∞

with the Luxemburg norm

u Lp(x)₍₎=inf

λ>  :ρp(x)

λ–u≤,

thenLp(x)_{() is a Banach space, and whenpis bounded, we have the following relations:}

min u p–

Lp(x)₍₎, u p+ Lp(x)₍₎

≤ρp(x)(u)≤max

u p–

Lp(x)₍₎, u p+ Lp(x)₍₎

.

That is, ifpis bounded, then norm convergence is equivalent to convergence with respect

to the modularρp(x). For a bounded exponent the dual space (Lp(x)())can be identified

withLp(x)_{(), where the conjugate exponentp(x) is defined byp(x) =} p(x)

p(x)–for eachx∈.

If  <p–_≤p+_{<∞, thenL}p(x)_{() is separable and reflexive.}

In the variable exponent Lebesgue space, Hölder’s inequality is still valid; see [], Theo-rem .. For allu∈Lp(x)_(),v∈Lp(x)_{() withp(x)∈(,∞) the following inequality holds:}

|uv|dx≤

 p– +

 (p)–

u _Lp(x)₍₎ v _Lp(x)₍₎≤ u _Lp(x)₍₎ v _Lp(x)₍₎.

If  <||<∞andp,qare variable exponents such thatp(x)≤q(x) for eachx∈, then there exists a continuous embeddingLp(x)_()→Lq(x)_().

Definition .(see []) We say that a bounded exponentp:→Ris log-Hölder

contin-uous if there is a constantC>  such that

p(y) –p(z)log|y–z| ≤C

for all pointsy,z∈.

The variable exponent Sobolev spaceWk,p(x)_{() is defined as}

Wk,p(x)() =u∈Lp(x)() :Dαu∈Lp(x)(),|α| ≤k

and equipped with the norm

u _Wk,p(x)₍₎=

|α|≤k _Dα

u_Lp(x)₍₎,

then the spaceWk,p(x)_{() is a Banach space. The spaceW}k,p(x)

 () is defined as the closure

ofC_∞() with the above norm. If  <p–_≤p+_{<∞, then the spaceW}k,p(x)_{() is separable}

Theorem .(see []) Let⊂RN_{be a bounded domain and assume that p:R}N _→(,∞)

is a bounded log-Hölder continuous exponent such that p–_{> ,then for any u∈W},p(x)  ()

we have

u _Lp(x)₍₎≤c ∇u _Lp(x)₍₎,

where the constant c only depends on the dimension N,||and the log-Hölder constant of p.

Theorem .(see []) Letbe a bounded domain with smooth boundary.Assume that p:→(,∞)is a bounded log-Hölder continuous function with p+_<N

k and q:→(,∞)

is a bounded measurable function with q(x)≤p∗=_NNp(x)_–kp(x).Then there is a continuous em-bedding

Wk,p(x)()→Lq(x)(),

where the embedding constant depends on||,N,q+_{and the log-Hölder constant of p.}

Theorem .(see []) Let⊂RN_{be a bounded domain with smooth boundary.Suppose}

that p is a bounded log-Hölder continuous functions in,with p–_{> .Then there exists a}

constant C> depending only on N,and the log-Hölder constant of p such that for each u∈W_,p(x)(),the following inequality holds:

u _W,p(x)  ()≤C

u _Lp(x)₍₎.

Proposition .(see [, ]) Letbe a bounded domain inRN and let{ωi}∞_i= be an orthogonal basis in L_{().Then for anyε> ,there exists a positive number N}

εsuch that

u L₍₎≤ _Nε

i=

uωidx 



+ε u _W,p

 ()

for all u∈W_,p()where≤p<∞.

The following theorem gives a relation between almost everywhere convergence and weak convergence.

Theorem .(see []) Let p:→Rbe a bounded log-Hölder continuous function with p–_{> .If{un}}∞

n=is bounded in Lp(x)(QT)and un→u a.e.in QTas n→ ∞,then there exists

a subsequence of{un}still denoted by{un}such that unu weakly in Lp(x)(QT)as n→ ∞.

Denote by C(RN) (N≥) the closure of continuous functions in RN with compact

support. The dual of C(RN) can be identified with the spaceM(RN) of signed Radon

measures with finite mass via the pairing

μ,f=

A mapμ:E→M(RN_{) (E⊂R}N_{) is called weak∗measurable if the functionsx→ μ(x),f}

are measurable for allf ∈C(RN). We writeμxinstead ofμ(x).

Theorem .(see [], Theorem .) Let E⊂RN _{be a measurable set of finite measure}

and let zj:E→RNbe a sequence of measurable functions.Then there exist a subsequence

zjk and a weak∗measurable mapν:E→M(R

N_{)such that}

(i) νx≥, νx M(RN₎=

Rd dνx≤,for a.e.x∈E.

(ii) For allf∈C(RN)

f(zjk)νx,f weakly∗ inL

∞_(E).

(iii) LetK⊂RN _{be compact.Then}

suppνx⊂K ifdist(zjk,K)→in measure. (iv) Furthermore one has

i νx _M(RN₎= , for a.e.x∈E

if and only if the sequence does not escape to infinity,i.e. if

lim

L→∞sup_k

_{x∈E:|zjk| ≥L}= . (.)

(v) If(i)holds,ifA⊂Eis measurable andf ∈C(RN)and iff(zjk)is relatively weakly compact inL_(A),thenf(z

jk)νx,fweakly inL(A). (vi) If(i)holds,then in(iii)one can replace ‘if ’ by ‘if and only if ’.

Remark . If for somes>  and allj∈N

E

|zj|s≤C

then (.) holds.

3 Existence of weak solutions of problem (1.2)

In this section, letε∈(, ) fixed, we prove the existence of weak solutions for problem (.). Our main hypotheses are the following:

(H) p,q:→(,∞)are two log-Hölder continuous functions satisfying

max

, N

N+  <p

–_=inf

p(x)≤p+=sup

p(x) <N 

and

 <q–=inf

q(x)≤q(x) < Np(x)

N– p(x), forx∈.

(H) a∈C([,∞))and there exists a constanta> such that

a(s)≥a> , fors∈[,∞).

(H) u∈W,p(x)()∩W,(),u∈L(),f ∈L(QT).

Definition . A functionuε:QT→Ris called a weak solution of (.), if ⎧

⎨ ⎩

uε∈L∞(,T;W_,p(x)())∩L∞(,T;Lq(x)())∩C(,T;W_,()), ∂uε

∂t ∈L∞(,T;L

_())∩L_(,T;W,  ()),

and

–

∂uε(x,τ)

∂t ϕ(x,τ)dx–

QT ∂uε

∂t ∂ϕ ∂t dx dt+

QT

|uε|p(x)–uεϕ

+ε∂uε

∂t ϕdx dt+

QT a

|uε|q(x)dx

∂uε

∂t ϕdx dt

=

QT

fϕdx dt+

uϕ(x, )dx,

for allϕ∈C_(,T;C∞

 ()) andτ∈(,T].

We choose a sequence{ωj}∞_j=⊂C_∞() such thatC_∞()⊂∞_n=Vn C(¯)

and{ωj}∞_j=is a complete orthogonal basis inL_{(), whereV}

n=span{ω,ω, . . . ,ωn}; see [, ].

Since∞_n=Vnis dense inC(¯), we have the following lemma.

Lemma .(see []) For the function u∈W,p(x)()∩W,(),there exists a sequence

ψnwithψn∈Vnsuch thatψn→uin W,p(x)()∩W,()as n→ ∞.

Proof Foru∈W,p(x)()∩W,(), there exists a sequence{vn}inC∞() such thatvn→

uinW,p(x)()∩W,(). Since{vn} ⊂C∞()⊂ ∞

m=Vm C(¯)

, we can find a sequence

{vk_{n} ⊂}∞

m=Vmsuch that, for eachn∈N, we havevkn→uninC() ask→ ∞. For _n, there existskn≥ such that vknn–un C₍₎≤ 

n. Thus

vkn

n –u_W,p(x)_()∩W,  ()≤C

vkn

n –vn_C₍₎+ vn–u _W,p(x)_()∩W,  ().

That is,vkn

n →uinW,p(x)()∩W,() asn→ ∞. Denoteun=vknn. Sinceun∈ _∞

m=Vm,

there existsVmnsuch thatun∈Vmn; without loss of generality, we assume thatVm⊂Vm asm≤m. We assume thatm>  and defineψnas follows:ψn(x) = ,n= , . . . ,m– ;

ψn=u,n=m, . . . ,m– ;ψn=u,n=m, . . . ,m– ; . . . , then we obtain the sequence {ψn}andψn→uinW,p(x)()∩W,() asn→ ∞.

The existence of weak solutions of problem (.) is proved by Galerkin’s approximation. We shall find the sequence of approximate solutions in the form

un(x,t) = n

j=

ηn(t)

The unknown functions (ηn(t))jare determined by ordinary differential equations in the

following.

We first define a vector-valued functionPn(t,μ,ν) : [,T]×Rn×Rn→Rnas follows:

Pn(t,μ,ν) i = n j=

μjωj p(x)– _n j=

μjωj

ωi+ n j=

μjωj q(x)– _n j=

μjωj

ωidx

+ ε n j=

νjωjωi+a n j=

νjωj q(x) dx

ωidx, i= , . . . ,n,

whereμ= (μ, . . . ,μn) andν= (ν, . . . ,νn). Now we consider the following Cauchy problem

of second-order ordinary differential equations:

⎧ ⎨ ⎩

η(t) +Pn(t,η(t),η(t)) =Fn(t),

η() =Un, η() =Un,

(.)

where (Un)i=

ψnωidx, (Un)i=

φnωidx, Fn=

fnωidx, ψn∈Vn, φn∈Vn, fn∈

C_∞(QT), andψn→ustrongly inW,p(x)()∩W,() (ψnfrom Lemma .),φn→u

strongly inL_(),f

n→f strongly inL(QT) (sinceC∞ (QT) is dense inL(QT)).

Letη(t) =X(t),Y(t) = (η(t),X(t)), andHn(t,Y) = (X,Fn–Pn(t,η,X)). Then the problem

(.) is transformed into the following problem:

⎧ ⎨ ⎩

Y(t) =Hn

t,Y(t),

Y() = (Un,Un).

(.)

The assumption (H) implies

Pn(t,η,X)X=Pn

t,η,ηη

=

|un|p(x)–un

∂un

∂t +|un|

q(x)–_u n

∂un

∂t dx+ε

∂un ∂t

∂un

∂t dx

+ a

|un|q(x)dx∂un ∂t dx ≥ d dt

|un|p(x)

p(x) +

|un|q(x)

q(x) dx+ε

∂un

∂t

dx+a

∂un

∂t

dx.

From (.) and Young’s inequality, we obtain

YY+ d dt

|un|p(x)

p(x) +

|un|q(x)

q(x) dx+ε

∂un

∂t

dx+a 

∂un

∂t dx ≤C  |Y|

₊

|un|p(x)

p(x) dx+

f_ndx

Thus,

d dt

 |Y|

₊

|un|p(x)

p(x) +

|un|q(x) q(x) dx

≤C

 |Y|

₊

|un|p(x)

p(x) dx+

f_ndx

.

Gronwall’s inequality andfn→f strongly inL(QT) imply

|Y|+

|un|p(x)

p(x) dx≤C, (.)

whereCis a constant independent ofnandε. Thus,|Y–Y()| ≤√C. We denote

Ln= max

(t,Y)∈[,T]×B(Y(),√C)

Hn(t,Y), τn=min

T,

√

C Ln

,

whereB(Y(), √C) is the ball of radius √Cwith center at the pointY() inRn_{. From}

the definition ofH(t,Y),H(t,Y) is continuous with respect to (t,Y). By Peano’s theorem, we know that (.) admits aCsolution on [,τn], that is, (.) has aCsolution on [,τn]

denoted byη

n(t). Letη(τn), ∂η∂(τtn) be the initial value of problem (.), then we can repeat

the above process and get aC _solutionη

n(t) on [τn, τn]. Without loss of generality, we

assume thatT= [_τT n]τn+ (

T

τn)τn,  < (

T

τn) < , where [

T

τn] is the integer part of

T τn, (

T τn) is the

decimal part of T

τn. We can divide [,T] into [(i– )τn,iτn],i= , . . . ,L, and [Lτn,T] where L= [_τT_n], then there exists aCsolutionη_ni(t) in [(i– )τn,iτn],i= , . . . ,L, andηnL+(t) in

[Lτn,T]. Therefore, we get a solutionηn(t)∈C([,T]) defined by

ηn(t) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

η_n(t), ift∈[,τn],

η

n(t), ift∈(τn, τn], · · ·

ηL

n(t), ift∈((L– )τn,Lτn],

ηL+_n (t), ift∈(Lτn,T].

Lemma .(A prioriestimates) The estimates

∂un(x,t)

∂t

dx+

un(x,t)p(x)+un(x,t)q(x)dx+ε

un(x,t)dx≤C,

∀t∈[,T],

QT

∂un

∂t

dx dt+

QT

|un|p(x)+|un|q(x)dx dt+ε

QT

∂un

∂t

dx dt≤C

hold uniformly with respect to n.

Proof By (.), we have

_un(x,t)

+∂un(x,t) ∂t

dx+

un(x,t) p(x)

+un(x,t) q(x)

dx≤C,

Further, integrating the inequality (.) with respect totover [,T], we obtain

QT

ε∂un ∂t

+∂un

∂t

dx dt≤C.

Moreover, for eacht∈[,T],

un(x,t) 

dx≤T

T

 ∂un

∂t

dx dt+ 

un(x, ) 

dx≤C ε.

Thus, this lemma is proved

By Lemma ., we have the following.

Lemma . The estimate

un

L∞(,T;W,p(x)())+

_|un|p(x)–un_Lp(x)_(Q

T)

+|un|q(x)–un_Lp(x)_(Q

T)+

a

|un|q(x)dx

∂un

∂t

L_(Q

T)

≤C

holds uniformly with respect to n andε.

Proof By Theorem ., we have

un _W,p(x)  ()≤C

un _Lp(x)₍₎≤C.

Thus, un

L∞(,T;W_,p(x)())≤C. By Lemma ., we obtain

QT

unp(x)–∇unp

_(x)

dx dt≤

QT

|∇un|p(x)dx dt≤C.

Thus,

_|un|p(x)–∇un_Lp(x)_(Q

T)

≤max

|un|p(x)dx

p–– p–

,

|un|p(x)dx

p+– p+

≤C.

Similarly, |un|q(x)–_un Lq(x)_(Q

T)≤C. Sincea∈C([,∞)) and

|un(x,t)|q(x)dx≤C, we

have

a

|un|q(x)dx

∂un

∂t

L_(Q

T)

≤C.

This lemma is proved.

Proof By Lemma . and Lemma ., there exist a subsequence of{un}(still denoted by

{un}),uε,ξ,η, andζsuch that ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

∂un

∂t ∂uε

∂t weakly ∗ inL∞(,T;L()),

unuε weakly∗ inL∞(,T;W,p(x)()∩Lq(x)())

∩L∞(,T;W_,()),

∂un

∂t ∂uε

∂t weakly inL

_(,T;W,  ()), |un|p(x)–_u

n ξ weakly inLp

_(x)

(QT), |un|q(x)–_u

n η weakly inLq

_(x)

(QT),

a(|un|q(x)_dx)∂un

∂t ζ weakly inL(QT).

Sinceun∈L∞(,T;W,()) and ∂un

∂t ∈L(QT), by the Lions-Aubin lemma, there exists a

subsequence of{un}still denoted by{un}such thatun→uεstrongly inL(QT) and a.e.

onQT. Further,|un|q(x)–un→ |uε|q(x)–uεa.e. onQT. In view of Theorem ., we obtain

η=|uε|q(x)–uε.

Next, we prove that there exists a subsequence of{un}(still denoted by{un}) such that

∂un

∂t → ∂uε

∂t strongly inL _(Q

T).

Since (η_n(t))j=

∂un

∂t ωjdx, by Lemma ., (ηn(t))jis uniformly bounded on [,T]. For ∀≤t<t≤T, integrating (.) with respect totfromttot, we have

∂un(x,t)

∂t ωjdx–

∂un(x,t)

∂t ωjdx+

t

t

|un|p(x)–unωj+|un|q(x)–unωj

+ε∂un ∂t ωj+a

|un|q(x)dx

∂un

∂t ωjdx dt=

t

t

fnωjdx dt.

Hölder’s inequality, Lemma ., and Lemma . imply

ηn(t)

j–

ηn(t)

j

≤|un|p(x)–un_Lp(x)_(Q

T) ωj Lp(x)(Qtt)

+|un|q(x)–un_Lq(x)_(Q

T) ωj Lq(x)(Qtt)

+∂un ∂t

L_(Q

T) ωj

L_(Qt

t_)+

a

|un|q(x)dx

∂un

∂t

L_(Q

T) ωj

L_(Qt

t_)

≤C ωj

Lp(x)_(Qt

t)

+ ωj

L_(Qt

t)

+ ωj

L_(Qt

t)

+ ωj

Lq(x)_(Qt

t)

≤max|t–t| 

p–

,|t–t| 

p+

,|t–t| 

_,|_t–t| 

q–

,|t–t| 

q+

×

|ωj|p(x)dx



p–

+

|ωj|p(x)dx



p+

+

|ωj|dx

 

+

|ωj|q(x)dx



q– +

|ωj|q(x)dx



q+ +

|ωj|dx

 

≤C(j)max|t–t| 

p–

,|t–t| 

p+

,|t–t| 

_,|_t–t| 

q–

,|t–t| 

q+

,

whereQtt=×(t,t). Thus, the sequence{(ηn(t))j}∞n=is uniformly bounded and

diagonal procedure, there exists a subsequence of{(ηn)j}still denoted by{(ηn)j}such that

(ηn(t))jconverges uniformly on [,T] to some continuous functionλεj(t) for each fixed

j= , , . . . .

Forr≤nwithr∈N, by Lemma ., we have

r

j=

η_n(t)_j≤

∂un

∂t

dx≤C, ∀t∈[,T].

Lettingn→ ∞, we get

r

j=

λε_j(t)≤C, ∀t∈[,T].

Then lettingr→ ∞, we obtain

∞

j=

λε_j(t)≤C, ∀t∈[,T].

Setuε(x,t) = ∞

j=λεj(t)ωj(x), thensup≤t≤T uε(x,t) L₍₎≤C(T) and, for eachj∈N, we

have

lim

n→∞

∂un

∂t ωjdx=

uεωjdx (.)

uniformly on [,T]. For eachδ>  andφ∈L(), by the completeness of{ωj}, there exists

am>  such that φ– m

i=(

φωidx)ωi L()≤δ. Thus,

∂un

∂t –u¯ε

φdx≤∂un ∂t –u¯ε

L₍₎ φ–

m

i=

φωidx

ωi

L₍₎

+

∂u ∂t –u¯ε

m

i=

φωidx

ωidx

≤Cδ+

∂un

∂t –u¯ε

m

i=

φωidx

ωidx

. (.)

Forδ> , by (.), there exists aM>  such that

∂un

∂t –u¯ε

ωidx ≤ δ

m

, forn>Mandi= , . . . ,m.

By (.) and the Hölder inequality, we have

∂un

∂t –u¯ε

φdx≤∂un ∂t –u¯ε

L₍₎ φ–

m

i=

φωidx

ωi

L₍₎

+

∂u ∂t –u¯ε

m

i=

φωidx

≤Cδ+ m

i=

φωidx

∂un

∂t –u¯ε

ωidx

≤C+ φ L₍₎δ_, forn>M. (.)

It follows from (.) and the arbitrariness ofδthat

∂un

∂t uε weakly inL

_{(). (.)}

uniformly on [,T] asn→ ∞. For eachϕ∈C_∞(QT), by Lebesgue’s dominated

conver-gence theorem, we obtain

lim

n→∞

QT

∂un

∂t –uε

ϕdx dt= .

Hence,

lim

n→∞

QT ∂un

∂t ϕdx dt=

QT

uεϕdx dt.

On the other hand, by integration by parts, we get

QT ∂un

∂t ϕdx dt= –

QT un

∂ϕ ∂t dx dt.

Lettingn→ ∞in above equality, we have

QT

uεϕdx dt= –

QT uε

∂ϕ

∂t dx dt, forϕ∈C

∞

 (QT).

Thus, we obtainu=∂_∂uε_t . Moreover, for eachj∈N, Lemma ., and Lebesgue’s dominated convergence theorem yield

lim

n→∞ T



∂un

∂t – ∂uε

∂t

ωjdx 

dt= .

Thus, forδ> , by Proposition ., there exists a positive numberNδ independent ofn

such that

∂un

∂t – ∂uε

∂t

L_(Q

T)

≤

Nδ

i= T



∂un

∂t – ∂uε

∂t

ωidx 

dt+ δ

T

 ∂un

∂t – ∂uε

∂t



W_,()

dt.

A similar discussion to (.) shows that there is aM( δ) >  such that

∂un

∂t – ∂uε

∂t

L_(Q

T)

Thus, ∂un

∂t → ∂uε

∂t strongly inL(QT). Further, there exists a subsequence of{un}still

de-noted by{un}such that∂un

∂t → ∂uε

∂t a.e. onQT.

For∀ϕ∈C(,T;C_∞()), we can choose a sequenceϕk∈C(,T;Vk) such thatϕk→ϕ

inC,_(Q

T). Here forv∈C,(QT) equipped with the norm v =sup|α|≤,(x,t)∈QT{|D

α_v|, |∂v

∂t|}. For∀τ∈(,T], we have

lim

k→∞nlim→∞

Qτ

∂_u n

∂t ϕkdx dt

= lim

k→∞nlim→∞

∂un(x,τ)

∂t ϕk(x,τ)dx–

∂un(x, )

∂t ϕk(x, )dx

–

Qτ

∂un

∂t ∂ϕk

∂t dx dt

= lim

k→∞

∂uε(x,τ)

∂t ϕk(x,τ)dx–

uϕ(x, )dx–

Qτ

∂uε

∂t ∂ϕk

∂t dx dt

=

∂uε(x,τ)

∂t ϕ(x,τ)dx–

uϕ(x, )dx–

Qτ

∂uε

∂t ∂ϕ

∂t dx dt

= lim

n→∞

Qτ

∂un

∂t ϕdx dt,

whereQτ=×(,τ). Replacingωiin (.) byϕk, we obtain

Qτ

∂_u n

∂t ϕkdx dt+

Qτ

|un|p(x)–unϕk+|un|q(x)–unϕk+ε

∂un

∂t ϕk

+a

t,

|un|q(x)dx

∂un

∂t ϕkdx dt=

Qτ

fnϕkdx dt.

Thus, we have

lim

n→∞

Qτ

∂_u n

∂t ϕdx dt=

Qτ

fϕ–ξ ϕ–|uε|q(x)–uεϕ–ε

∂uε

∂t ϕ–ζ ϕdx dt. (.)

Furthermore, for anyψ(x)∈C_∞(), we get

∂uε(x,τ)

∂t –u

ψdx

= lim

n→∞

∂un(x,τ)

∂t –

∂un(x, )

∂t

ψ(x)dx

= lim

n→∞ τ



∂un

∂t ψ(x)dx dt

=

Qτ

fϕ–ξ ϕ–|uε|q(x)–uεϕ–ε

∂uε

∂t ϕ–ζ ϕdx dt→

asτ →. Similarly, fort∈[,T], we have

lim

τ→

∂uε(x,τ)

∂t –

∂uε(x,t)

∂t

Furthermore, we obtain ∂uε(x,)_∂t =u. Since uε∈L∞(,T;W_,()) and ∂_∂uε_t ∈ L(,T;

W_,()), we can assume that uε∈ C(,T;W,()). Lemma . and the embedding

W_,()→L_{() imply that} u



n(x,T)dx≤C(T). Thus, there exist a subsequence of {un}still denoted by {un}and a functionusuch thatun(x,T)uweakly inL(). For

eachϕ∈C_∞() andη∈C_{([,T]), we have}

QT ∂un

∂t ϕηdx dt=

un(x,T)ϕη(T) –un(x, )ϕη()dx–

QT

unϕη(t)dx dt.

Lettingn→ ∞, we get

QT ∂uε

∂t ϕηdx dt=

uϕη(T) –uϕη()dx–

QT

uεϕη(t)dx dt.

By integration by parts, we have

uε(x,T) –u

ϕη(T)dx=

uε(x, ) –u

ϕη()dx.

Choosingη(T) = ,η() =  orη(T) = ,η() = , we obtainu=uε(x,T) anduε(x, ) =

u(x) for x∈. Similarly, we can prove that uε(x, ) =u, un(x,T) uε(x,T)

weakly inL_{() (up to a subsequence) and}

uε(x,T) 

dx≤lim inf

n→∞

un(x,T) 

dx. (.)

Further, by the compact embedding W_,()→→L_{(), we getu}

n(x,T)→uε(x,T)

strongly inL_().

Takingϕ=ukin (.) and lettingk→ ∞, we get

∂uε(x,T)

∂t uε(x,T)dx–

uudx–

QT

∂uε

∂t

dx dt+

QT

ξ uε+|uε|q(x)

+ε∂uε

∂t uε+ζuεdx dt=

QT

fuεdx dt. (.)

Multiplying (.) by (ηn)jand summing upjfrom  ton, then integrating with respect to

tover [,T], we have

T



∂un

∂t undx dt+ T



|un|p(x)+|un|q(x)+ε∂un ∂t un

+a

|un|q(x)dx

∂un

∂t undx dt=

T



fnundx dt. (.)

Thus,

≤

T



|un|p(x)–un–|uε|p(x)–uε

(un–uε)dx dt

=

T



fnun–|un|q(x)–a

|un|p(x)dx

∂un

∂t un–ε ∂un

–

∂un(x,T)

∂t un(x,T)dx+

∂un(x, )

∂t un(x, )dx+

T



∂un

∂t

dx dt

–

T



|un|p(x)–unuε+|uε|p(x)–uε(un–uε)dx dt.

By (.) and (.), we get

lim sup

n→∞ T



|un|p(x)–un–|uε|p(x)–uε

(un–uε)dx dt

≤ T



fuε–|uε|p(x)–a

|uε|q(x)dx

∂uε

∂t uε–ξ uεdx dt

–ε



uε(x,T) 

dx+ε 

uε(x, ) 

dx–

uε(x,T)

∂t uε(x,T)dx

+

uudx+ T



∂uε

∂t

dx dt

= .

It follows that

lim

n→∞ T



|un|p(x)–un–|uε|p(x)–uε

(un–uε)dx dt= .

Following the ideas of [], we setQ={(x,t)∈QT:p(x)≥}andQ={(x,t)∈QT:  <

p(x) < }, then, asn→ ∞,

Q

|un–uε|p(x)dx dt

≤C

Q

|un|p(x)–un–|uε|p(x)–uε

(un–uε)dx dt

→

and

Q

|un–uε|p(x)dx dt

≤C|un|p(x)–un–|uε|p(x)–uε

(un–uε) p(x)

 L



p(x)_(Q)

×|un|p(x)+|uε|p(x) –p(x)

 L

 –p(x)_(Q)

→.

Therefore, we obtainun→uεstrongly inLp(x)(Qτ). Thus, there exists a subsequence

of{un}still denoted by{un}such thatun→uεa.e. onQT. Further,

In view of Theorem ., we getξ =|uε|p(x)–uε. Similarly, we can prove thatun→uε

strongly inLq(x)_(Q

T). Thus, there exists a subsequence of{un}still denoted{un}such that

lim

n→

un(x,t) –uε(x,t) q(x)

dx= , for a.e.t∈[,T].

Furthermore, we have

lim

n→

un(x,t)q(x)dx=

uε(x,t) q(x)

dx, for a.e.t∈[,T].

Thus,a(|un|q(x)dx)∂un

∂t →a(

|uε|

q(x)_dx)∂uε

∂t a.e. on QT. By Theorem ., we obtain

ζ =a(|uε|q(x)dx)∂_∂uε_t . It follows from (.) that the theorem is proved.

Remark . Obviously, in this section, the two inequalities in (H) can be replaced by  <p–_≤p+_{<∞and  <q}–_≤q+_{<∞, respectively.}

4 Existence of Young measure solutions for problem (1.2)

In this section, from the sequence of approximate solutions{uε}<ε<of problem (.), we

shall prove that the limit ofuε(asε→+) is a Young measure solution of problem (.).

Definition . A pair (u,ν) is called a Young measure solution of problem (.) if

u∈L∞,T;W_,p(x)()∩L∞,T;Lq(x)()∩W,∞,T;L(),

ν={νx,t}x,tis a probability measure,

and

∂u(x,τ) ∂t ϕdx–

uϕ(x, )dx–

QT ∂u

∂t ∂ϕ

∂t dx dt+

QT

R|A|

p(x)–_{A dν(A)ϕ}

+|u|q(x)–uϕdx dt+

QT a

|u|q(x)dx

∂u

∂tϕdx dt=

QT

fϕdx dt,

for allϕ∈C_(,T;C∞

 ()) andτ∈(,T].

Theorem . Under conditions(H)-(H),problem(.)has a Young measure solution.

Proof For eachτ∈(,T] andϕ∈C_(,T;C∞

 ()), we have by (.)

∂uε(x,τ)

∂t ϕ(x,τ)dx–

uϕ(x, )dx–

Qτ

∂uε

∂t ∂ϕ

∂t dx dt+

Qτ

|uε|p(x)–uεϕ

+|uε|q(x)–uεϕ+a

|uε|q(x)dx

∂uε

∂t ϕ+ε ∂uε

∂t ϕdx dt=

Qτ

fϕdx dt. (.)

Since the constant in Lemma . is independent ofnandε, by the convergence ofunand

∂un

∂t in Section , we have

∂uε(x,t)

∂t

dx+

uε(x,t) p(x)

+uε(x,t) q(x)

dx+ε

uε(x,t) 

for a.et∈[,T] and

QT

∂uε

∂t

dx dt+

QT

|uε|p(x)+|uε|q(x)dx dt+ε

QT

∂uε

∂t

dx dt≤C.

Similarly, by Lemma ., we have

uε _L∞_(,T;W,p(x)  ())

+|uε|p(x)–uε_Lp(x)_(Q

T)

+|uε|q(x)–uε_Lq(x)_(Q

T)+

a

|uε|q(x)dx

∂uε

∂t

L_(Q

T)

≤C. (.)

Thus, there exists a subsequence of{uε}<ε<still denoted by{uε}<ε<such that ⎧

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

uεu weakly∗ inL∞(,T;W,p(x)()∩Lq(x)()),

uε u weakly inLp(x)(QT),

uεu weakly inLq(x)(QT),

∂uε ∂t

∂u

∂t weakly∗ inL∞(,T;L()),

|uε|q(x)–uε α weakly inLq

_(x)

(QT),

a(|uε|q(x)dx)∂_∂uε_t β weakly inL(QT).

Sincep–>max{,_NN_+}, the embeddingW_,p(x)()→L() is compact. Further, asuε∈

L∞(,T;W,p(x)()) and ∂∂uεt ∈L∞(,T;L

_{()), by the Lions-Aubin lemma, there exists a}

subsequence of{uε}<ε<still denoted by{uε}<ε<such thatuε→ustrongly inL(QT) and

a.e. onQT. Thus,|uε|q(x)–uε→ |u|q(x)–ua.e. onQT. In view of Theorem ., we obtainα= |u|q(x)–_{u. By assumption (H), we haveμ=inf}

( Np(x)

N–p(x)–q(x)) > . For each measurable

subsetU⊂QTwith|U| ≤, by Hölder’s inequality, Theorem ., and Theorem ., we

obtain

|uε|q(x)≤|uε|q(x) L

Np(x)

(N–p(x))q(x)_(U)  _L

Np(x)

Np(x)–Nq(x)+p(x)q(x)_(U)≤C|U| μ(N–p+)

Np+

.

Thus, the sequence{|uε–u|q(x)}<ε<is equi-integrable onL(QT). The Vitali convergence

theorem implies that_Q

T|uε–u|

q(x)_{dx dt→, that is to say, we obtainu}

ε→ustrongly in

Lq(x)(QT). Thus, there exists a subsequence of{uε}<ε<still labeled by{uε}<ε<such that

lim

ε→

|uε–u|q(x)dx= , for a.e.t∈[,T].

Furthermore,

lim

ε→

_|uε|q(x)–|u|q(x)dx= , for a.e.t∈[,T].

Hence, we find by the continuity ofathata(|uε|q(x)dx)→a(

|u|

q(x)_{dx) for a.e.t∈}

[,T]. Since|uε|q(x)dx≤Cfor a.e.t∈[,T] anda∈C([,∞)), for eachϕ∈L(QT), by

Lebesgue’s dominated convergence theorem, we have

a

|uε|q(x)dx

ϕ→a

|u|q(x)dx

Further, by the weak convergence of ∂_∂uε_t inL_(Q

T), we get

lim

ε→

QT a

|uε|q(x)dx

∂uε

∂t ϕdx dt=εlim→

QT ∂uε

∂t

a

|uε|q(x)dx

ϕ

dx dt

=

QT a

|u|q(x)dx

∂u ∂tϕdx dt.

Thus, a(|uε|q(x)dx)∂_∂uε_t a(

|u|q(x)dx) ∂u

∂t weakly inL(QT). The uniqueness of the

limit implies thatβ=a(|u|q(x)_dx)∂u ∂t.

Finally, we prove that the sequence{uε}<ε<generates a Young measure{νx,t}x,t such

that

|uε|p(x)–uε

R|A|

p(x)–_{A dν}

x,t(A) weakly inL(QT). (.)

Following Theorem ., we first verify that the Young measureνx,t generated by the

se-quence{uε}<ε<is a probability measure for a.e. (x,t)∈QT. Indeed, fors≤p–, we have

QT

|uε|sdx≤ ||T+

QT

|uε|p(x)dx≤C.

It follows from (iv) in Theorem . that νx,t is a probability measure. Set H(x,A) =

|A|p(x)–_{A. Next, we prove that the sequence{H(x,u}

ε)}εis weakly relatively compact in

L_(Q

T). It is clear that{H(x,uε)}εwill be weakly relatively compact inL(QT), if we prove

that{H(x,uε)}εis uniformly bounded and equi-integrable onL(QT); see [],

Proposi-tion .. Indeed, for each measurable subsetU⊂QT with|U| ≤, by (.) and Hölder’s

inequality, we have

U

_H(x,uε)dx dt=

U

|uε|p(x)–dx dt≤ uε _Lp(x)_(U)  _Lp(x)_(U)≤C|U| 

p+_.

Thus, the sequence{H(x,uε)}εis equi-integrable. Similarly, the sequence{H(x,uε)}ε

is uniformly bounded onL_(Q

T). Therefore, the convergence property (.) holds.

The estimate (.) implies

ε∂uε

∂t → strongly inL

_(Q T).

From the same procedures as in Section , we can prove there exists a subsequence of

{uε}<ε<still denoted by{uε}<ε<such that∂uε(x,t)∂t ∂u(x,t)

∂t weakly inL

_{() uniformly on}

[,T]. Takingε→ in Definition ., we obtain

∂u(x,τ)

∂t ϕ(x,τ)dx–

uϕ(x, )dx–

QT ∂u ∂t

∂ϕ ∂t dx dt

+

QT

R|A|

p(x)–_{A dν}

x,t(A)ϕ+|u|q(x)–uϕ+a

|u|q(x)dx

∂u ∂tϕdx dt

=

QT

fϕdx dt,

for allϕ∈C_(,T;C∞

By Theorem ., we have the following corollary.

Corollary . Suppose that f(x,t)≡and(H)and(H)are satisfied.Then for a given u∈W,p(x)()∩W,(),there exist a function u:×(,∞)→Rand a Young measure

νx,tsuch that for∀T> ,

u∈L∞,T;W_,p(x)()∩L∞,T;Lq(x)()∩W,∞,T;L()

and

∂u(x,τ)

∂t ϕ(x,τ)dx–

∂u(x, )

∂t ϕ(x, )dx–

QT ∂u ∂t

∂ϕ ∂t dx dt

+

QT

R|A|

p(x)–_Adν

x,t(A)ϕ+|u|q(x)–uϕ+a

|u|q(x)dx

∂u

∂tϕdx dt= ,

for allϕ∈C_(,T;C∞

 ())andτ∈(,T].

5 Energy decay of Young measure solutions

In this section, we give the decay estimates of weak solutions obtained by Corollary .. First we give a lemma by Nakao []

Lemma .(see []) Let: (,∞)→Rbe a bounded nonnegative function.If there exist two constantsα> andβ≥such that

sup

t≤s≤t+

+β(s)≤α(t) –(t+ ), for∀t≥,

then there exist positive constants C andγ such that

⎧ ⎨ ⎩

(t)≤Ce–γt_{, ∀t≥,asβ= ,}

(t)≤C(t+ )–β_, ∀_t≥_{,asβ> .}

Theorem . Let p–_{≥.Then there exist constants C,γ > such that the weak solutions}

obtained by Corollary.have the following estimates:If p+_{= ,then}

∂u(x,t)

∂t

dx+

_u(x,t)q(x)

dx≤Ce–γt, for a.e. t≥.

If p+> ,then

∂u(x,_∂t t)dx+

u(x,t)q(x)dx≤C(t+ )– p+

p+–_{, for a.e. t≥.}

Proof We define

In(t) =

 

∂un

∂t

dx+

|un|p(x)

p(x) +

|un|q(x)

q(x) dx.

The definition ofIn(t) and equality (.) implyIn(t) is nonnegative and uniformly bounded.

(H) that

d

dtIn(t) +ε

∂un

∂t

dx+a

∂un

∂t

dx≤. (.)

This impliesIn(t) is a nonincreasing function. PuttingJn(t) =In(t) –In(t+ ) and integrating

(.) over (t,t+ ), we get

J_n(t)≥ε

t+

t

∂un(x,τ) ∂t

+a

∂un(x,τ)

∂t

dx dτ

≥a t+

t

∂un(x,τ)

∂t

dx dτ. (.)

By the mean value theorem and (.), there existt∈[t,t+_] andt∈[t+_,t+ ] such

that

∂un(x,ti)

∂t

dx≤  a

J_n(t), i= , . (.)

From (.), we have

|un|p(x)+|un|q(x)dx

= –

∂_u n

∂t un+ε

∂un

∂t un+a

|un|q(x)dx

∂un

∂t undx. (.)

Integrating (.) fromttot, we obtain

t

t

un(x,τ) p(x)

dx dτ+

t

t

|un|q(x)dx dτ

= –

∂un(x,t)

∂t un(x,t)dx+

∂un(x,t)

∂t un(x,t)dx+

t

t

∂un(x,τ)

∂t

dx dτ

–ε

t

t

∂un(x,τ)

∂t un(x,τ)dx dτ

–

t

t

a

|un|q(x)dx

∂un

∂t undx dτ. (.)

The Hölder inequality, (.), Theorem ., Theorem ., andIn(t) being decreasing imply

∂un(x,ti)

∂t un(x,ti)dx

≤un(x,ti)_L₍₎

∂un(x,ti)

∂t

L₍₎ ≤Cun(x,ti)_Lp(x)₍₎Jn(t)

≤C

|un(x,ti)|p(x)

p(x) dx



p+

Jn(t)

≤C

In(t) 

p+_J