Renorm alized Solu tions of Non linear Parabolic Equations in Weigthed Variab le-Exponen t Space

YOUSSEFAkd im,CHAKIRA llalouand NEZHA Elgorch

Laboratory LSI,Faculty Polydisciplinary ofTaza.University SidiM ohamed Ben Abdellah,P.O.Box 1223TazaGare,Marocco.

Received 28M arch 2015;Accep ted 5 Ju ly 2015

Renorm alized Solu tions of Non linear Parabolic Equations in Weigthed Variab le-Exponen t Space

YOUSSEFAkd im,CHAKIRA llalou?and NEZHA Elgorch

Laboratory LSI,Faculty Polydisciplinary ofTaza.University SidiM ohamed Ben Abdellah,P.O.Box 1223TazaGare,Marocco.

Received 28M arch 2015;Accep ted 5 Ju ly 2015

.This article is devoted to study the existence of renorm alized so lutions for

Weighted variable exponent Lebesgue Sobolev space;Young’s inequality;renorm alized so lution;parabolic p rob lem s.

1 In troduction

In the p resent paper w e establish the existence of renorm alized solu tions for a class of nonlinear p(x)-parabolic equation of the type:

In the p roblem(P),? is a bounded dom ain in RN,N ≥ 1,T is a positive real num ber, w hile u0∈ L1(?),f∈ L1(Q).

Theoperator-d iv(a(x,t,u,?u)isaWeigh ted Leray–Lionsoperatordefi ned on Lp?(0,T; W 1,p(x)

0(?,ω))(see assum p tions(3.1)-(3.3)of Section 3)w hich is coercive and w here H is a nonlinear low er order term,satisfying som egrow th cond ition bu tno sign cond ition orthe coercivity cond ition and the criticalgrow th cond ition on H isw ith respect to ?u and no grow th w ith respect to u.

For the degenerated parabolic equations the existence of w eak solu tions have been p roved by Aharouch et al.[1]in the case w here a is strictly m onotone,H=0 and f ∈and the p roblem(P)is stud ied by Akd im et al.[2]in the degenerated Weighted Sobolev spacew ith u=b(x,u).

Thisnotion w asadapted to the study w ith H=0 of som e nonlinear ellip tic p roblem s on Sobolev spacesa exponentvariab lew ith D irich letboundary cond itionsby Boccardo et al.[3]and Lions[4].In the case w here p(x)is a constant,som e resu lts have been p roved by Akd im etal.[5].

This paper is organized as follow s.In Section 2,w e state som e basic resu lts for the w eighted variable exponent Lebesgue-Sobolev spacesw hich is given in[8].In Section 3,w e give ou r basic assum p tion and the definition of a renorm alized solu tion of the problem(P)forw hich our p roblem hasa solu tion.In Section 4,w eestablish theexistence of such a so lu tion in Theorem 4.1.In Section 5,w e give the p roof o f Theorem 4.2,Lemm a 4.2 and Proposition 4.2(seeappend ix).

2 Prelim inaries

In this section,w e state som eelem entary p roperties for theWeighted Variable Exponent Lebesgue–Sobolev spaces Lp(x)(?,ω)w hich w illbe used in the nextsections.The basic p ropertieso f the variab le exponent Lebesgue–Sobolev spaces W1,p(x)(?,ω),that is,w hen ω(x)≡ 1 can be found from[9,10].

Let? be a bounded open subsectof RN(N ≥2).Set

For any p ∈ C+(?),w e define

Forany p∈C+(?),w e introduce thew eighted variableexponentLebesguespace Lp(x)(?,ω) that consists of allm easu rable real-valued functions u such that

Then,Lp(x)(?,ω)endow ed w ith the Luxem burg norm

becom es a norm ed space.W hen ω(x)≡ 1,w e have Lp(x)(?,ω) ≡ Lp(x)(?)and w e use the notation|u|Lp(x)(?)instead of|u|Lp(x)(?,ω).The follow ing H ¨older type inequality is usefu l for the nextsections.Thew eighted variab le exponent Sobolev space W1,p(x)(?,ω) is defined by

w here the norm is

or,equivalently

for all u ∈ W1,p(x)(?,ω).

It is signifi cant thatsm ooth functionsare notdense in W1,p(x)(? )w ithou tadd itional assum p tions on the exponent p(x).This featu re w as observed by Zhikov[11]in connection w ith the Lavrentiev phenom enon.How ever,if the exponent p(x)is log-H ¨older continuous,i.e.,there is a constan t C such that

for every x,y w ith|x?y|≤ 1/2,then sm ooth functions are dense in variable exponent Sobolev spacesand there is no confusion in defining the Sobolev spacew ith zero boundary values W1,p(x)(?),as the com p letion of C∞0(?)w ith respect to the norm ‖u‖W1,p(x)(?) (see[12]).

0(?,ω )is defined as the com p letion of C∞0(? )w.r.t.the norm ‖ u ‖W1,p(x)(?,ω ). Th roughou t the paper,w eassum e that p∈C+(?)and ω isam easurable positiveand a.e. finite function in ?.

Lemm a 2.1.([9,10])(Generalised H ¨older inequality).

i)Forany functionsu ∈ Lp(x)(?)and v ∈ Lp′(x)(?),wehave

ii)Forall p,q∈ C+( ˉ?)such that p(x)≤ q(x)a.e.in ?,wehave

Lq(x)■→ Lp(x)and the embedding is continuous.

Lemm a 2.2.([10])Denote ρ(u)=R?ω(x)|u(x)|p(x)d x forallu ∈ Lp(x)(?,ω).Then

Rem ark 2.1.([8])Ifw e set

Then,follow ing the sam e argum ent,w e have

Letω bea w eight function satisfying that

The reasons thatw eassum e(W1)and(W2)can be found in[8].

Rem ark 2.2.([8])(i)If ω isa positivem easu rable and finite function,then Lp(x)(?,ω)is a refl exive Banach space.(ii)Moreover,if(W1)holds,then W1,p(x)(?,ω )is a reflexive Banach space.

For p,s∈ C+(?),denote

w here s(x)is given in(W2).Assum e thatw e fi x the variab le exponent restrictions

for alm ost all x ∈ ?.These definitions p lay a key role in ou r paper.We shall frequently m akeuseof the follow ing(com pact)im bedd ing theorem for thew eigh ted variableexponent Lebesgue–Sobolev space in the next sections.

Lemm a 2.3.([8])Let p,s ∈ C+(?)satisfy the log-H¨older continuity condition(2.2)and let (W1)and(W2)be satisfied.Ifr ∈ C+(?)and 1 ＜ r(x) ≤ p?s(x)then,we obtain the continuous imbedding

M oreover,wehave the compact imbedding

provided that 1 ＜ r(x)＜ p?s(x)forall x ∈ ?.

From Lemm a 2.3,w ehave Poincar′e-type inequality imm ed iately.

Proposition 2.1.([8])Let p ∈ C+(?)satisfy the log-H¨older continuity condition(2.2).If(W1)

and(W2)hold,then the estimate

holds,for every u ∈ C∞0(?)w ith apositive constantC independent ofu.

Throughou t this paper,let p∈C+(?)satisfy the log-H¨older con tinuity cond ition(2.2) and X:=W 1,p(x)0(?,ω)be the w eighted variable exponent Sobolev space that consists of all real valued functions u from W1,p(x)(?,ω)w hich vanish on the boundary ??,endow ed w ith thenorm

w hich is equivalen t to the norm(2.1)due to Corollary 2.1.The follow ing proposition gives the characterization of the dual space(W0k,p(x)(?,ω))?,w hich is analogous to[9,

Theorem 3.16].We recall that the dualspace ofw eighted Sobolev spaces W01,p(x)(?,ω)is equivalent to W?1,p′(x)(?,ω),w here ω?= ω1?p′(x).

Lemm a2.4.([7])Letg∈Lp(x)(Q,ω)and letgn∈Lp(x)(Q,ω),with ‖gn‖Lp(x)(Q,ω)≤c,1＜r(x)＜∞.Ifgn(x)→ g(x)a.e.in Q,then gn? g in Lp(x)(Q,ω),where ? denotesweak convergence and ω isaweight function on Q.

Wew illalso use the standard notation for Bochner spaces,i.e.,if q ≥ 1 and X is a Banach space then Lq(0,T;X)denotes thespaceofstronglym easu rable function u:(0,T)→X for w hich t→ ‖u(t)‖X∈ Lq(0,T)M orever,C([0;T];X)denotes the space of con tinuous function u:[0;T]→ X endow ed w ith the norm ‖u‖C([0;T];X)=m axt∈[0;T]‖u‖X,

and w e defi ne the space

w here the norm is defined by:

We in troduce the functionalspace see[7]

w hich endow ed w ith thenorm

or,the equivalen tnorm

is a separable and refl exive Banach space.The equivalence of the tw o norm s is an easy consequence of the con tinuous em bedd ing Lp(x)(Q)■→ Lp?(0,T;Lp(x)(?))and the Poincar′e inequality.We state som e fu rther p ropertiesof V in the follow ing lemm a.

Lemm a 2.5.Let V be defined as in(2.6)and itsdualspacebedenote by V ? .Then,

(i)Wehave thefollow ing continuous dense embeddings:

In particular,since D(Q)is dense in Lp+(0,T;W1,p(x)0(?,ω);it is dense in V and for the corresponding dualspaces,wehave

Note that,wehave thefollowing continuousdenseembeddings

(ii)Onecan represent theelementsofV?asfollows:ifT∈V?;then thereexists F=(f1,....,fN)∈(Lp′(x)(Q))Nsuch that T=divXF and

forany ξ∈ V.M oreover,wehave

Rem ark 2.3.The space V ∩ L∞(Q),is endow ed w ith the norm definieby the form u la:

Is a Banach space.In fact,it is the dual space of the Banach space V?+L1(Q)endow ed w ith the norm:

Som e techn ical resu lts

Lemm a 2.6.Assume(3.1)-(3.3)and let(un)nbe a sequence insuch that un? u weakly inand

Then,un→ u strongly in Lp?(0,T,W01

,p(·)(?,ω)). Proof.Letthanks to(3.4),w e have Dnis a positive function and by(2.7),Dn→ 0 in L1(Q)as n → ∞.

Extracting a subsequence,still denoted by un,w e can w rite un? u a.e.in Q and since Dn→ 0 a.e.in Q.There exists a subset B in Q w ith m easu re zero such that for all (t,x)∈QB,

Taking ξn=?unand ξ=?u,w e have

w here Cx,tdepend ing on x,bu tdoesnotdepend on n.(Since un(x,t)→ u(x,t)then,(un)nisbounded),w eobtain

by the standard argum ent(ξn)nisbounded alm osteveryw here in Q.Indeed,if|ξn|→ ∞in am easu rable subset E ∈ Q,then,

w hich is absu rd since Dn(x,t)→ 0 in L1(Q)).Let ξ?an accum u lation point of(ξn)n,w e have|ξ?|＜ ∞ and by continuity of a(.,.,.,.),w eobtain

thanks to(3.2),w e have ξ?= ξ,the uniqueness of the accum u lation point im p lies that?un(x,t)→?u(x,t)a.e.in Q.Since sequence a(x,t,u,?un)isbounded in(Lp′(x)(Q,ω?))Nand a(x,t,u,?un)→ a(x,t,u,?u)a.e.in Q,Lemm a 2.4 im p lies

Letus taking ˉyn=a(x,t,u,?un)?unand ˉy=a(x,t,u,?u)?u,then ˉyn→ ˉy in L1(Q),according to the cond ition(3.3),w ehave

Let zn=|?un|p(x)ω,z=|?u|p(x)ω and yn= ˉyn/α,y= ˉy/α.Then,by Fatou’s Lemm a,w e obtain

this im p lies

w e deduce that

w hich com p letesou r p roof.

Defi n ition 2.1.A monotonemap T:D(T)→ X? is calledmaximalmonotone ifitsgraphis not a proper subset ofany monotone set in X×X?.Let us consider the operator ?/?twhich inducesa linearmap L from thesubset

Definition 2.2.Amapping S iscalled pseudo-monotonew ith

thatwehave

3 Essen tialassum p tion

Th roughou t the paper,w e assum e that the follow ing assum p tion hold true.

Assum p tion(H 1)

Let ? is a bounded open set of RN(N ≥ 2),p ∈ C+( ˉ?),T ＞ 0 is given and w e set Q=? × [0,T].We consider a Leray-Lions operator defi ned by the form u la: w here a:?×[0,T]×R×RN→R isa Caratheodory function,i.e.,(m easu rablew ith respect to x in ? for every(s,ξ)in R×RNand continuousw ith respect to(s,ξ)in R×RN,for alm ostevery x in ?)w hich satisfies the follow ing cond itions thereexist k ∈ Lp′(x)(Q)and α ＞ 0,β ＞ 0 such that for alm ostevery(x,t)∈ Q all(s,ξ)∈ R×RN

Assum p tion(H 2)

Let H:?×[0,T]×R×RN→ R isa Carath′eodory function such that for a.e.(x,t)∈ Q and for all s∈ R,ξ∈ RN,thegrow th cond itionis satisfi ed,w here g:R → R+is a bounded continuous positive function that belongs to L1(R)w hile γ ∈ L1(Q).We recall that,for k ＞ 0and s ∈ R,the truncation function Tk(·)defined by

Definition 3.1.Let f ∈ L1(Q)and u0∈ L1(?).A real-valued function u defined on Q is renormalized solutions ofproblem(P)if

forall S∈W2,∞(R)which arepiecewise C1and such that S′hasa compact support in R,where

Rem ark 3.1.Eq.(3.7)is form ally obtained th rough pointw isem u ltip lication of problem (P)by S′(u).How ever,w h ile a(x,t,u,?u)and H(x,t,u,?u)do not in generalm ake sense in(P),all the term s in(3.7)have am eaning in D′(Q).Indeed,if M is such that supp S ? [?M,M],the follow ing iden tifications arem ade in(3.7):

? S(u)belongsto V ∩ L∞(Q).Since S isa bounded function.

? S′(u)a(x,t,u,?u)identifiesw ith S′(u)a(x,t,TM(u),?TM(u))a.e.in Q,

for any ? ∈ D(Q),using H ¨older inequality

w here M ＞ 0 is that supp S′? [?M,M].As D(Q)is dense in V,w e deduce that

? S′′(u)a(x,t,u,? u)? u identifiesw ith S′′(u)a(x,u,TM(u),? TM(u))? TM(u)and

? S′(u)H(x,t,u,?u)identifiesw ith S′(u)H(x,t,TM(u),?TM(u))a.e.in Q.Since

w e see from(3.4)that S′(u)H(x,t,TM(u),?TM(u))∈ L1(Q).

? S′(u)f belongs to L1(Q).

Theabove considerations show that Eq.(3.7)hold in D′(Q)and

It follow s that u belongs to C(0,T;L1(?))so that the initial cond ition(3.8)m ake sense.

4 Existence resu lts

In this section w e establish the follow ing existence theorem:

Theorem 4.1.Let f∈ L1(Q),p(·)∈C+( ˉ?)and u0∈ L1(?).Assume that(H 1)and(H 2)hold true.Then,there existsa renormalized solution u ofproblem(P)in the sense ofDefinition 3.1. Proof.The p roof is in six steps.

Step 1:App roxim ate p rob lem:

For n ＞ 0,letus define the follow ing respectiveapp roxim ation of H,f,and u0,

Note that

|Hn(x,t,s,ξ)|≤ |H(x,t,s,ξ)|, |Hn(x,t,s,ξ)|≤ n for all(s,ξ)∈ R×RN.

and select fn,u0nso that

Letusnow consider theapp roxim ate problem

Theorem 4.2.Let fn∈ Lp′?(0,T;W?1,p′(x)(? ,ω?)),p(·) ∈ C+( ˉ?)forfixed n,theapproximate problem(Pn)hasat least oneweak solution un∈ Lp?(0,T;W1,p(x)0(?,ω)).

Proof.See Append ix.

In view of Theorem 4.2,for the problem(Pn),there exists at least onew eak solu tion un∈ Lp?(0,T;W1,p(·)0(?,ω)).See[4].

Step 2:A PrioriEstim ates:

Proposition 4.1.Let unbe a solution of the approximate problem(Pn).Then,there exists a constantC(which doesnotdepend on then and k)such that

the function g appears in(3.4),w e have

In view o f(3.4),w e obtain

By using(3.3),w e obtain

for all ? ∈ Lp?(0,T;W1,p(·)0(?,ω))∩ L∞(Q),w ith ? ＞ 0.

On theother hand,taking v=exp(?G(un))? asa test function in(Pn),w e deduceas in(4.3),that

for all ? ∈ Lp?(0,T;W1,p(x)0(?,ω))∩ L∞(Q),w ith ? ＞ 0.

Letting ? =Tk(un)+χ(0,τ),for every τ ∈ [0,T]in(4.3),w ehave

w h ich gives

Since G(un)≤ ‖g‖L1(R)/α,w e have|?k(r)|≤ k exp(‖g‖L1(R)/α)|r|,w here

Consequen tly,

Then,w e deduce that

Since a satisfies(3.3)and by using the fact ?k(un(τ))≥ 0,w e get ? n ＞ 0

w here C1is a positive constant.

Sim ilarly to(4.6),w e take ? =Tk(un)?χ(0,τ)in(4.4),w e deduce that

Com bining(4.6),(4.7)and Rem ark 2.1,w e conclude that

w here C1,C2,C3and C4are constants independent o f n.Thus,Tk(un)is bounded in Lp?(0,T;W1,p(x)0(?,ω))independently of n for any k ＞ 0.Now w e tu rn to p roving that (un)nis a Cauchy sequence in m easu res.

Let k＞0 largeenough and BRbeaballof?.Using(4.8)and app lying H ¨older inequality and Poincar`e’s inequality,w eobtain that

w here

w hich im p lies that,

So,w e have

then,w e obtain for allδ ＞ 0

Since Tk(un)isbounded in Lp?(0,T;W1,p(x)0(?,ω)).Then Tk(un)→vkstrongly in Lp(x)(Q,ω) and alm osteveryw here in Q.Hence(Tk(un))nis a cauchy sequence inm easu re in Q.

Let∈＞ 0,then by(4.10),there exist k(∈)＞ 0 such that

This p roves that(un))nis a cauchy sequence inm easu res in BR. Consider a non decreasing function gk∈ C2(R)such that

M u ltip lying theapp roxim ate equation by g′k(un),w eget

in the sense of d istributions.This im p lies,thanks to the fact g′khas com pact support, that gk(un)is bounded in Lp?(0,T;W1,p(x)0(?,ω)),w hile it’s tim e derivative ?gk(un)/?t is bounded in L1(Q)+V?.Due to the choice of gk,w e conclude that for each k,the sequence Tk(un)converges alm ost everyw here in Q,w hich im p lies that the sequence unconverge alm osteveryw here to som em easu rable function v in Q.Thus by using the sam e argum en tas in[14–16],w e can show the follow ing lemm a.

Lemm a 4.1.Let unbea solution oftheapproximateproblem(Pn)then,

We can deducefrom(4.8)that

which implies,by using(3.3),thatforall k ＞ 0 there exists ?k∈ (Lp′(x)(Q,ω?))Nsuch that

Lemm a 4.2.Let unbea solu tion of the app roxim ate p roblem(Pn).Then,

Proof.See Append ix.

Step 3:A lm osteveryw here convergence o f the grad ien ts:

This step is devoted to p rove the strong convergence of truncation of Tk(un)thatw ew ill use the follow ing function ofone realvariable s,w hich is defineasw here m ＞ k

Let ψi∈ D(?)be a sequence w hich converges strong ly to u0in L1(?).Set νiμ= (Tk(u))μ+e?μtTk(ψi)w here(Tk(u))μis them ollifi cation of Tk(u)w ith respect to tim e. Note that ωiμisa sm ooth function having the follow ing properties:

The very defin ition of the sequence ωiμm akes itpossible to establish the follow ing lemm a.

Lemm a 4.3.(See[2,16])Fork≥ 0,wehave

Z

Proposition 4.2.Thesubsequenceofunsolution ofproblem(Pn)satisfiesforany k≥0 follow ing assertion:

Proof.See Append ix.

Thanks to the Lemm a 2.6,w e have

and ?un→ ?u a.e.in Q,w hich im p lies that

Step 4:Equi-in tegrability of the non linearity sequence:

We shallnow p rove that:Hn(x,t,un,?un)→ H(x,t,u,?u)strongly in L1(Q).by using Vitali’s thReorem.Since Hn(x,t,un,?un)→ H(x,t,u,?u)a.e.in Q,considering now ? =d s as a test function in(4.3),w e obtain

w here θh(r)=Rr0ρh(τ)dτ,w hich im p lies,since θh≥ 0 and(3.3)

and since g ∈ L1(R),w e deduce that

Sim ilarly,taking ? = ρh(un)=R0ung(s)χ{s＜?h}d s as a test function in(4.4),w e conclude that

Consequen tly,

W hich im p lies,for h large enough and for a subset E o f Q,

so g(un)|?un|p(x)ω(x)isequi-integrable.Thusw ehave show n that

Consequently,by using(3.4),w e conclude that

Step 5:Convergence of un∈ C([0,T];L1(?)):

Proposition 4.3.The sequence(un)is a Cauchy sequence in C([0,T];L1(?)),moreover,u ∈C([0,T];L1(?))and unconverges to u ∈ C([0,T];L1(?)).

Proof.Let m and n be tw o integers,since unand umare the so lu tionso f the p rob lem(Pn).

Taking ? =T1(un? um)χ[0,s[in(Pn),w ith s≤ T,w ew ellhave ? ∈ Lp?(0,T;W1,p(·)0(?,ω))∩L∞(Q),hence

Which im p lies that,

Recall that ?1isa p rim itive function of T1,thus

Because of un∈ C([0,T];H),w e can w rite

Rem ark that,?T1(un? um)= ?(un? um)χ{|un?um|≤1}and also

due to them onotonicity of a,then w e deduce from(4.18)that

On the other hand,since ?1(r)≤ |r|,w e get for all s≤ T

Denoting αn,mfor the second term in(4.19)and observing that

w e have

Thus,by(4.19)-(4.21),w eget

Tanks to(4.17)and the fact fnand u0nconverge in L1,w e conclude that,

Hence,unisa Cauchy sequence in C([0,T];L1(?)),also u ∈ C([0,T];L1(?))and for s≤ T, w e have un(s)→ u(s)in L1(?).This achieves the p roo f o f the p roposition.

Step 6:Passing to the lim it:

a)Proofthatu satisfies(3.6).For any fi xed m ≥ 0,onehas

Accord ing to(4.15)and(4.16),one can pass to the lim it as n → ∞ for fi xed m ≥ 0 and to obtain

Taking the lim it as m → ∞ in(4.22)and using estim ate(4.12)show s that u satisfies(3.6).

b)Proofthatu satisfies(3.7).Let S bea function in W2,∞(R)w ith S′hasa com pactsupport in R.Let M ＞ 0 such thatsupp(S′)? [?M,M].Poin tw isem u ltip lication of the approxim ate

problem(Pn)by S(un)leads to

In w hat follow sw e pass to the lim it in(4.23)as n tends to ∞.

? Lim itof ?S(un)/?t.

Since S is bounded and continuous,un→ u a.e.in Q im p lies that S(un)converges to S(u)a.e.in Q and L∞w eak ly.Then

Since supp(S′)? [?M,M],w ehave for n ≥ M

The poin tw ise convergence of unto u and(4.16)and the bounded character of S′perm it us to conclude that as n → ∞ S′(u)a(x,t,TM(u),?TM(u))has been denoted by S′(u)a(x,t,u,?u)in Eq.(3.7).

? Lim it of S′′(un)a(x,t,un,?un)?un.

Consider the “energy” term

The pointw ise convergence of S′(un)to S′(u)and(4.15)and(4.16)as n → ∞ and the bounded character of S′′perm etus to conclude that

Recall that

? Lim it of S′(un)Hn(x,t,un,?un).

From supp(S′)? [?M,M]and(4.17),w e have

? Lim it of S′(un)fn.

Since un→u a.e.in Q,and(4.1),w ehave S′(un)fn→S′(u)f strong ly in L1(Q),as n→∞. Asa consequenceof theabove convergence resu lt,w eare in a position to pass to the lim it as n → ∞ in Eq.(4.23)and to conclude that u satisfies(3.7).Asa consequence,an Aubin’s type Lemm a(see,e.g,[17]im p lies that S(un)lies in a com pactsetof C0([0,T],L1(?)).

It follow s that on the hand,the sm oothness of S im p lies that S(un)(|t=0)=S(u0n)in? converge to S(u)|t=0strong ly in L1(?)im p lies that S(u)(|t=0)=S(u0)in ?.

As a conclusion o f steps 1 to step 6,the p roo f o f Theorem 4.1 is com p lete.

5 Append ix

Proof of Theorem 4.2.

We define the operator

by

accord ing to the H¨older inequality,w e have

We defi ne the operator Gn:by

Thanks to the H ¨older inequality,w e have that for u,v ∈ Lp?(0,T;W1,p(·)0(?,ω))

w ith

Lemm a5.1.Let Bn: .Theoperator Bn= A+Gnisa)coercive;b)pseudo-monotone;c)bounded and dem icontinuous. Proof.a)For the coercivity,w ehave for any

then

w hich is due to Poincar′e inequality w ith

Consequen tly,

w hich leads to

Hencew e have

Therefore,Bnis coercive.

b)It rem ains to show that Bnis pseudo-m onotone.

Let(uk)ka sequence in Lp?(0,T;W1,p(x)0(?,ω))such that

then,w e have p rove that

By the definition of theoperator Lndefined in Definition 2.1,w eobtain that ukisbounded inand since(Q)then uk→u in, then the grow th cond ition(3.1′)(a(x,t,uk,?uk))kis bounded intherefore, thereexistsa function ? ∈ (Lp(x)(Q,ω?))Nsuch that

Sim ilarly,using cond ition(3.4)(Hn(x,t,uk,?uk))kis bounded in(L1(Q))w e know n that thereexistsa function ψn∈ L1(Q)such that

Using(5.3)and(5.6),w e obtain

Thanks to(5.5),w e have

Therefore,

On theother hand,using(3.2),w e have

Then,

and by(5.4),w e get

This im p lies,thanks to(5.9)that

Now by(5.11),w e can obtain

In view of the Lemm a 2.6,w e obtain

Then,

We deduce that

w hich im p lies

com p leting the p roof o f assertion(b).

c)Using H ¨older′s inequality and the grow th cond ition(3.1),w e can show that the operator A isbounded,and by using(5.2),w e conclude that Bnis bounded.For to show that Bnis dem icontinuous.

Let uk→ u in Lp?(0,T;W1,p(·)0(?,ω))and prove that

Since a(x,t,uk,?uk)→ a(x,t,u,?u)as k→ ∞ a.e.in Q,then by thegrow th cond ition(3.1) and Lemm a 2.4

and for all ? ∈ Lp?(0,T;W1,p(x)0(?,ω)),〈Auk,?〉→ 〈Au,?〉as k→ ∞.

Sim ilarly,Gnuk→Gnu as k→∞ a.e.in Q then,by the(3.4)and Lemm a 2.4Gnuk?Gnu in Lp′(x)(Q,ω?)and for all φ ∈ Lp?(0,T;W1,p(·)

0(?,ω)),

w hich im p lies Bnis dem icon tinuous.

Proof of Lemm a 4.2.

Set?=T1(un?Tm(un))+=αm(un)in(4.3),this function isadm issible since ?∈Lp?(0,T; W 1,p(x)

0(?,ω))and ?≥0.Then,w ehave

Which gives,by setting

Since ?m(un)(T)＞0,γ∈ L1(?),fn→ f in L1(Q)and u0n→u0in L1(?)then,by Lebesgue’s theorem,w e conclude that

On the other hand,taking ? =T1(un?Tm(un))?as a test function in(4.4)and reasoning as in the p roof(5.12),w e deduce that

By using(5.12)and(5.13),w e have

Proo f o f Proposition 4.2.

For m ＞ k,let ? =(Tk(un)? νiμ)+hm(un)∈ Lp?(0,T;W1,p(·)0(?,ω ))∩ L∞(Q)and ? ≥ 0.If w e take this function in(4.3),w eobtain

Observe that

Tanks to(4.12)the third and fou rth integrals on the righ t hand side tend to zero as n and m tend to in finity and by Lebesgue’s theorem,w e deduce that the right hand side converges to zero as n,m and μ tend to infinity.Since

and strongly in Lp?(0,T;W01,p(·)(?,ω))and(Tk(un)?νμi)+hm(un)?0in L∞(Q)and strongly in Lp?(0,T;W01,p(·)(?,ω))as μ → ∞,it follow s that the fi rst and second integrals on the right-hand side of(5.15)converge to zeros as n,m,μ → ∞,using[19]Lemm a 4.3 and Lemm a 2.6 the p roo f of Proposition 4.2 is com p lete.

A cknow ledgm en ts

Theau thorsw ould like to thank the referees for theirm any valuable comm entsand suggestions for the im p rovem en tof the paper.

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?Correspond ing au thor.Email addresses:akdimyoussef@yahoo.f r(Y.Akd im),chakir.al lalou@yahoo.f r (C.A llalou),nezhaelgor ch@gmai l.com(N.Elgorch)

thenon linear p(x)-parabolicp rob lem in theWeighted-Variable-ExponentSobolev spaces, w ithou t the sign cond ition and the coercivity cond ition.

AM S Sub ject C lassifi cations:35J15,35J70,35J85

Chinese Library Classifi cations:O175.27