Existence of Renormalized Solutions for Nonlinear Parabolic Equations

AKDIMY.,BENKIRANEA.,ELMOUMNIM.andREDWANEH.3,?Facult′e Poly-disciplinaire de Taza,B.P 3 Taza Gare,Maroc.

2Laboratory of Mathematical Analysis and Applications,Department of Mathematics,Faculty of Sciences Dhar El Mehraz,University Sidi Mohamed Ben Abdellah,P.O.Box 1796,Atlas-F`es,Morocco.

3Facult′e des sciences juridiques,Economiques et Sociales,Universit′e Hassan 1 B.P. 784,Settat,Morocco.

Received 5 February 2013;Accepted 24 October 2013

Existence of Renormalized Solutions for Nonlinear Parabolic Equations

AKDIMY.1,BENKIRANEA.2,ELMOUMNIM.2andREDWANEH.3,?1Facult′e Poly-disciplinaire de Taza,B.P 1223 Taza Gare,Maroc.

2Laboratory of Mathematical Analysis and Applications,Department of Mathematics,Faculty of Sciences Dhar El Mehraz,University Sidi Mohamed Ben Abdellah,P.O.Box 1796,Atlas-F`es,Morocco.

3Facult′e des sciences juridiques,Economiques et Sociales,Universit′e Hassan 1 B.P. 784,Settat,Morocco.

Received 5 February 2013;Accepted 24 October 2013

.We give an existence result of a renormalized solution for a class of nonlinear parabolic equations

where the right side belongs to Lp′(0,T;W?1,p′(?))and where b(x,u)is unbounded function of u andwhere?div(a(x,t,u,?u))isaLeray-Lionstypeoperatorwith growth |?u|p?1in?u.The critical growth condition on g is with respect to?u and no growth condition with respect to u,while the function H(x,t,?u)grows as|?u|p?1.

AMS Subject Classifications:35K10,47D20,46E35

Chinese Library Classifications:O175.23,O175.26

Nonlinear parabolic equations;renormalized solutions;Sobolev spaces.

1 Introduction

In the present paper,we study a nonlinear parabolic problem of the type

where ? is a bounded open s?ubset of RN?,N≥1,T＞0,p＞1 and QTis the cylinder?×(0,T).The operator?diva(x,t,u,?u)is a Leray-Lions operator which is coercive and grows like|?u|p?1with respect to?u,the function b(x,u)is an unbounded on u. T (sh e ee A fu sns cutmi opn s ti o gn a(nHd2)H) . aF rien a tw llyo t th he e d Ca a ta ra f th is e′o indo L rpy′( f0 u ,Tnc ;Wtio?n1s,p′w(?i th )). su Wite abalree a in ssteu rmespt e tidonins proving an existence result to(1.1).The difficulties connected to this problem are due to the data and the presence of the two terms g and H which induce a lack of coercivity.

For b(x,u)=u,the existence of a weak solution to Problem(1.1)(which belongs to Lm(0,T;W01,m(?))with p＞2?1/(N+1)and m＜(p(N+1)?N)/N+1 was proved in[1] (see also[2])when g=H=0,and in[3]when g=0,and in[4-6]when H=0.In the present paper we prove the existence of renormalized solutions for a class of nonlinear parabolic problems(1.1).The notion of renormalized solution was introduced by Diperna and Lions[7]in their study of the Boltzmann equation.This notion was then adapted t′o an elliptic version of(1.1)by Boccardo et al.[8]when the right hand side is in W?1,p(?), by Rakotoson[9]when the right hand side is in L1(?),and finally by Dal Maso,Murat, Orsina and Prignet[10]for the case of right hand side is general measure data.

In the case where H=0 and where the function g(x,t,u,?u)≡g(u)is independent on the(x,t,?u)and g iscontinuous,theexistenceofarenormalized solutiontoProblem(1.1) isprovedin[11].Thecase H=0 isstudiedbyAkdimetal.(see[12,13]).Thecase H=0and where g depends on(x,t,u)is investigated in[14].In[15]the authors prove the existence of a renormalized solution for the complete operator.The case g(x,t,u,?u)≡div(φ(u)) and H=0 is studied by Redwane in the classical Sobolev spaces W1,p(?)and Orlicz spaces see[16,17],and where b(x,u)=u(see[18]).

The aim of the present paper we prove an existence result for renormalized solutions to a class of problems(1.1)with the two lower order terms.It is worth noting that for the analogous elliptic equation with two lower order terms(see e.g.[19,20]).The plan of the article is as follows.In Section 2 we make precise all the assumptions on b,a,g,H,f and give the definition of a renormalized solution of(1.1).In Section 3 we establish the existence of such a solution(Theorem 3.1).

2 Basic assumptions on the data and definition of arenormalized solution

Throughout the paper,we assume that the following assumptions hold true:

Assumption(H1)

Let ? be a bounded open set of RN(N≥1),T＞0 is given and we set QT=?×(0,T),and

such that for every x∈?,b(x,.)is a strictly increasing C1-function with b(x,0)=0.

Next,for any k＞0,there exists λk＞0 and functions Ak∈L∞(?)and Bk∈Lp(?)such that

for almost every x∈?,for every s such that|s|k,we denote bythe gradient ofdefined in the sense of distributions.Also,

for a.e.(x,t)∈QT,all(s,ξ)∈R×RN,some positive function k(x,t)∈Lp′(QT)and β＞0.

where α is a strictly positive constant.

Assumption(H2)

Furthermore,let g(x,t,s,ξ):QT×R×RN→R and H(x,t,ξ):QT×RN→R are two Carath′eodory functions which satisfy,for almost every(x,t)∈QTand for all s∈R,ξ∈RN, the following conditions

where L1:R+→R+is a continuous increasing function,while L2(x,t)is positive and belongs to L1(QT).

where h(x,t)is positiveTWe recall that,for k＞1 and s in R,the truncation is defined as

Definition 2.1.A real-valued function u defined on QTis a renormalized solution of problem (1.1)if for all functions S∈W2,∞(R)which are piecewiseC1and such that S′has a compact support in R,and

Remark 2.1.Eq.(2.12)is formally obtained through pointwise multiplication of(1.1)by S′(u).However,while a(x,t,u,?u),g(x,t,u,?u)and H(x,t,?u)doesnot in generalmake sense inD′(QT),all the terms in(2.12)have a meaning inD′(QT).Indeed,if M is such that suppS′?[?M,M],the following identifications are made in(2.12):

?BS(x,u)belongs to L∞(QT)because|BS(x,u)|≤‖AM‖L∞(?)‖S‖L∞(R).

?S′(u)a(x,t,u,?u)identifies with S′(u)a(x,t,TM(u),?TM(u))a.e.in QT.Since

|TM(u)|≤M a.e.in QTand S′(u)∈L∞(QT),we obtain from(2.3)and(2.10)that

?S′(u)a(x,t,u,?u)?u identifies with S′′(u)a(x,t,TM(u),?TM(u))?TM(u)and

?S′(u)(g(x,t,u,?u)+H(x,t,?u))=S′(u)?g(x,t,TM(u),?TM(u))+H(x,t,?TM(u))) a.e.in QT.Since|TM(u)|≤M a.e.in QTand S′(u)∈L∞(QT),we obtain from (2.3),(2.7)and(2.9)that S′(u)(g(x,t,TM(u),?TM(u))+H(x,t,?TM(u)))∈L1(QT).

?In view of(2.6)and(2.10),we have S′(u)f belongs to Lp′(0,T;W?1,p′(?)). The above considerations show that(2.12)holds inD′(QT)and that

for almost every x∈? and for every r,r′∈R.

Now we state the proposition is a slight modification of Gronwall’s lemma(see[21]).

Proposition 2.1.Given the function λ,γ,?,ρ defined on[a,+∞[,suppose that a≥0,λ≥0, γ≥0 and that λγ,λ? and λρ belong to L1([a,+∞[).If for almost every t≥0 we have

then

for almost every t≥0.

3 Main results

In this section we establish the following existence theorem.

Theorem 3.1.Assume that(H1)-(H2)hold true.Then,there exists a renormalized solution u of problem(1.1)in the sense of Definition 2.1.

Proof.The proof of this theorem is done in five steps.

Step 1:Approximate problem and a priori estimates.

For n＞0,let us define the following approximation of b,g and H.First,set

In view of(3.1),bnis a Carath′eodory function and satisfies(2.2),there exist λn＞0 and functions An∈L∞(?)and Bn∈Lp(?)such that

a.e.in ?,s∈R.Next,set

Let us now consider the approximate problem

Note that gn(x,t,s,ξ)and Hn(x,t,ξ)are satisfying the following conditions

Moreover,since f∈Lp′(0,T;W?1,p′(?)),proving existence of a weak solution un∈Lp(0,T;W1,p0(?))of(3.2)is an easy task(see e.g.[22]).For ε＞0 and s≥0,we define

We choose v=?ε(un)as test function in(3.2),we have

where

Using

(2.9)and H¨older’s inequality,we obtain

Observe that,

Because,

By(2.5)and(3.3),we deduce that

Letting ε go to zero,we obtain

where{s＜|un|}denotes the set{(x,t)∈QT,s＜|un(x,t)|}andμ(s)stands for the distribution function of un,that isμ(s)=|{(x,t)∈QT,|un(x,t)|＜s}|for all s≥0.

Now,we recall the following inequality(see for example[23]),we have for almost every s＞0

Using(3.6),we have

which implies that,

Now,we consider two functions B(s)and F(s)(see[24,Lemma 2.2])defined by

From(3.8),(3.9)and(3.10)becomes

From Proposition 2.1,we obtain

Raising to the power p′,integrating between0 and+∞and by a variable change we have

Using H¨older’s inequality and(3.11),then we get

where c1is a positive constant independent of n.Then there exists u∈Lp(0,T;W01,p(?)) such that,for some subsequence

we conclude that

We deduce from the above inequality,(2.2)and(3.14),that

where

Now,we turn to prove the almost every convergence of unand bn(x,un).Consider now a function non decreasing gk∈C2(R)such that gk(s)=s for|s|≤k/2 and gk(s)=k for|s|≥k.Multiplying the approximate equation by,we obtain

where

Due to the choice of gk,we conclude that for each k,the sequence Tk(un)converges almost everywhere in QT,which implies that unconverges almost everywhere to some measurable function u in QT.Thus by using the same argument as in[11,25]and[26], we can show

We can deduce from(3.14)that

Which implies,by using(2.3),for all k＞0 that there exists a function a∈(Lp′(QT))N,such that

We now establish that b(.,u)belongs to L∞(0,T;L1(?)).Using(3.17)and passing to the limit inf in(3.15)as n tends to+∞,we obtain that for almost any τ in(0,T).Due to the definitionof BTk(x,s)and the fact that1kBTk(x,u)converges pointwise to b(x,u),as k tendsto+∞,shows that b(x,u)belong to L∞(0,T;L1(?)).Lemma 3.1.Let unbe a solution of the approximate problem(3.2).Then

Proof.Considering the function ?=T1(un?Tm(un))+=αm(un)in(3.2)this function is admissible since ?∈Lp(0,T;W1,p0(?))and ?≥0.Then,we have

Which,by setting

(2.8)and(2.9)gives

Using this H¨older’s inequality and(3.12),we deduce

Similarly,since b∈Lr(QT)(with r≥p),we obtain

We conclude that

On the other hand,let ?=T1(un?Tm(un))?as test function in(3.2)and reasoning as in the proof of(3.24)we deduce that

Thus(3.21)follows from(3.24)and(3.25).

Step 2:Almost everywhere convergence of the gradients.

This step is devoted to introduce for k≥0 fixed a time regularization of the function Tk(u)in order to perform the monotonicity method.This kind of regularization has been first introduced by R.Landes(see[27,Lemma 6,proposition 3 and proposition 4]).For k＞0 fixed,and let ?(t)=teγt2,γ＞0.It is will known that when γ＞(L1(k)/2α)2,one has

Let ψi∈D(?)be a sequence which converge strongly to u0in L1(?).

where(Tk(u))μis the mollification with respect to time of Tk(u).Note that wiμis a smooth function having the following properties:

We introduce the following function of one real:

which implies since gn(x,t,un,?un)?(Tk(un)?wiμ)hm(un)≥0 on{|un|＞k}:

In the sequel and throughout the paper,we will omit for simplicity the denote ε(n,μ,i,m) all quantities(possibly different)such that

and this will be the order in which the parameters we use will tend to infinity,that is,first n,thenμ,i and finally m.Similarly we will write only ε(n),or ε(n,μ),···to mean that the limits are made only on the specified parameters.

We will deal with each term of(3.29).First of all,observe that

since ?(Tk(un)?wiμ)hm(un)converges to ?(Tk(u)?(Tk(u))μ+e?μtTk(ψi))hm(u)strongly in Lp(QT)and weakly??in L∞(QT)as n→∞and finally ?(Tk(u)?(Tk(u))μ+e?μtTk(ψi)) ×hm(u)converges to 0 strongly in Lp(QT)and weakly??in L∞(QT)asμ→∞. On the one hand.The definition of the sequence wi

μmakes it possible to establish the following Lemma 3.2.

Lemma 3.2.For k≥0 we have

Proof.(see Blanchard and Redwane[28]).

On the other hand,the second term of the left hand side of(3.29)can be written

since m＞k and hm(un)=1 on{|un|≤k},we deduce that Using(2.3),(3.20)and Lebesgue theorem we have a(x,t,Tk(un),?Tk(u))converges to a(x,t,Tk(u),?Tk(u))strongly in(Lp′(QT))Nand?Tk(un)converges to?Tk(u)weakly in(Lp(QT))N,then K2=ε(n).Using(3.20)and(3.28)we have

For what concerns K4can be written,since hm(un)=0 on{|un|＞m+1}

and,as above,by letting n→∞

so that,by lettingμ→∞

We conclude then that

To deal with the third term of the left hand side of(3.29),observe that

Thanks to(3.21),we obtain

We now turn to fourth term of the left hand side of(3.29),can be written

since L2(x,t)belong to L1(QT)it is easy to see that

On the other hand,the second term of the right hand side of(3.34),write as

and,as above,by letting first n then finallyμgo to infinity,we can easily see,that each one of last two integrals is of the form ε(n,μ).This implies that

Combining(3.29),(3.31),(3.32),(3.33)and(3.35),we get

and so,thanks to(3.26),we have

Hence by passing to the limit sup over n,we get

This implies that

Now,observe that for every σ＞0,

then as a consequence of(3.37)we have that?unconverges to?u in measure and therefore,always reasoning for a subsequence,

which implies

Step 3:Equi-integrability of Hnand gn.

We shall now prove that Hn(x,t,?un)converges to H(x,t,?u)and gn(x,t,un,?un) converges to g(x,t,u,?u)strongly in L1(QT)by using Vitali’s theorem.

Since Hn(x,t,?un)→H(x,t,?u)a.e.QTand gn(x,t,un,?un)→g(x,t,u,?u)a.e.QT, thanks to(2.7)and(2.9),it suffices to prove that Hn(x,t,?un)and gn(x,t,un,?un)are uniformly equi-integrable in QT.We will now prove that Hn(x,?un)is uniformly equiintegrable,we use H¨older’s inequality and(3.12),we have

which is small uniformly in n when the measure of E is small.

To prove the uniform equi-integrability of gn(x,t,un,?un).Forany measurable subset E?QTand m≥0,

For fixed m,we get

We chooses ψm(un)as a test function for m＞1 in(3.2),we obtain

where

By(3.12),we have

Thus we proved that the second term of the right hand side of(3.41)is also small,uniformly in n and in E when m is sufficiently large.Which shows that gn(x,t,un,?un)and Hn(x,t,?un)are uniformly equi-integrable in QTas required,we conclude that

Step 4:In this step we prove that u satisfies(2.11).

Lemma 3.3.The limit u of the approximate solution unof(3.2)satisfies

Proof.Note that for any fixed m≥0,one has

According to(3.37)and(3.39),one can pass to the limit as n→+∞for fixed m≥0,to obtain

Takingthe limit as m→+∞in(3.44)and using the estimate(3.21),we showthat u satisfies (2.11)and the proof is complete.

Step 5:In this step we prove that u satisfies(2.12)and(2.13).

Let S be a function inW2,∞(R)such that S′has a compact support.Let M be a positive real number such that support of S′is a subset of[?M,M].Pointwise multiplication of the approximate equation(3.2)by S′(un)leads to

Passing to the limit,as n tends to+∞,we have

?Since S is bounded and continuous,un→u a.e.in QTimplies that BnS(x,un)converges to BS(x,u)a.e.in QTand L∞weak?.Then?BnS(x,un)/?t converges to?BS(x,u)/?t inD′(QT)as n tends to+∞.

?Since supp(S′)?[?M,M],we have for n≥M,

The pointwise convergence of unto u and(3.39)as n tends to+∞and the bounded character of S′permit us to conclude that

as n tends to+∞.S′(u)a(x,t,TM(u),?TM(u))has been denoted by S′(u)a(x,t,u,?u)in Eq.(2.12).

?Regarding the‘energy’term,we have

The pointwise convergence of S′(un)to S′(u)and(3.39)as n tends to+∞and the bounded character of S′permit us to conclude that S′′(un)an(x,t,un,?un)?unconverges to S′′(u)a(x,t,TM(u),?TM(u))?TM(u)weakly in L1(QT).Recall that

?Since supp(S′)?[?M,M],by(3.43),we have

strongly in L1(QT),as n tends to+∞.

As a consequence of the above convergence result,we are in a position to pass to the limit as n tends to+∞in equation(3.45)and to conclude that u satisfies(2.12).

It remains to show that BS(x,u)satisfies the initial condition(2.13).To this end, firstly remark that,S being bounded,BnS(x,un)is bounded in L∞(QT).Secondly,(3.45) and the above considerations on the behavior of the terms of this equation show that?BnS(x,un)/?t is bounded in Lp′(0,T;W?1,p′(?)).As a consequence,an Aubin’s type lemma(see,e.g,[29])implies that BnS(x,un)lies in a compact set of C0([0,T],L1(?)).It follows that on the one hand,BnS(x,un)(t=0)=BnS(x,0)=0 converges to BS(x,u)(t=0) strongly in L1(?).On the other hand,the smoothness of S implies that BS(x,u)(t=0)=0 in ?.

As a conclusion,steps 1-5 complete the proof of Theorem 3.1.

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?Corresponding author.Email addresses:akdimyoussef@yahoo.fr(Y.Akdim),abd.benkirane@gmail.com (A.Benkirane),mostafaelmoumni@gmail.com,(M.EL Moumni)redwane hicham@yahoo.fr(H.Redwane)