(0, T)×∂Ω, (1.2) bi(ui)(t= 0) =bi(ui,0) in Ω, (1.3) wherei= 1,2

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu

EXISTENCE OF SOLUTIONS FOR NONLINEAR PARABOLIC SYSTEMS VIA WEAK CONVERGENCE OF TRUNCATIONS

ELHOUSSINE AZROUL, HICHAM REDWANE, MOHAMED RHOUDAF

Abstract. We prove an existence result for a class of nonlinear parabolic systems. Without assumptions on the growth of some nonlinear terms, we prove the existence of a renormalized solution.

1. Introduction

Let Ω be a bounded open subset of R^N, (N ≥1), T >0 and let Q:= (0, T)× Ω. We prove the existence of a renormalized solution for the nonlinear parabolic systems

(bi(ui))t−div

a(x, ui, Dui) + Φi(ui)

+fi(x, u1, u2) = 0 inQ, (1.1)

ui= 0 on Γ := (0, T)×∂Ω, (1.2)

bi(ui)(t= 0) =bi(ui,0) in Ω, (1.3) wherei= 1,2. Here, the vector field

a: Ω×R×R^N →R^N is a Carath´eodory function such that (1.4)

• There existsα >0 with

a(x, s, ξ).ξ≥α|ξ|^p (1.5)

for almost everyx∈Ω, for every s∈R, for everyξ∈R^N.

• For each K > 0, there exists βK > 0 and a function CK in L^p0(Ω) such that

|a(x, s, ξ)| ≤CK(x) +βK|ξ|^p−1 (1.6) for almost everyx∈Ω, for everyssuch that|s| ≤K, and for everyξ∈R^N.

• The vector fieldais monotone inξ; i.e.,

[a(x, s, ξ)−a(x, s, ξ⁰)][ξ−ξ⁰]≥0, (1.7) for anys∈R, for any (ξ, ξ⁰)∈R^2N and for almost everyx∈Ω.

2000Mathematics Subject Classification. 47A15, 46A32, 47D20.

Key words and phrases. Nonlinear parabolic systems; existence; truncations;

renormalized solutions.

c

2010 Texas State University - San Marcos.

Submitted March 2, 2010. Published May 17, 2010.

1

Moreover, we suppose that fori= 1,2,

Φ_i:R→R^N is a continuous function, (1.8) b_i:R→Ris a strictly increasingC¹-function withb_i(0) = 0, (1.9) fi: Ω×R×R→Ris a Carath´eodory function with

f₁(x,0, s) =f₂(x, s,0) = 0 a.e. x∈Ω,∀s∈R. (1.10) and for almost everyx∈Ω, for every s1, s2∈R,

sign(si)fi(x, s1, s2)≥0. (1.11) The growth assumptions onf_i are as follows: For eachK >0, there existsσ_K >0 and a functionF_K in L¹(Ω) such that

|f1(x, s1, s2)| ≤FK(x) +σK |b2(s2)| (1.12) a.e. in Ω, for alls₁ such that|s1| ≤K, for alls₂∈R.

For eachK >0, there existsλ_K >0 and a functionG_K in L¹(Ω) such that

|f2(x, s1, s2)| ≤GK(x) +λK |b1(s1)| (1.13) for almost every x ∈ Ω, for every s₂ such that |s2| ≤ K, and for every s₁ ∈ R. Finally, we assume the following condition on the initial datau_i,0:

ui,0is a measurable function such that bi(ui,0)∈L¹(Ω), fori= 1,2. (1.14) The main difficulty when dealing with problem (1.1)-(1.3) is due to the fact that the functionsa(x, u_i, Du_i),Φ_i(u_i) andf_i(x, u₁, u₂) are not in (L¹_loc(Q))^N in general, since the growth of a(x, u_i, Du_i),Φ_i(u_i) and f_i(x, u₁, u₂) are not controlled with respect toui, so that proving existence of a weak solution (i.e. in the distribution meaning) seems to be an arduous task. To overcome this difficulty, we use in this paper the framework of renormalized solutions due to Lions and DiPerna [20] for the study of Boltzmann equations (see also Lions [21] for a few applications to fluid mechanics models). This notion was then adapted to the elliptic version of (1.1)-(1.3) in Boccardo, Diaz, Giachetti, Murat [11], in Lions and Murat [22] and Murat[22, 23]. At the same the equivalent notion of entropy solutions have been developed independently by B´enilan and al. [1] for the study of nonlinear elliptic problems.

The particular case wherebi(ui) =ui and Φi= Φ, i= 1,2 has been studied in Redwane [25] and for the parabolic version of (1.1)-(1.3), existence and uniqueness results are already proved in [4] (see also [30] and [24]) in the case wheref_i(x, u₁, u₂) is replaced byf+ div(g) wheref ∈L¹(Q) andg∈L^p0(Q)^N.

In the case wherea(t, x, s, ξ) is independent of s, Φi = 0 and g = 0, existence and uniqueness are established in [2]; in [3], and in the case where a(t, x, s, ξ) is independent of s and linear with respect to ξ, existence and uniqueness are established in [7].

In the case where Φi = 0 and the operator ∆pu= div|∇u|^p−2∇u) p-Laplacian replaces a nonlinear term diva(x, s, ξ)), existence of a solution for nonlinear par- abolic systems (1.1)-(1.3) is investigated in [26, 27], in [28] and in [29], where an existence result of as (usual) weak solution is proved.

This article is organized as follows: in Section 2, we specify the notation and give the definition of a renormalized solution of (1.1)-(1.3). Then, in Section 3, we establish the existence of such a solution (see Theorem 3.1).

2. Notation

In this paper, for K > 0, we denote by TK : r 7→ min(K, max(r,−K)) the truncation function at height K. For any measurable subset E of Q, we denote by meas(E) the Lebesgue measure of E. For any measurable function v defined on Qand for any real number s, χ_{v<s} (respectively, χ_{v=s}, χ_{v>s}) denote the characteristic function of the set {(x, t)∈Q ; v(x, t)< s} (respectively, {(x, t)∈ Q;v(x, t) =s},{(x, t)∈Q;v(x, t)> s}).

Definition 2.1. A couple of functions(u₁, u₂)defined onQis called a renormalized solution of (1.1)-(1.3)if for i= 1,2 the functionui satisfies

TK(ui)∈L^p(0, T;W₀^1,p(Ω)) and bi(ui)∈L^∞(0, T;L¹(Ω)), (2.1) for any K≥0.

Z

{(t,x)∈Q; n≤|u_i(x,t)|≤n+1}

a(x, u_i, Du_i)Du_idx dt→0 asn→+∞, (2.2) and if, for every function S in W^2,∞(R) which is piecewise C¹ and such that S⁰ has a compact support, we have

∂b_i,S(u_i)

∂t −divS⁰(u_i)a(x, u_i, Du_i)) +S⁰⁰(u_i)a(x, u_i, Du_i)Du_i

−divS⁰(ui)Φi(ui)) +S⁰⁰(ui)Φi(ui)Dui+fi(x, u1, u2)S⁰(ui) = 0 inD⁰(Q), (2.3) and

bi,S(ui)(t= 0) =bi,S(ui,0) inΩ, (2.4) wherebi,S(r) =Rr

0 b⁰_i(s)S⁰(s)ds.

Remark 2.2. Equation (2.3) is formally obtained through pointwise multiplication of equation (1.1) by S⁰(ui). Note that in Definition 2.1, the gradient Dui is not defined even as a distribution, but that due to (2.1) each term in (2.3) has a meaning inL¹(Q) +L^p0(0, T;W^−1,p0(Ω)).

Indeed ifK >0 is such that suppS⁰ ⊂[−K, K], the following identifications are made in (2.3):

• b_i,S(u_i) belong toL^∞(Q)∩L^p(0, T;W₀^1,p(Ω)). Indeed Dbi,S(u) =S⁰(ui)b⁰_i(TK(ui))DTK(ui)∈(L^p(Ω))^N and

|bi,S(ui)| ≤ Z |u_i|

0

|S⁰(s)b⁰_i(s)|ds≤K max

|r|≤K|S⁰(r)b⁰_i(r)|.

• S⁰(ui)a(x, ui, Dui) can be identified withS⁰(ui)a(x, TK(ui), DTK(ui)) a.e.

in Q. Indeed, since |TK(u_i)| ≤ K a.e. in Q, assumptions (1.4) and (1.6) imply that

a(x, TK(ui), DTK(ui))

≤CK(t, x) +βK|DTK(ui)|^p−1 a.e. inQ.

As a consequence of (2.1) and ofS⁰(ui)∈L^∞(Q), it follows that S⁰(ui)a(x, TK(ui), DTK(ui))∈(L^p0(Q))^N.

• S⁰⁰(ui)a(x, ui, Dui)Dui can be identified with S⁰⁰(u_i)a(x, T_K(u_i), DT_K(u_i))DT_K(u_i) and in view of (1.4), (1.6) and (2.1) one has

S⁰⁰(ui)a(x, TK(ui), DTK(ui))DTK(ui)∈L¹(Q).

• S⁰(ui)Φi(ui) andS⁰⁰(ui)Φi(ui)Dui respectively identify with

S⁰(ui)Φi(TK(ui)) andS⁰⁰(ui)Φ(TK(ui))DTK(ui). Due to the properties of Sand (1.8), the functionsS⁰, S⁰⁰and Φ◦T_K are bounded onRso that (2.1) implies thatS⁰(u_i)Φ_i(T_K(u_i))∈(L^∞(Q))^N andS⁰⁰(u_i)Φ_i(T_K(u_i))DT_K(u_i) belongs toL^p(Q).

• S⁰(u_i)f_i(x, u₁, u₂) identifies withS⁰(u_i)f₁(x, T_K(u₁), u₂) a.e. inQ

(or S⁰(u_i)f₂(x, u₁, T_K(u₂)) a.e. in Q). Indeed, since|T_K(u_i)| ≤K a.e. in Q, assumptions (1.12) and (1.13) imply that

f1(x, TK(u1), u2)

≤FK(x) +σK |b2(u2)| a.e. inQ and

f₂(x, u₁, T_K(u₂))

≤G_K(x) +σ_K |b₁(u₁)| a.e. inQ.

As a consequence of (2.1) and ofS⁰(ui)∈L^∞(Q), it follows that S⁰(u1)f1(x, TK(u1), u2)∈L¹(Q) and S⁰(u2)f2(x, u1, TK(u2))∈L¹(Q).

The above considerations show that (2.3) takes place in D⁰(Q) and that ^∂bi,S_∂t^(ui) belongs toL^p0(0, T;W^−1,p0(Ω))+L¹(Q), which in turn implies that^∂bi,S_∂t^(ui)belongs to L¹(0, T;W^−1,s(Ω)) for alls < inf(p⁰,_N^N₋₁). It follows that bi,S(ui) belongs to C⁰([0, T];W^−1,s(Ω)) so that the initial condition (2.4) makes sense.

3. Existence result

This section is devoted to the proof of the following existence theorem.

Theorem 3.1. Under assumptions(1.4)-(1.14), there exists at least a renormalized solution (u₁, u₂)of Problem (1.1)-(1.3).

Proof. The proof is divided into 9 steps. In step1, we introduce an approximate problem and step 2 is devoted to establish a few a prioriestimates. In step 3, we prove some properties of the limituiof the approximate solutionsu^ε_i. In step 4, we define a time regularization of the fieldTK(ui) and we establish Lemma 3.2 which allows to control the parabolic contribution that arises in the monotonicity method when passing to the limit. In step 5, we prove an energy estimate (see Lemma 3.3) which is a key point for the monotonicity arguments that are developed in Step 6 and Step 7. In Step 8, we prove that ui satisfies (2.2) and finally, in step 9, we prove thatui satisfies properties (2.3) and (2.4) of Definition 2.1.

Step 1. Let us introduce the following regularization of the data: for ε >0 and i= 1,2

bi,ε(s) =bi(T1

ε(s)) +ε s ∀s∈R, (3.1)

aε(x, s, ξ) =a(x, T1

ε(s), ξ) a.e. in Ω,∀s∈R,∀ξ∈R^N, (3.2) Φ_i,ε is a Lipschitz continuous bounded function fromRintoR^N (3.3)

such that Φ^ε_i converges uniformly to Φi on any compact subset of Rasεtends to 0.

f₁^ε(x, s₁, s₂) =f₁(x, T1 ε(s₁), T1

ε(s₂)) a.e. in Ω,∀s1, s₂∈R, (3.4) f₂^ε(x, s1, s2) =f2(x, T1

ε(s1), T1

ε(s2)) a.e. in Ω,∀s1, s2∈R, (3.5) u^ε_i,0∈C₀^∞(Ω), bi,ε(u^ε_i,0)→bi(ui,0) in L¹(Ω) asεtends to 0. (3.6) Let us now consider the regularized problem

∂bi,ε(u^ε)

∂t −div aε(x, u^ε, Du^ε) + Φi,ε(u^ε)

+f_i^ε(x, u^ε₁, u^ε₂) = 0 inQ, (3.7)

u^ε_i = 0 on (0, T)×∂Ω, (3.8)

b_i,ε(u^ε_i)(t= 0) =b_i,ε(u^ε_i,0) in Ω. (3.9) In view of (1.9) and (3.1), fori= 1,2, we have

b⁰_i,ε(s)≥ε, |bi,ε(s)| ≤ max

|s|≤¹_ε

|bi(s)|+ 1 ∀s∈R,

In view of (1.6), (1.12) and (1.13), aε, f₁^ε and f₂^ε satisfy: There exists Cε ∈ L^p0(Ω), F_ε∈L¹(Ω), G_ε∈L¹(Ω) andβ_ε>0, σ_ε>0, λ_ε>0, such that

|aε(x, s, ξ)| ≤Cε(x) +βε|ξ|^p−1 a.e. inx∈Ω,∀s∈R,∀ξ∈R^N.

|f₁^ε(x, s₁, s₂)| ≤F_ε(x) +σ_ε max

|s|≤¹_ε

|b_i(s)| a.e. inx∈Ω,∀s₁, s₂∈R,

|f₂^ε(x, s1, s2)| ≤Gε(x) +λε max

|s|≤¹_ε

|bi(s)| a.e. inx∈Ω,∀s1, s2∈R.

As a consequence, proving the existence of a weak solutionu^ε_i ∈L^p(0, T;W₀^1,p(Ω)) of (3.7)-(3.9) is an easy task (see e.g. [29, 26, 27]).

Step 2. The estimates derived in this step rely on usual techniques for problems of type (3.9)-(3.13) and we just sketch the proof of them (the reader is referred to [2, 3, 7, 10, 4, 5] or to [11, 22, 23] for elliptic versions of (3.9)-(3.13)).

UsingT_K(u^ε_i) as a test function in (3.7) leads to Z

Ω

b^K_i,ε(u^ε_i)(t)dx+ Z t

0

Z

Ω

aε(x, u^ε_i, Du^ε_i)DTK(u^ε_i)dx ds +

Z t

0

Z

Ω

Φ_i,ε(u^ε_i)DT_K(u^ε_i)dx ds+ Z t

0

Z

Ω

f_i^ε(x, u^ε₁, u^ε₂)T_K(u^ε_i)dx ds

= Z

Ω

b^K_i,ε(u^ε_i,0)dx

(3.10)

fori= 1,2, for almost everytin (0, T), and whereb^K_i,ε(r) =Rr

0 TK(s)b⁰_i,ε(s)ds. The Lipschitz character of Φi,ε, Stokes formula together with the boundary condition (3.8) allow to obtain obtain

Z t

0

Z

Ω

Φ_i,ε(u^ε_i)DT_K(u^ε_i)dx ds= 0, (3.11) for almost anyt∈(0, T). Now, as 0≤b^K_i,ε(u^ε_i,0)≤K|bi,ε(u^ε_i,0)| a.e. in Ω, it follows that 0≤R

Ωb^K_i,ε(u^ε_i,0)dx ≤KR

Ω|bi,ε(u^ε_i,0)|dx. Since aε satisfies (3.2), f_i^ε satisfies (3.4), (3.5), we deduce from (3.14) ( taking into account the properties ofb^K_i,ε and u^ε_i,0 ) that

T_K(u^ε_i) is bounded inL^p(0, T;W₀^1,p(Ω)) (3.12)

independently ofεfor anyK≥0.

Proceeding as in [3, 7, 4], we prove that for any S ∈W^2,∞(R) such that S⁰ is compact (suppS⁰⊂[−K, K])

S(bi,ε(u^ε_i)) is bounded inL^p(0, T;W₀^1,p(Ω)), (3.13) and

∂S(b_i,ε(u^ε_i))

∂t is bounded inL¹(Q) +L^p0(0, T;W^−1,p0(Ω)), (3.14) independently ofε, as soon asε < _K¹. Due to the definition (3.1) ofbε, it is clear that

{−K≤b_i,ε(u^ε_i)≤K} ⊂ {−K≤b_i(u^ε_i)≤K}={b⁻¹_i (−K)≤u^ε_i ≤b⁻¹_i (K)}

as long asε < _K¹. As a first consequence we have DS(b_i,ε(u^ε_i)) =S⁰(b_i,ε(u^ε_i))b⁰_i,ε(T_K^∗

i(u^ε_i))DT_K^∗

i(u^ε_i) a.e. inQ. (3.15) as long as ε < _K¹, and K_i^∗ = max(|b⁻¹_i (−K)|, b⁻¹_i (K)). Secondly, the following estimate holds true

kS⁰(b_i,ε(u^ε_i))b⁰_i,ε(T_K^∗

i(u^ε_i))k_L∞(Q)≤ kS⁰k_L∞(R) max

|r|≤K_i^∗(b⁰_i(r)) + 1 ,

as long asε < _K¹.

As a consequence of (3.12), (3.15) we obtain (3.13). To show that (3.14) holds, we multiply the equation foru^εin (2.3) byS⁰(bi,ε(u^ε_i)) to obtain

∂S(b_i,ε(u^ε))

∂t

= div

S⁰(bε(u^ε_i))aε(x, u^ε_i, Du^ε_i)Du^ε_i

−S⁰⁰(b_i,ε(u^ε_i))b⁰_i,ε(u^ε_i)a_ε(x, u^ε_i, Du^ε_i)Du^ε_iDu^ε_i + div Φ_i,ε(u^ε)S⁰(b_i,ε(u^ε)))

−S⁰⁰(bi,ε(u^ε_i))b⁰_i,ε(u^ε_i)Φi,ε(u^ε_i)Du^ε_i+f_i^ε(x, u^ε₁, u^ε₂)S⁰(bi,ε(u^ε_i)) = 0,

(3.16)

in D⁰(Q). Since suppS⁰ and suppS⁰⁰ are both included in [−K, K], u^ε_i may be replaced byT_K^∗

i(u^ε_i) in each of these terms, whereK_i^∗= max(|b⁻¹_i (−K)|, b⁻¹_i (K)).

As a consequence, each term in the right hand side of (3.16) is bounded either in L^p0(0, T;W^−1,p0(Ω)) or in L¹(Q). (see [4, 7]). As a consequence of (3.12), (3.16) we then obtain (3.14).

Now for fixedK >0 :aε(x, TK(u^ε_i), DTK(u^ε_i)) =a(x, TK(u^ε_i), DTK(u^ε_i)) a.e. in Qas long asε < _K¹, while assumption (1.6) gives

aε(x, TK(u^ε_i), DTK(u^ε_i))

≤CK(x) +βK|DTK(u^ε_i)|^p−1 whereβK>0 andCK ∈L^p0(Q). In view of (3.12), we deduce that

a x, TK(u^ε_i), DTK(u^ε_i)

is bounded in (L^p0(Q))^N. (3.17) independently ofεforε <_K¹.

For any integer n ≥ 1, consider the Lipschitz continuous function θ_n defined through

θ_n(r) =T_n+1(r)−T_n(r)

We remark thatkθnkL^∞(R)≤1 for any n≥1 and thatθ_n(r)→0 for anyr when ntends to infinity.

Using the admissible test functionθn(u^ε) in (3.7) leads to Z

Ω

bⁿ_i,ε(u^ε_i)(t)dx+ Z t

0

Z

Ω

a_ε(x, u^ε_i, Du^ε_i)Dθ_n(u^ε_i)dx ds +

Z t

0

Z

Ω

Φi,ε(u^ε_i)Dθn(u^ε_i)dx ds+ Z t

0

Z

Ω

f_i^ε(x, u^ε₁, u^ε₂)θn(u^ε_i)dx ds

= Z

Ω

bⁿ_i,ε(u^ε_i,0)dx,

(3.18)

for almost anyt in (0, T) and wherebⁿ_i,ε(r) =Rr

0 b⁰_i,ε(s)θ_n(s)ds.

The Lipschitz character of Φ_ε, Stokes formula together with the boundary con- dition (3.8) allow to obtain

Z t

0

Z

Ω

Φi,ε(u^ε)Dθn(u^ε_i)dx ds= 0. (3.19) Sincebⁿ_i,ε(.)≥0, f_i^ε satisfies (1.11), we have

Z t

0

Z

Ω

a(x, u^ε_i, Du^ε_i)Dθn(u^ε_i)dx ds≤ Z

Ω

bⁿ_i,ε(u^ε_i,0)dx, (3.20) for almostt∈(0, T) and forε < _n+1¹ .

Step 3. Arguing again as in [3, 7, 4, 5], estimates (3.13)and (3.14) imply that for a subsequence still indexed byε,

u^ε_i converges almost every where tou_i inQ (3.21) and thanks to (3.12),

T_K(u^ε_i) converges weakly to T_K(u_i) inL^p(0, T;W₀^1,p(Ω)), (3.22) θ_n(u^ε_i)* θ_n(u_i) weakly inL^p(0, T;W₀^1,p(Ω)) (3.23) aε

x, TK(u^ε_i), DTK(u^ε_i)

* Xi,K weakly in (L^p0(Q))^N. (3.24) as ε tends to 0 for any K > 0 and any n ≥ 1. Here, for any K > 0 and for i= 1,2, Xi,K belongs to (L^p0(Q))^N.

We now establish thatb_i(u_i) belongs toL^∞(0, T;L¹(Ω)). Indeed using ¹_σT_σ(u^ε_i) as a test function in (3.7) leads to

1 σ

Z

Ω

b^σ_i,ε(u^ε_i)(t)dx+ 1 σ

Z t

0

Z

Ω

a_ε(x, u^ε_i, Du^ε_i)DT_σ(u^ε_i)dx ds + 1

σ Z t

0

Z

Ω

Φi,ε(u^ε_i)DTσ(u^ε_i)dx ds+ 1 σ

Z t

0

Z

Ω

f_i^ε(x, u^ε₁, u^ε₂)Tσ(u^ε_i)dx ds

= 1 σ

Z

Ω

b^σ_i,ε(u^ε_i,0)dx,

(3.25)

for almost anyt in (0, T). Where,bⁿ_i,ε(r) =Rr

0 b⁰_i,ε(s)T_σ(s)ds.

The Lipschitz character of Φε, Stokes formula together with the boundary con- dition (3.8) allow to obtain

1 σ

Z t

0

Z

Ω

Φi,ε(u^ε_i)DTσ(u^ε_i)dx ds= 0. (3.26)

Sinceaεsatisfies (1.5) andf_i^εsatisfies (1.11), lettingσgo to zero, it follows that Z

Ω

|b_i,ε(u^ε_i)(t)|dx≤ kb_i,ε(u^ε_i,0)k_L1(Ω) (3.27) for almostt∈(0, T). Recalling (3.6), (3.21) and (3.27) makes it possible to pass to the limit-inf and we show thatb_i(u_i) belongs toL^∞(0, T;L¹(Ω)).

We are now in a position to exploit (3.20). The pointwise convergence ofu^ε to uandbi,ε(u^ε₀) tobi(u0) imply that

lim sup

ε→0

Z t

0

Z

Ω

a(x, u^ε_i, Du^ε_i)Dθn(u^ε_i)dx ds≤ Z

Ω

bⁿ_i(ui,0)dx, (3.28) Sinceθn converge to zero everywhere asngoes to zero, the Lebesgue’s convergence theorem permits to conclude that

n→+∞lim lim sup

ε→0

Z

{n≤|u^ε_i|≤n+1}

aε(x, u^ε_i, Du^ε_i)Du^ε_idx dt= 0. (3.29) Step 4. This step is devoted to introduce for K≥0 fixed, a time regularization of the functionT_K(u_i) in order to perform the monotonicity method which will be developed in Step 5 and Step 6. This kind of regularization has been first introduced by Landes (see Lemma 6 and Proposition 3, p. 230 and Proposition 4, p. 231 in [18]). More recently, it has been exploited in [9] and [16] to solve a few nonlinear evolution problems withL¹ or measure data.

This specific time regularization ofTK(ui) (for fixedK≥0) is defined as follows.

let us consider the unique solution TK(ui)µ ∈ L^∞(Q)∩L^p(0, T;W₀^1,p(Ω)) of the monotone problem:

∂T_K(u_i)_µ

∂t +µ

T_K(u_i)_µ−T_K(u_i)

= 0 in D⁰(Q). (3.30) TK(ui)µ(t= 0) = 0 in Ω. (3.31) We remark that forµ >0 andK≥0,

∂TK(ui)µ

∂t ∈L^p(0, T;W₀^1,p(Ω)). (3.32) The behavior ofTK(ui)µ asµ→+∞is investigated in [18] (see also [16] and [17]) and we just recall here that (3.30)-(3.31) imply that

TK(ui)µ →TK(ui) a.e. inQ , (3.33) and inL^∞(Q) weak? and strongly inL^p(0, T;W₀^1,p(Ω)) asµ→+∞.

kTK(ui)µk_L^∞_(Q)≤K (3.34) for anyµand anyK≥0.

Letv_i,j∈C₀^∞(Ω), such thatv_i,j converges almost everywhere tou_i,0in Ω. And let us consider

TK(ui)_µ,j =TK(ui)_µ+e^−µtTK(vi,j)

is a smooth approximation ofTK(ui). We remark that forµ >0, j >0 andK≥0, we have|TK(ui)µ,j| ≤K and

∂TK(ui)µ,j

∂t =µ

TK(ui)−TK(ui)µ,j

, (3.35)

TK(ui)µ,j(0) =TK(vi,j), (3.36) TK(ui)µ,j→TK(ui) strongly inL^p(0, T;W₀^1,p(Ω)), (3.37) asµtends to infinity.

We denote byw(ε, µ, j) the quantities such that

j→+∞lim lim

µ→+∞lim

ε→0w(ε, µ, j) = 0.

The main estimate is as follows.

Lemma 3.2. LetK≥0be fixed. LetS be an increasingC^∞(R)-function such that S(r) =rfor|r| ≤K andsupp(S⁰) is compact. Then

lim inf

µ→+∞lim

ε→0

Z T

0

Z s

0

∂b_i,S(u^ε_i)

∂t , TK(u^ε_i)−(TK(ui))µ

dt ds≥0

where h,i denotes the duality pairing between L¹(Ω) +W^−1,p0(Ω) and L^∞(Ω)∩ W₀^1,p(Ω). and where bi,S(r) =Rr

0 b⁰_i(s)S(s)ds.

The proof of the above Lemma can be found in [24].

Step 5. In this step we prove the following Lemma which is the key point in the monotonocity arguments that will be developed in Step 6.

Lemma 3.3. The subsequence ofu^εdefined is Step 3 satisfies: For any K≥0, lim sup

ε→0

Z T

0

Z t

0

Z

Ω

a(u^ε_i, DTK(u^ε_i))DTK(u^ε_i)dx ds dt

≤ Z T

0

Z t

0

Z

Ω

Xi,KDTK(ui)dx ds dt

(3.38)

Proof. We first introduce a sequence of increasingC^∞(R)-functions Sn such that, for anyn≥1

Sn(r) =rfor|r| ≤n, supp(S_n⁰)⊂[−(n+ 1),(n+ 1)], kS_n⁰⁰k_L^∞_(R)≤1. (3.39) Pointwise multiplication of (3.7) byS_n⁰(u^ε_i) (which is licit) leads to

∂bi,S_n(u^ε_i)

∂t −div

Sn(u^ε_i)aε(x, u^ε_i, Du^ε_i)

+S_n⁰⁰(u^ε_i)aε(x, u^ε_i, Du^ε_i)Du^ε_i

−div

Φ_i,ε(u^ε_i)S⁰_n(u^ε_i)

+S⁰⁰_n(u^ε_i)Φ_i,ε(u^ε_i)Du^ε_i+f_i^ε(x, u^ε₁, u^ε₂)S⁰_n(u^ε_i) = 0

(3.40)

in D⁰(Q). We use the sequence TK(u)µ of approximations of TK(u) defined by (3.30), (3.31) of Step 4 and plug the test functionTK(u^ε)−TK(u)µ (forε >0 and µ >0) in (3.40). Through setting, for fixedK≥0,

W_i,µ^ε =T_K(u^ε_i)−T_K(u_i)_µ (3.41)

we obtain upon integration over (0, t) and then over (0, T), Z T

0

Z t

0

∂bi,S_n(u^ε_i)

∂t , W_i,µ^ε ds dt

+ Z T

0

Z t

0

Z

Ω

S_n⁰(u^ε_i)aε(x, u^ε_i, Du^ε_i)DW_i,µ^ε dx ds dt +

Z T

0

Z t

0

Z

Ω

S_n⁰⁰(u^ε_i)W_i,µ^ε aε(x, u^ε_i, Du^ε_i)Du^ε_idx ds dt +

Z T

0

Z t

0

Z

Ω

Φi,ε(u^ε_i)S_n⁰(u^ε_i)DW_i,µ^ε dx ds dt +

Z T

0

Z t

0

Z

Ω

S_n⁰⁰(u^ε_i)W_i,µ^ε Φ_i,ε(u^ε_i)Du^ε_idx ds dt +

Z T

0

Z t

0

Z

Ω

f_i^ε(x, u^ε₁, u^ε₂)S⁰_n(u^ε_i)W_i,µ^ε dx ds dt= 0

(3.42)

Next we pass to the limit as εtends to 0, then µtends to +∞and thenn tends to +∞, the real numberK≥0 being kept fixed. In order to perform this task we prove below the following results for fixedK≥0:

lim inf

µ→+∞lim

ε→0

Z T

0

Z t

0

∂b_i,Sn(u^ε_i)

∂t , W_i,µ^ε

ds dt≥0 for anyn≥K, (3.43)

µ→+∞lim lim

ε→0

Z T

0

Z t

0

Z

Ω

S_n⁰(u^ε_i)Φi,ε(u^ε_i)DW_i,µ^ε dx ds dt= 0 for anyn≥1, (3.44)

µ→+∞lim lim

ε→0

Z T

0

Z t

0

Z

Ω

S_n⁰⁰(u^ε_i)W_i,µ^ε Φi,ε(u^ε_i)Du^ε_idx ds dt= 0 for anyn, (3.45)

n→+∞lim lim

µ→+∞ lim

ε→0

Z T

0

Z t

0

Z

Ω

S_n⁰⁰(u^ε_i)W_i,µ^ε aε(u^ε_i, Du^ε_i)Du^ε_idx ds dt

= 0, (3.46)

µ→+∞lim lim

ε→0

Z T

0

Z t

0

Z

Ω

f_i^ε(x, u^ε₁, u^ε₂)S_n⁰(u^ε_i)W_i,µ^ε dx ds dt= 0 for anyn≥1. (3.47) Proof of (3.43). In view of (3.41) of W_i,µ^ε , Lemma 3.2 applies with S=Sn for fixedn≥K. As a consequence (3.43) holds.

Proof of (3.44). For fixedn≥1, we have

S_n⁰(u^ε_i)Φ_i,ε(u^ε_i)DW_i,µ^ε =S_n⁰(u^ε_i)Φ_i,ε(T_n+1(u^ε_i))DW_i,µ^ε (3.48) a.e. inQ, and for allε≤ _n+1¹ , and where suppS_n⁰ ⊂[−(n+ 1), n+ 1].

SinceS_n⁰ is smooth and bounded, (1.8), (3.5) and (3.22) lead to

S⁰_n(u^ε_i)Φ_i,ε(T_n+1(u^ε_i))→S_n⁰(u_i)Φ_i(T_n+1(u_i)) (3.49) a.e. inQand inL^∞(Q) weak?, asεtends to 0. For fixed µ >0, we have

W_i,µ^ε * T_K(u_i)−T_K(u_i)_µ weakly inL^p(0, T;W₀^1,p(Ω)) (3.50)

and a.e. inQand in L^∞(Q) weak?, as εtends to 0. As a consequence of (3.48), (3.49) and (3.50) we deduce that

ε→0lim Z T

0

Z t

0

Z

Ω

S⁰_n(u^ε_i)Φ_i,ε(u^ε_i)DW_i,µ^ε dx ds dt

= Z T

0

Z t

0

Z

Ω

S_n⁰(ui)Φi(ui)

DTK(ui)−DTK(ui)µ

dx ds dt

(3.51)

for any µ > 0. Appealing now to (3.33) and passing to the limit as µ →+∞ in (3.51) allows to conclude that (3.44) holds.

Proof of (3.45). For fixedn≥1, and by the same arguments as those which lead to (3.48), we have

S_n⁰⁰(u^ε_i)Φ_i,ε(u^ε_i)Du^ε_iW_i,µ^ε =S_n⁰⁰(u^ε_i)Φ_i,ε(T_n+1(u^ε_i))DT_n+1(u^ε_i)W_i,µ^ε a.e. inQ.

From (1.8), (3.3) and (3.22), it follows that for anyµ >0,

ε→0lim Z T

0

Z t

0

Z

Ω

S_n⁰⁰(u^ε_i)Φi,ε(u^ε_i)Du^ε_iW_i,µ^ε dx ds dt

= Z T

0

Z t

0

Z

Ω

S_n⁰⁰(ui)Φi(Tn+1(ui))DTn+1(ui)Wi,µ

DTK(ui)−DTK(ui)µ

dx ds dt

with the help of (3.37) passing to the limit, asµtends to +∞, in the above equality, we find (3.45).

Proof of (3.46). For any n ≥ 1 fixed, we have supp(S_n⁰⁰)⊂ [−(n+ 1),−n]∪ [n, n+ 1]. As a consequence

Z T

0

Z t

0

Z

Ω

S_n⁰⁰(u^ε_i)aε(x, u^ε_i, Du^ε_i)Du^ε_iW_i,µ^ε dx ds dt

≤TkS_n⁰⁰kL^∞(R)kW_i,µ^ε kL^∞(Q)

Z

{n≤|u^ε_i|≤n+1}

aε(x, u^ε_i, Du^ε_i)Du^ε_idx dt,

for anyn≥1, and anyµ >0. The above inequality together with (3.34) and (3.39) make it possible to obtain

lim sup

µ→+∞

lim sup

ε→0

Z T

0

Z t

0

Z

Ω

S_n⁰⁰(u^ε_i)aε(u^ε_i, Du^ε_i)Du^ε_iW_i,µ^ε dx ds dt

≤Clim sup

ε→0

Z

{n≤|u^ε_i|≤n+1}

aε(u^ε_i, Du^ε_i)Du^ε_idx dt,

(3.52)

for anyn≥1, whereCis a constant independent ofn. Using (3.29) we pass to the limit asntends to +∞in (3.52) and establish (3.46).

Proof of (3.47). For fixedn≥1, we have,

f₁^ε(x, u^ε₁, u^ε₂)S_n⁰(u^ε₁) =f1(x, Tn+1(u^ε₁), T1

ε(u^ε₂))S_n⁰(u^ε₁), f₂^ε(x, u^ε₁, u^ε₂)S_n⁰(u^ε₂) =f2(x, T1

ε(u^ε₁), Tn+1(u^ε₂))S_n⁰(u^ε₂)

a.e. in Q, and for all ε ≤ _n+1¹ . In view of (1.10), (3.21) and (3.22), Lebesgue’s convergence theorem implies that for anyµ >0 and anyn≥1

ε→0lim Z T

0

Z t

0

Z

Ω

f₁^ε(x, u^ε₁, u^ε₂)S_n⁰(u^ε_i)W_µ^εdx ds dt

= Z T

0

Z t

0

Z

Ω

f1(x, u1, u2)S_n⁰(ui)

TK(ui)−TK(ui)µ

dx ds dt.

Now for fixedn≥1, using (3.33) permits to pass to the limit asµtends to +∞in the above equality to obtain (3.47).

We now turn back to the proof of Lemma 3.3, due to (3.43), (3.44), (3.45), (3.46) and (3.47), we are in a position to pass to the lim-sup when εtends to zero, then to the limit-sup when µtends to +∞ and then to the limit asn tends to +∞ in (3.42). We obtain using the definition ofW_µ^εthat for anyK≥0,

n→+∞lim lim sup

µ→+∞

lim sup

ε→0

Z T

0

Z t

0

Z

Ω

S_n⁰(u^ε_i)a_ε(u^ε_i, Du^ε_i) DT_K(u^ε_i)

−DT_K(u_i)_µ

dx ds dt≤0.

SinceS_n⁰(u^ε_i)a_ε(u^ε_i, Du^ε_i)DT_K(u^ε_i) =a(u^ε_i, Du^ε_i)DT_K(u^ε_i) forε≤ _K¹ andK≤n.

The above inequality implies that forK≤n, lim sup

ε→0

Z T

0

Z t

0

Z

Ω

aε(x, u^ε_i, Du^ε_i)DTK(u^ε_i)dx ds dt

≤ lim

n→+∞lim sup

µ→+∞

lim sup

ε→0

Z T

0

Z t

0

Z

Ω

S_n⁰(u^ε_i)aε(x, u^ε_i, Du^ε_i)DTK(ui)µdx ds dt (3.53)

The right hand side of (3.53) is computed as follows: In view of (3.2) and (3.40), we have forε≤ _n+1¹ ,

S_n⁰(u^ε_i)aε(x, u^ε_i, Du^ε_i) =S_n⁰(u^ε_i)a

x, Tn+1(u^ε_i), DTn+1(u^ε_i)

a.e. inQ.

Due to (3.24), it follows that for fixedn≥1,

S_n⁰(u^ε_i)aε(u^ε_i, Du^ε_i)* S_n⁰(ui)Xi,n+1 weakly in (L^p0(Q))^N, whenεtends to 0.

The strong convergence ofT_K(u_i)_µ toT_K(u_i) inL^p(0, T;W₀^1,p(Ω)) asµtends to +∞, allows then to conclude that

µ→+∞lim lim

ε→0

Z T

0

Z t

0

Z

Ω

S⁰_n(u^ε_i)aε(x, u^ε_i, Du^ε_i)DTK(ui)µdx ds dt

= Z T

0

Z t

0

Z

Ω

S_n⁰(u_i)X_i,n+1DT_K(u_i)dx ds dt

= Z T

0

Z t

0

Z

Ω

Xi,n+1DTK(ui)dx ds dt

(3.54)

as long asK≤n, sinceS_n⁰(r) = 1 for|r| ≤n. Now forK≤n, we have a x, T_n+1(u^ε_i), DT_n+1(u^ε_i)

χ_{|uε

i|<K}=a x, T_K(u^ε_i), DT_K(u^ε_i) χ_{|uε

i|<K}, a.e. inQ. Passing to the limit asεtends to 0, we obtain

X_i,n+1χ_{|ui|<K}=X_i,Kχ_{|ui|<K} a.e. inQ− {|ui|=K}forK≤n. (3.55)

As a consequence of (3.55), forK≤n, we have

Xn+1DTK(ui) =XKDTK(ui) a.e. inQ. (3.56) Taking into account (3.53), (3.54) and (3.56), we conclude that (3.38) holds true and the proof of Lemma 3.3 is complete.

Step 6. In this step, we prove the following monotonicity estimate.

Lemma 3.4. The subsequence ofu^ε_i defined in step 3 satisfies: For any K≥0,

ε→0lim Z T

0

Z t

0

Z

Ω

a(T_K(u^ε_i), DT_K(u^ε_i))−a(T_K(u^ε_i), DT_K(u_i))

×

DTK(u^ε_i)−DTK(ui)

dx ds dt= 0.

(3.57)

Proof. LetK≥0 be fixed. The monotone character (1.7) ofa(s, ξ) with respect to ξimplies that

Z T

0

Z t

0

Z

Ω

a(TK(u^ε_i), DTK(u^ε_i))−a(TK(u^ε_i), DTK(ui))

×

DTK(u^ε_i)−DTK(ui)

dx ds dt≥0,

(3.58)

In order to pass to the limit-sup as ε tends to 0 in (3.58), let us recall first that (1.4), (1.6) and (3.21) imply

a(T_K(u^ε_i), DT_K(u_i))→a(T_K(u_i), DT_K(u_i)) a.e. inQ, asεtends to 0, and that

a(TK(u^ε_i), DTK(ui))

≤CK(t, x) +βK|DTK(ui)|^p−1 a.e. inQ, uniformly with respect toε. It follows that whenεtends to 0,

a TK(u^ε_i), DTK(ui)

→a TK(ui), DTK(ui)

strongly in (L^p0(Q))^N. (3.59) Using (3.38) of Lemma 3.3, (3.22), (3.24) and (3.59), we can pass to the lim-sup as εtends to zero in (3.58) to obtain (3.57) of Lemma 3.4.

Step 7. In this step we identify the weak limitXi,K and we prove the weakL¹ convergence of the “truncated” energy a T_K(x, u^ε_i), DT_K(u^ε_i)

DT_K(u^ε_i) as εtends to 0.

Lemma 3.5. For fixedK≥0, asεtends to0, we have X_i,K=a x, T_K(u^ε_i), DT_K(u^ε_i)

a.e. inQ. (3.60)

Also, asε tends to0, a TK(u^ε_i), DTK(u^ε_i)

DTK(u^ε_i)* a TK(ui), DTK(ui)

DTK(ui), (3.61) weakly inL¹(Q).

Proof. The proof is standard once we remark that for anyK ≥0, any 0< ε < _K¹ and anyξ∈R^N

aε(x, TK(u^ε_i), ξ) =a(x, TK(u^ε_i), ξ) =a1

K(x, TK(u^ε_i), ξ) a.e. inQ