EXISTENCE OF A RENORMALISED SOLUTION FOR A CLASS OF NONLINEAR DEGENERATED PARABOLIC PROBLEM WITH UNBOUNDED NONLINEARITIES

EXISTENCE OF A RENORMALISED SOLUTION FOR A CLASS OF NONLINEAR DEGENERATED PARABOLIC PROBLEM

WITH UNBOUNDED NONLINEARITIES

Y. AKDIM, J. BENNOUNA, M. MEKKOUR and H. REDWANE

Communicated by Daniel Tataru

In this work, we study the existence of renormalized solutions for a class of nonlinear degenerated parabolic problem in the form

(0.1) ∂b(x, u)

∂t −div(a(x, t, u, Du)) + div(φ(u)) =f in Q,

where b(x, u) is unbounded function on u, the Caratheodory function a satis- fying the coercivity condition, the general growth condition and only the large monotonicity, the function φis assumed to be continuous on Rand not belong to(L¹_loc(Q))^N. The data belongs to L¹(Q).

AMS 2010 Subject Classication: A7A15, A6A32, 47D20.

Key words: weighted Sobolev spaces; truncations; time-regularization; renormal- ized solutions.

1. INTRODUCTION

LetΩbe a bounded open set ofR^N,pbe a real number such that2< p <

∞,Q= Ω×]0, T[and w={w_i(x), 0≤i≤N}be a vector of weight functions (i.e., every component wi(x) is a measurable function which is positive a.e. in Ω) satisfying some integrability conditions. The objective of this paper is to study the following problem in the weighted Sobolev space:

(1.1)

∂b(x, u)

∂t −div(a(x, t, u, Du)) + div(φ(u)) =f inQ, b(x, u)(t= 0) =b(x, u₀) in Ω

u= 0 in ∂Ω×]0, T[.

The data f and b(x, u0) lie in L¹(Q) and L¹(Ω), respectively. The func- tions φis just assumed to be continnous ofRwith values inR^N. The operator div(a(x, t, u, Du))is a Leray-Lions operator which is coercive, and which grows like |Du|^p−1 with respect to |Du|, but which is not restricted by any growth

REV. ROUMAINE MATH. PURES APPL. 60 (2015), 2, 129154

condition with respect to u and only the large monotonicity (see assumption (H₂)) and b(x, u) is unbounded function on u.

Let us point out, the diculties that arise in problem (1.1) are due to the following facts: the data f and u₀ only belong to L¹, a satises the large monotonicity that is

[a(x, t, s, ξ)−a(x, t, s, η)](ξ−η)≥0 for all (ξ, η)∈R^N ×R^N.

and the functionφ(u)does not belong to(L¹_loc(Q))^N (because the function φis just assumed to be continuous onR). To overcome this diculty, we will apply Landes's technical (see [14, 24]) and the framework of renormalized solutions.

This notion was introduced by Diperna and P.-L. Lions [20] in their study of the Boltzmann equation. This notion was then adapted to an elliptic version of (1.1) by L. Boccardo et al. [8] when the right hand side is in W^−1,p0(Ω), by J.-M. Rakotoson [27] when the right hand side is in L¹(Ω), and nally by G.

Dal Maso, F. Murat, L. Orsina and A. Prignet [19] for the case of right hand side is general measure data.

For the parabolic equation (1.1) the existence of weak solution has been proved by J.-M. Rakotoson [26] with the strict monotonicity and a measure data, the existence and uniqueness of a renormalized solution has been proved by D. Blanchard and F. Murat [10] in the case wherea(x, t, u, Du) is indepen- dent of u,φ= 0,b(x, u) =u, and by D.Blanchard, F. Murat and H. Redwane [11] with the large monotonicity on a.

For the degenerated parabolic equations the existence of weak solutions have been proved by L. Aharouch et al. [3] in the case where a is strictly monotone, φ= 0, b(x, u) = u and f ∈ L^p0(0, T, W^{−1, p0}(Ω, w^∗)). See also the existence of renormalized solution by Y. Akdim et al. [7] in the case where a(x, t, u, Du) is independent ofu andφ= 0,b(x, u) =u.

Note that, this paper can be seen as a generalization of [3, 29] in weighted case and as a continuation of [7].

The plan of the paper is as follows. In Section 2 we give some preliminaries and the denition of weighted Sobolev spaces. In Section 3 we make precise all the assumptions ona, φ, f andu0. In Section 4 we give some technical results.

In Section 5 we give the denition of a renormalized solution of (1.1) and we establish the existence of such a solution (Theorem 5.3). Section 6 is devoted to an example which illustrates our abstract result.

2. PRELIMINARIES

Let Ω be a bounded open set of R^N, p be a real number such that 2 <

p <∞ andw={w_i(x),0≤i≤N} be a vector of weight functions, i.e., every

component w_i(x) is a measurable function which is strictly positive a.e. in Ω. Further, we suppose in all our considerations that, there exits

r₀>max(N, p) such that w

−r0 r0−p

i ∈L¹_loc(Ω), (2.1)

wi∈L¹_loc(Ω), (2.2)

w

−1 p−1

i ∈L¹(Ω), (2.3)

for any 0 ≤ i ≤ N. We denote by W^1,p(Ω, w) the space of all real-valued functions u∈L^p(Ω, w₀) such that the derivatives in the sense of distributions fulll

∂u

∂x_i ∈L^p(Ω, w_i) for i= 1, . . . , N.

Which is a Banach space under the norm (2.4) kuk_1,p,w=hZ

Ω

|u(x)|^pw₀(x)dx+

N

X

i=1

Z

Ω

|∂u(x)

∂x_i |^pw_i(x)dxi1/p

.

The condition (2.2) implies that C₀^∞(Ω)is a subspace of W^1,p(Ω, w) and consequently, we can introduce the subspaceV =W₀^1,p(Ω, w) of W^1,p(Ω, w) as the closure ofC₀^∞(Ω)with respect to the norm(2.4). Moreover, condition(2.3) implies that W^1,p(Ω, w) as well asW₀^1,p(Ω, w) are reexive Banach spaces.

We recall that the dual space of weighted Sobolev spaces W₀^1,p(Ω, w) is equivalent to W^−1,p0(Ω, w^∗), where w^∗ = {w^∗_i = w_i^1−p0, i = 0, . . . , N} and wherep⁰ is the conjugate of pi.e. p⁰ = _p−1^p (see [23]).

3. BASIC ASSUMPTIONS

Assumption (H1). For2≤p <∞, we assume that the expression

(3.1) k|u|k_V =X^N

i=1

Z

Ω

|∂u(x)

∂xi

|^pw_i(x)dx1/p

is a norm dened on V which equivalent to the norm (2.4), and there exist a weight functionσ onΩ such that,

σ ∈L¹(Ω) and σ⁻¹∈L¹(Ω).

We assume also the Hardy inequality,

(3.2) Z

Ω

|u(x)|^qσ dx1/q

≤cX^N

i=1

Z

Ω

|∂u(x)

∂xi

|^pw_i(x)dx1/p

,

holds for every u ∈V with a constant c >0 independent of u, and moreover, the imbedding

(3.3) W^{1, p}(Ω, w),→,→L^q(Ω, σ),

expressed by the inequality (3.2) is compact. Note that(V,k|.|k_V)is a uniformly convex (and thus, reexive) Banach space.

Remark 3.1. If we assume thatw₀(x)≡1and in addition the integrability condition: There exists ν∈]^N_p,+∞[∩[_p−1¹ ,+∞[such that

(3.4) w^−ν_i ∈L¹(Ω) and w

N N−1

i ∈L¹_loc(Ω) for all i= 1, . . . , N.

Note that the assumptions (2.2) and (3.4) imply that,

(3.5) k|uk|=X^N

i=1

Z

Ω

|∂u

∂x_i|^pw_i(x)dx1/p

,

is a norm dened onW₀^1,p(Ω, w)and its equivalent to(2.4)and that, the imbed- ding

(3.6) W₀^1,p(Ω, w),→L^p(Ω),

is compact for all1≤q≤p^∗₁ifp.ν < N(ν+1)and for allq ≥1ifp.ν≥N(ν+1) wherep1 = _ν+1^pν andp^∗₁ is the Sobolev conjugate ofp1 (see [22], pp. 3031).

Assumption (H2).

(3.7) b: Ω×R→ R

is a Caratheodory function such that for every x ∈ Ω, b(x, .) is a strictly in- creasing C¹−function with b(x,0) = 0.

Next, for any k > 0, there exist λ_k > 0 and functions A_k ∈ L¹(Ω) and Bk∈L^p(Ω)such that

(3.8) λ_k≤ ∂b(x, s)

∂s ≤A_k(x) and

D_x

∂b(x, s)

∂s

≤B_k(x) for almost everyx∈Ω, for everyssuch that|s| ≤k, we denote byDx

∂b(x,s)

∂s

the gradient of ^∂b(x,s)_∂s dened in the sense of distributions.

For i = 1, ..., N and for any k > 0 there exist β_k > 0 and a function C_k(x, t)∈L^p0(Q) such that,

(3.9) |a_i(x, t, s, ξ)| ≤β_kw

1 p

i (x)[C_k(x, t) +

N

X

j=1

w

1 p0

j (x)|ξ_j|^p−1],

for almost everyx∈Ω, for everyssuch that|s| ≤kand ξ ∈R^N.

[a(x, t, s, ξ)−a(x, t, s, η)](ξ−η)≥0 for all(ξ, η)∈R^N ×R^N, (3.10)

a(x, t, s, ξ).ξ≥α

N

X

i=1

w_i|ξ_i|^p, (3.11)

φ: R→ R^N is a continuous function, (3.12)

f is an element of L¹(Q), (3.13)

u0is measurable function dened on Ωsuch thatb(x, u0)∈L¹(Ω).

(3.14)

Where α is strictly positive constant. We recall that, for k >1 and s in R, the truncation is dened as,

T_k(s) =

(s if |s| ≤k k_|s|^s if |s|> k.

4. SOME TECHNICAL RESULTS

Characterization of the time mollication of a functionu. In order to deal with time derivative, we introduce a time mollication of a function u belonging to a some weighted Lebesgue space. Thus, we dene for all µ ≥ 0 and all(x, t)∈Q,

uµ=µ Z t

∞

˜

u(x, s) exp(µ(s−t))ds, where u(x, s) =˜ u(x, s)χ_(0,T)(s).

Proposition 4.1 ([3]).

1) Ifu∈L^p(Q, w_i) then u_µ is measurable in Q and ^∂u_∂t^µ =µ(u−u_µ) and, ku_µk_Lp(Q,w

i)≤ kuk_Lp(Q,wi).

2) Ifu∈W₀^1,p(Q, w), then uµ→u in W₀^1,p(Q, w) as µ→ ∞.

3) Ifun→u in W₀^1,p(Q, w) , then (un)µ→uµ in W₀^1,p(Q, w).

Some weighted embedding and compactness results. In this sec- tion, we establish some embedding and compactness results in weighted Sobolev spaces, some trace results, Aubin's and Simon's results [30].

LetV=W₀^{1, p}(Ω, w), H=L²(Ω, σ)and letV^∗=W^−1,p0,with(2≤p <∞).

LetX=L^p(0, T;W₀^{1, p}(Ω, w)).The dual space ofXisX^∗ =L^p0(0, T, V^∗) where ¹_p+_p¹0 = 1 and denoting the spaceW_p¹(0, T, V, H) ={v∈X: v⁰ ∈X^∗} endowed with the norm

kuk_W1

p =kuk_X + u⁰

X^∗,

which is a Banach space. Hereu⁰ stands for the generalized derivative ofu, i.e., Z T

0

u⁰(t)ϕ(t)dt=− Z T

0

u(t)ϕ⁰(t)dt for all ϕ∈C₀^∞(0, T).

Lemma 4.2 ([31]).

1) The evolution triple V ⊆H ⊆V^∗ is veried.

2) The imbedding W_p¹(0, T, V, H)⊆C(0, T, H) is continuous.

3) The imbedding W_p¹(0, T, V, H)⊆L^p(Q, σ) is compact.

Lemma 4.3 ([3]). Letg∈L^r(Q, γ)and let g_n∈L^r(Q, γ), withkg_nk_Lr(Q,γ)

≤C, 1< r <∞. Ifgn(x)→g(x) a.e. in Q, then gn* g in L^r(Q, γ).

Lemma 4.4 ([3]). Assume that,

∂v_n

∂t =α_n+β_n in D⁰(Q)

whereα_n andβ_nare bounded respectively inX^∗ and in L¹(Q).If v_nis bounded in L^p(0, T;W₀^{1, p}(Ω, w)), then v_n→v in L^p_loc(Q, σ).

Further vn→v strongly in L¹(Q).

Denition 4.5. A monotone mapT :D(T)→X^∗is called maximal mono- tone if its graph

G(T) ={(u, T(u))∈X×X^∗ for all u∈D(T)}

is not a proper subset of any monotone set in X×X^∗.

Let us consider the operator _∂t^∂ which induces a linear map L from the subset D(L) ={v∈X :v⁰ ∈X^∗, v(0) = 0} ofX intoX^∗ by

hLu, vi_X = Z T

0

hu⁰(t), v(t)_Vdti u∈D(L), v ∈X.

Lemma 4.6 ([31]). L is a closed linear maximal monotone map.

In our study we deal with mappings of the form F = L+S where L is a given linear densely dened maximal monotone map from D(L) ⊂X to X^∗ and S is a bounded demicontinuous map of monotone type from X to X^∗.

Denition 4.7. A mapping S is called pseudo-monotone with un * u, Lu_n* Luand lim

n→∞suphS(u_n), u_n−ui ≤0,that we have

n→∞lim suphS(u_n), un−ui= 0 and S(un)* S(u) asn→ ∞.

5. MAIN RESULTS Consider the problem

(5.1)

b(x, u0)∈L¹(Ω), f ∈L¹(Q)

∂b(x, u)

∂t −div(a(x, t, u, Du)) + div(φ(u)) =f inQ u= 0 on ∂Ω×]0, T[,

b(x, u(x,0)) =b(x, u₀) on Ω

Denition 5.1. Let f ∈ L¹(Q) and b(x, u0)∈ L¹(Ω). A real-valued func- tion udened on Ω×]0, T[is a renormalized solution of problem (5.1) if (5.2)

Tk(u)∈L^p(0, T;W₀^{1, p}(Ω, w)) for all (k≥0) and b(x, u)∈L^∞(0, T;L¹(Ω));

(5.3) Z

{m≤|u|≤m+1}

a(x, t, u, Du)Dudxdt→0 as m→ +∞;

∂BS(x, u)

∂t −div S⁰(u)a(u, Du)

+S⁰⁰(u)a(u, Du)Du (5.4) +div(S⁰(u)φ(u))−S⁰⁰(u)φ(u)Du=f S⁰(u) in D⁰(Q);

for all functionsS ∈W^2,∞(R)which compact support in R, where B_S(x, z) = Rz

0

∂b(x,r)

∂r S⁰(r)dr and

(5.5) B_S(x, u)(t= 0) =B_S(x, u₀) in Ω.

Remark 5.2. Equation (5.4) is formally obtained through pointwise mul- tiplication of equation (5.1) by S⁰(u). However, while a(u, Du) and φ(u) does not in general make sense in (5.1), all the terms in (5.4) have a meaning in D⁰(Q).

Indeed, if M is such that supp(S⁰) ⊂ [−M, M], the following identica- tions are made in (5.4):

• S(u)∈L^∞(Q) since S is a bounded function.

• S⁰(u)a(u, Du)identies with S⁰(u)a(T_M(u), DT_M(u)) a.e. inQ. Since

|T_M(u)| ≤M a.e. inQ, assumptions (3.9) imply that

|a_i(x, t, TM(u), DTM(u))| ≤βMw

1 p

i (x) CM(x, t) +

N

X

i=1

w

1 p0

j (x)

∂T_M(u)

∂xj

p−1! .

We obtain that S⁰(u)a(TM(u), DTM(u))∈

N

Y

i=1

L^p0(Q, w_i^∗).

• S⁰⁰(u)a(u, Du)Du identies withS⁰⁰(u)a(T_M(u), DT_M(u))DT_M(u) we have S⁰⁰(u)a(T_M(u), DT_M(u))DT_M(u)∈L¹(Q).

• S⁰⁰(u)φ(u)Du and S⁰(u)φ(u)respectively identify withS⁰⁰(u)φ(T_M(u)) DT_M(u) and S⁰(u)φ(T_M(u)). Due to the properties of S⁰ and to (3.12), the functions S⁰, S⁰⁰ and φoTM are bounded on R so that (5.2) implies that S⁰(u)φ(T_M(u))∈(L^∞(Q))^N,andS⁰⁰(u)φ(T_M(u))DT_M(u)∈L^p(Q, w)

• S⁰(u)f belongs toL¹(Q) .

The above considerations show that equation (5.4) holds in D⁰(Q) and

that ∂B_S(x, u)

∂t ∈L^p0(0, T;W^{−1, p0}(Ω, w_i^∗)) +L¹(Q).

Due to the properties of S and (5.4), ^∂S(u)_∂t ∈L^p0(0, T;W^{−1, p0}(Ω, w_i^∗)) + L¹(Q), which implies thatS(u)∈C⁰([0, T];L¹(Ω))so that the initial condition (5.5) makes sense, since, due to the properties of S (increasing) and (3.8), we have

(5.6)

BS(x, r)−BS(x, r⁰)

≤Ak(x)

S(r)−S(r⁰)

for all r, r⁰ ∈R.

Theorem 5.3. Let f ∈L¹(Q) and b(x, u₀) ∈ L¹(Ω). Assume that (H1) and (H2) hold true. then, there exists at least a renormalized solution u of the problem (5.1) (in the sense of Denition 5.1).

Remark 5.4. The statement of Theorem 5.3 generalized in weighted case the analogous in [29] and [7] (with b(x, u) =u).

Remark 5.5. Since, the function φ(u) does note belong to L¹_loc(Q)N

. Then the problem (5.1) can have a renormalized solution, but not a weak solution.

Proof. Step 1: The approximate problem.

Forn >0,let us dene the following approximation of b, a, φ, f andu₀; (5.7) b_n(x, r) =b(x, T_n(r)) + 1

nr for n >0,

In view of (5.7), b_n is a Caratheodory function and satises (3.8), there exist λn>0 and functions An∈L¹(Ω)andBn∈L^p(Ω)such that

λ_n≤ ∂b_n(x, s)

∂s ≤A_n(x) and

D_x

∂b_n(x, s)

∂s

≤B_n(x) a.e.in Ω, s∈R. (5.8) an(x, t, s, d) =a(x, t, Tn(s), d) a.e. in Q, ∀s∈R, ∀d∈R^N,

In view of (5.8),an satisfy (3.11) and (3.9), there existsCn∈L^p0(Q) and βn>0such that

(5.9)

|aⁿ_i(x, t, s, ξ)| ≤β_nw

1 p

i (x)[C_n(x, t) +

N

X

j=1

w

1 p0

j (x)|ξ_j|^p−1], for all(s, ξ)∈R×R^N,

(5.10) φ_n is a Lipschitz continuous bounded function from R into R^N, such thatφ_n uniformly converges toφ on any compact subset ofR asntends to +∞,

(5.11)

f_n∈L^p0(Q) and f_n→f a.e. in Q and strongly in L¹(Q) as n→+∞, u0n∈D(Ω) : kb_n(x, u0n)k_L1 ≤ kb(x, u₀)k_L1,

(5.12) bn(x, u0n)→b(x, u0) a.e. in Ω and strongly in L¹(Ω).

Let us now consider the approximate problem:

(5.13)

∂b_n(x, u_n)

∂t −div(a_n(x, t, u_n, Du_n)) + div(φ_n(u_n)) =f_n in D⁰(Q), un= 0 in (0, T)×∂Ω,

bn(x, un(t= 0)) =bn(x, u0n) in Ω.

As a consequence, proving existence of a weak solution u_n ∈ L^p(0, T; W₀^{1, p}(Ω, w))of (5.13) is an easy task (see e.g. [25, 28]).

Step 2: The estimates derived in this step rely on standard techniques for problems of type (5.13).

Using in (5.13) the test function T_k(u_n)χ_(0,τ), we get, for everyτ ∈[0, T].

(5.14)

h∂b_n(x, u_n)

∂t , T_k(u_n)χ_(0,τ)i+ Z

Qτ

a(x, t, T_k(u_n), DT_k(u_n))DT_k(u_n)dxdt +

Z

Qτ

φn(un)DTk(un)dxdt= Z

Qτ

fnTk(un)dxdt, which implies that,

(5.15) Z

Ω

B_kⁿ(x, u_n(τ))dx+ Z τ

0

Z

Ω

a(x, t, T_k(u_n), DT_k(u_n))DT_k(u_n)dxdt +

Z

Qτ

φ_n(u_n)DT_k(u_n)dxdt= Z

Qτ

f_nT_k(u_n)dxdt+ Z

Ω

B_kⁿ(x, u_0n)dx, whereB_kⁿ(x, r) =Rr

0 Tk(s)^∂bn_∂s^(x,s)ds.The Lipschitz character ofφnand Stokes' formula together with the boundary condition 2 of problem (5.13) give

(5.16)

Z τ 0

Z

Ω

φn(un)DT_k(un)dxdt= 0.

Due to the denition of B_kⁿwe have (5.17) 0≤

Z

Ω

B_kⁿ(x, u0n)dx≤k Z

Ω

|b_n(x, u0n)|dx≤kkb(x, u₀)k_L1(Ω). Using (5.16), (5.17) and B_kⁿ(x, un)≥0, it follows from (5.15) that

(5.18) Z τ 0

Z

Ω

a(x, t, T_k(u_n), DT_k(u_n))DT_k(u_n)dxdt

≤k(kf_nk_L1(Q)+kb_n(x, u_0n)k_L1(Ω))≤Ck, Thanks to (3.11) we have

(5.19) α

Z

Q N

X

i=1

w_i(x)

∂T_k(u_n)

∂x_i

p

dxdt≤Ck, ∀k≥1.

We deduce from that above inequality (5.15) and (5.17) that (5.20) Z

Ω

Bⁿ_k(x, un)dx≤k(kfk_L1(Q)+kb(x, u₀)k_L1(Ω))≡Ck.

Then,T_k(un)is bounded inL^p(0, T;W₀^{1, p}(Ω, w)),T_k(un)* v_kinL^p(0, T; W₀^{1, p}(Ω, w)), and by the compact imbedding (3.6) gives,

T_k(u_n)→v_k strongly in L^p(Q, σ) and a.e. in Q.

Let k >0 large enough andB_R be a ball of Ω, we have, k meas({|u_n|> k} ∩BR×[0, T]) =

Z T 0

Z

{|un|>k}∩B_R

|T_k(un)|dxdt

≤ Z T

0

Z

BR

|T_k(un)|dxdt

≤ Z

Q

|T_k(u_n)|^pσdxdt

_p¹Z T 0

Z

BR

σ^1−p0dxdt _p¹0

≤T cR

Z

Q N

X

i=1

wi(x)

∂Tk(un)

∂x_i

p

dxdt

!¹_p

≤ck^1p, which implies that,

meas({|u_n|> k} ∩B_R×[0, T])≤ c₁ k^1−1p

, ∀k≥1.

So, we have

k→+∞lim (meas({|u_n|> k} ∩BR×[0, T])) = 0.

Now, we turn to prove the almost every convergence of unand bn(x, un). Consider now a function non decreasing g_k ∈ C²(R) such that g_k(s) = s for

|s| ≤ ^k₂ and gk(s) = k for |s| ≥ k. Multiplying the approximate equation by g_k⁰(bn(x, un)), we get

(5.21) ∂g_k(b_n(x, u_n))

∂t −div(a(x, t, u_n, Du_n)g⁰_k(b_n(x, u_n)))

+a(x, t, u_n, Du_n)g_k⁰⁰(b_n(x, u_n))D_x

∂b_n(x, u_n)

∂s

Du_n

−div(g⁰_k(bn(x, un))φn(un)) +g⁰⁰_k(bn(x, un))Dx

∂bn(x, un)

∂s

φn(un)Dun

=f_ng_k⁰(b_n(x, u_n)) in the sense of distributions, which implies that

(5.22) gk(bn(x, un)) is bounded in L^p(0, T;W₀^{1, p}(Ω, w)), and

(5.23) ∂g_k(b_n(x, u_n))

∂t is bounded in X^∗+L¹(Q),

independently ofnas soon ask < n. Due to Denition (3.7) and (5.7) ofbn, it is clear that

{|b_n(x, un)| ≤k} ⊂ {|u_n| ≤k^∗}

as soon as k < nand k^∗ is a constant independent ofn. As a rst consequence we have

(5.24)

Dgk(bn(x, un)) =g_k⁰(x, bn(un))Dx

∂bn(x, Tk^∗(un))

∂s

DTk^∗(un) a.e. in Q as soon as k < n. Secondly, the following estimate holds true

g_k⁰(b_n(x,u_n))D_x

∂b_n(x,T_k^∗(u_n))

∂s

_L∞(Q)

≤ g⁰_k

L^∞(Q)( max

|r|≤k^∗

D_x

∂b_n(x, s)

∂s

+1).

As a consequence of (5.19), (5.24) we then obtain (5.22). To show that (5.23) holds true, due to (5.21) we obtain

(5.25) ∂gk(bn(x, un))

∂t =div(a(x, t, un, Dun)g_k⁰(bn(x, un)))

−a(x, t, un, Dun)g⁰⁰_k(bn(un))Dx

∂b_n(x, u_n)

∂s

+div(g⁰_k(bn(x, un))φn(un)

−g_k⁰⁰(bn(un))Dx

∂bn(x, un)

∂s

φn(un)Dun+fng⁰_k(bn(x, un)).

Since suppg⁰_k and suppg_k⁰⁰ are both included in [−k, k], un may be re- placed by T_k^∗(u_n) in each of these terms. As a consequence, each term on the right-hand side of (5.25) is bounded either inL^p0(0, T;W^−1,p0(Ω, w^∗))or in L¹(Q). Hence, lemma 4.4 allows us to conclude that gk(bn(x, un)) is compact inL^p_loc(Q, σ).

Thus, for a subsequence, it also converges in measure and almost every where inQ, due to the choice of gk, we conclude that for each k, the sequence T_k(b_n(x, u_n))converges almost everywhere inQ(since we have, for everyλ >0,)

meas({|b_n(x, u_n)−b_m(x, u_m)|> λ} ∩B_R×[0, T])

≤meas({|b_n(x, un)|> k}∩B_R×[0, T])+meas({|b_m(x, um)|> k}∩B_R×[0, T]) +meas({|g_k(bn(x, un))−gk(bm(x, um))|> λ}).

Let ε >0,then, there exist k(ε)>0 such that, meas({|b_n(x, un)−bm(x, um)|> λ} ∩BR×[0, T])≤ε

for all n, m≥n0(k(ε), λ, R).

This proves that(bn(x, un))is a Cauchy sequence in measure inBR×[0, T], thus converges almost everywhere to some measurable function v. Then for a subsequence denoted again u_n,

(5.26) un→u a.e. in Q,

and

(5.27) b_n(x, u_n)→b(x, u) a.e. in Q, we can deduce from (5.19) that,

(5.28) T_k(un)* T_k(u) weakly in L^p(0, T;W₀^{1, p}(Ω, w)) and then, the compact imbedding (3.3) gives,

Tk(un)→Tk(u) strongly in L^q(Q, σ) and a.e. in Q.

Which implies, by using (3.9), for all k > 0 that there exists a function hk∈

N

Q

i=1

L^p0(Q, w_i^∗), such that

(5.29) a(x, t, T_k(un), DT_k(un))* h_k weakly in

N

Y

i=1

L^p0(Q, w^∗_i).

We now establish that b(x, u) belongs to L^∞(0, T;L¹(Ω)). Using (5.26) and passing to the limit-inf in (5.20) as ntends to+∞, we obtain that

1 k

Z

Ω

B_k(x, u)(τ)dx≤[kfk_L1(Q)+ku₀k_L1(Ω)]≡C,

for almost any τ in (0, T). Due to the denition ofB_k(x, s) and the fact that

1

kB_k(x, u) converges pointwise tob(x, u), as ktends to+∞, shows thatb(x, u) belong to L^∞(0, T;L¹(Ω)).

Step 3: This step is devoted to introduce for k≥0 xed a time regular- ization of the function T_k(u) and to establish the following limits:

(5.30)

a(x, t, T_k(u_n), DT_k(u_n))* a(x, t, T_k(u), DT_k(u)) weakly in

N

Y

i=1

L^p0(Q, w^∗_i),

asntends to+∞.This proof is devoted to introduce fork≥0xed, a time reg- ularization of the functionT_k(u)in order to perform the monotonicity method.

Firstly, we prove the following lemma:

Lemma 5.6.

(5.31) lim

m→+∞ lim

n→+∞

Z

{m≤|un|≤m+1}

a(x, t, u_n, Du_n)Du_ndxdt= 0, for any integer m≥1,

Proof. Taking T₁(u_n−T_m(u_n))as a test function in (5.13), we obtain (5.32)

∂bn(x, un)

∂t , T1(un−Tm(un))

+ Z

{m≤|un|≤m+1}

a(un, Dun)Dundxdt +

Z

Q

div Z un

0

φ(r)T₁⁰(r−T_m(r))

dxdt= Z

Q

f_nT₁(u_n−T_m(u_n)).

Using the fact that Run

0 φ(r)T₁⁰(r −Tm(r))dxdt ∈ L^p(0, T;W₀^{1, p}(Ω, w)) and Stokes' formula, we get

(5.33)

Z

Ω

B_n^m(x, u_n)(T)dx+ Z

{m≤|u_n|≤m+1}

a(u_n, Du_n)Du_ndxdt

≤ Z

Q

|f_nT₁(u_n−T_m(u_n))|dxdt+ Z

Ω

B_n^m(x, u_0n)dx, where B_n^m(r) = Rr

0

∂bn(x,s)

∂s T₁(s−T_m(s))ds. In order to pass to the limit as n tends to +∞ in (5.33), we use B_n^m(x, u_n)(T) ≥0 and (5.11),(5.12), we obtain that

(5.34)

m→+∞lim Z

{m≤|un|≤m+1}

a(un, Dun)Dundxdt

≤ Z

{|u(x)|>m}

|f|dxdt+ Z

{|u0(x)|>m}

|b(x, u₀(x))|dx.

Finally, by (3.14), (3.13) and (5.34) we get

(5.35) lim

m→+∞ lim

n→+∞

Z

{m≤|un|≤m+1}

a(u_n, Du_n)Du_ndxdt= 0.

The very denition of the sequence (T_k(u))_µ for µ > 0 (and xed k) we establish the following lemma.

Lemma 5.7. Let k ≥0 be xed. Let (T_k(u))µ the mollication of T_k(u). Let S be an increasing C^∞(R)-function such that S(r) =r f or |r| ≤k and supp S⁰ is compact. Then,

(5.36) lim

µ→+∞ lim

n→+∞

Z T 0

∂b_n(x, u_n)

∂t , S⁰(u_n)(T_k(u_n)−(T_k(u))_µ)

dxdt≥0,

where h., .i denotes the duality pairing between L¹(Ω) + W^−1,p0(Ω, w^∗) and L^∞(Ω)∩W₀^{1, p}(Ω, w).

Proof. See H. Redwane [29].

We prove the following lemma, which is the key point in the monotonicity arguments.

Lemma 5.8. The subsequence of un satises for any k≥0 (5.37)

lim sup

n→+∞

Z T 0

Z t 0

Z

Ω

a(Tk(un), DTk(un))DTk(un)dxdsdt≤ Z T

0

Z t 0

Z

Ω

hkDTk(u)dxdsdt, where hk is dened in (5.29).

Proof. In the following we adapt the above-mentionned method to prob- lem (5.1) and we rst introduce a sequence of increasing C^∞(R)-functions S_m such that

S_m(r) =r if |r| ≤m, suppS_m⁰ ⊂[−(m+ 1), m+ 1],

S_m⁰⁰

L^∞ ≤1, for any m≥1.

We use the sequenceT_k(u)_µof approximations ofT_k(u), and plug the test function S_m⁰ (un)(Tk(un)−(Tk(u))µ) (forn >0 andµ >0)in (5.13). Through setting, for xedk≤0,

W_µⁿ=T_k(u_n)−(T_k(u))_µ,

we obtain upon integration over (0, t)and then over (0, T): (5.38)

Z T 0

Z t 0

∂bn(x, un)

∂t , S_m⁰ (un)W_µⁿ

dtds+

Z T 0

Z t 0

Z

Ω

S_m⁰ (un)an(un, Dun)DW_µⁿdxdsdt +

Z _T

0

Z _t

0

Z

Ω

S_m⁰⁰(u_n)a_n(u_n, Du_n)Du_nW_µⁿdxdsdt−

Z _T

0

Z _t

0

Z

Ω

S_m⁰ (u_n)φ_n(u_n)DW_µⁿdxdsdt

− Z T

0

Z t 0

Z

Ω

S_m⁰⁰(un)φn(un)DunW_µⁿdxdsdt= Z T

0

Z t 0

Z

Ω

fnS_m⁰ (un)W_µⁿdxdsdt.

In the following we pass the limit in (5.38) as n tends to+∞, thenµtends to +∞ and then m tends to +∞, the real numberk≥0 being kept xed. In order to perform this task we prove below the following results for xedk≥0 : (5.39) lim inf

µ→+∞ lim

n→+∞

Z T 0

Z t 0

∂B_mⁿ(x, un)

∂t , W_µⁿ

dtds≥0, for any m≥k,

(5.40) lim

n→+∞ lim

µ→+∞

Z T 0

Z t 0

Z

Ω

S_m⁰ (un)φn(un)DW_µⁿdxdsdt= 0, for any m≥1,

(5.41)

n→+∞lim lim

µ→+∞

Z T 0

Z t 0

Z

Ω

S_m⁰⁰(u_n)φ_n(u_n)Du_nW_µⁿdxdsdt= 0, for any m≥1, (5.42)

m→+∞lim lim sup

µ→+∞ lim sup

n→+∞

Z T 0

Z t 0

Z

Ω

S_m⁰⁰(un)a(un, Dun)DunW_µⁿdxdsdt

= 0, m≥1

(5.43) lim

µ→+∞ lim

m→+∞

Z T 0

Z t 0

Z

Ω

fnS_m⁰ (un)W_µⁿdxdsdt= 0.

Proof of (5.39). The functionSm belongs toC^∞(R)and is increasing. We have for m≥k,S_m(r) =r f or |r| ≤kwhile suppS⁰_m is compact. In view of the denition of W_µⁿ, lemma 5.7 applies with S = S_m for xed m ≥ k. As a consequence (5.39) holds true.

Proof of (5.40). In order to avoid repetitions in the proofs of (5.43), let us summarize the properties of W_µⁿ. For xedµ >0

W_µⁿ* T_k(u)−(T_k(u))_µ weakly in L^p(0, T;W₀^{1, p}(Ω, w)), as n→+∞

W_µⁿ

L^∞(Q)≤2k, for any n >0 and for anyµ >0 we deduce that for xed µ >0

W_µⁿ→T_k(u)−(T_k(u))_µ a.e. in Q and in L^∞(Q)weak− ∗, as n→+∞

one has suppS⁰⁰_m⊂[−(m+ 1),−m]∪[m, m+ 1] for any xed m≥1,we have (5.44) S_m⁰ (u_n)φ_n(u_n)DW_µⁿ=S_m⁰ (u_n)φ_n(T_m+1(u_n))DW_µⁿ a.e. in Q, since suppS_m⁰ ⊂ [−m−1, m+ 1]. Since S_m⁰ is smooth and bounded, (3.12), (5.10), andu_n→u a.e. in Q lead to

(5.45)

S_m⁰ (u_n)φ_n(T_m+1(u_n))→S_m⁰ (u)φ(T_m+1(u)) a.e. in Q and in L^∞(Q)weak−∗, asntends to +∞. As a consequence of (5.47) and (5.45), we deduce that

(5.46) lim

n→+∞

Z T 0

Z t 0

Z

Ω

S_m⁰ (un)φn(un)DW_µⁿdxdsdt=

= lim

n→+∞

Z T 0

Z t 0

Z

Ω

S_m⁰ (u_n)φ(T_m+1(u)) (DT_k(u)−D(T_k(u))_µ)dxdsdt, for any µ > 0. Passing to the limit as µ → +∞ in (5.46) we conclude that (5.40) holds true.

Proof of (5.41). For xed m ≥1,and by the same arguments that those that lead to (5.47), we have

(5.47)

S_m⁰⁰(u_n)φ_n(u_n)Du_nW_µⁿ=S_m⁰⁰(u_n)φ_n(T_m+1(u_n))DT_m+1(u_n)W_µⁿ a.e. in Q.

From (3.12),u_n→u a.e. in Qand (5.28), it follows that for anyµ >0

n→+∞lim Z T

0

Z t 0

Z

Ω

S_m⁰⁰(un)φn(un)DunW_µⁿdxdsdt

= Z T

0

Z t 0

Z

Ω

S_m⁰⁰(u_n)φ(T_m+1(u)) (DT_k(u)−D(T_k(u))_µ)dxdsdt,

for any µ > 0. Passing to the limit as µ → +∞ in (5.46) we conclude that (5.41) holds true.

Proof of (5.42). One has suppS_m⁰⁰ ⊂[−(m+ 1),−m]∪[m, m+ 1] for any m≥1. As a consequence

Z T 0

Z t 0

Z

Ω

S_m⁰⁰(u_n)a(u_n, Du_n)Du_nW_µⁿdxdsdt

≤T

S_m⁰⁰(u_n) L^∞

W_µⁿ L^∞

Z

{m≤|un|≤m+1}

a(u_n, Du_n)Du_ndxdt, for any m≥1,any µ >0 and anyn≥1. It is possible to obtain

lim sup

µ→+∞ lim sup

n→+∞

Z T 0

Z t 0

Z

Ω

S_m⁰⁰(un)a(un, Dun)DunW_µⁿdxdsdt

≤Clim sup

n→+∞

Z

{m≤|un|≤m+1}

a(un, Dun)Dundxdt, for any m≥1,whereC is a constant independent of m.

Appealing now to (5.31) it possible to pass the limit as m tends to +∞

to establish (5.42).

Proof of (5.43). Lebesgue's convergence theorem implies that for any µ >0and any m≥1

n→+∞lim Z T

0

Z t 0

Z

Ω

f_nS_m⁰ (u_n)W_µⁿdxdsdt= Z T

0

Z t 0

Z

Ω

f S_m⁰ (u)(T_k(u)−(T_k(u)_µ))dxdsdt Now, for xedm≥1, using lemma 4.1 and passing to the limit asµ→+∞

in the above equality to obtain (5.43).

We now turn back to the proof of lemma 5.8. Due to (5.39)(5.42) and (5.43), we are in a position to pass the limit-sup when n tends to +∞, then to the limit-sup whenµtends +∞ and then to the limit as m tends to+∞ in (5.38). We obtain by using the denition ofW_µⁿ that for any k≥0

m→+∞lim lim sup

µ→+∞ lim sup

n→+∞

Z T 0

Z t 0

Z

Ω

S_m⁰ (un)an(un, Dun)(DTk(un)−D(Tk(u))µ)dxdsdt≤0.

Since S_m⁰ (u_n)a_n(u_n, Du_n)DT_k(u_n) = a(u_n, Du_n)DT_k(u_n) for k ≤ n and k≤m, the above inequality implies that for k≤m

(5.48)

lim sup

n→+∞

Z T 0

Z t 0

Z

Ω

a_n(u_n, Du_n)DT_k(u_n)dxdsdt

≤ lim

m→+∞lim sup

µ→+∞ lim sup

n→+∞

Z T 0

Z t 0

Z

Ω

S_m⁰ (un)an(un, Dun)D(T_k(u))µdxdsdt.

The right-hand side of (5.48) is computed as follows. We have for n ≥ m+ 1:

S_m⁰ (u_n)a_n(u_n, Du_n) =S⁰_m(u_n)a(T_m+1(u_n), DT_m+1(u_n)) a.e. inQ.

Due to the weak convergence of a(DT_m+1(u_n)) it follows that for xed m≥1

S_m⁰ (un)an(un, Dun)* S_m⁰ (un)hm+1 weakly in

N

Y

i=1

L^p0(Q, w^∗_i),

when n tends to +∞. The strong convergence of (T_k(u))_µ to T_k(u) in L^p(0, T;W₀^{1, p}(Ω, w))asµ tends to+∞, then we conclude that

(5.49)

µ→+∞lim lim

n→+∞

Z T 0

Z t 0

Z

Ω

S_m⁰ (un)an(un, Dun)D(Tk(u))µdxdsdt

= Z T

0

Z t 0

Z

Ω

S_m⁰ (u_n)h_m+1DT_k(u)dxdsdt,

as soon as k≤m, S_m⁰ (r) = 1 for |r| ≤m.Now for k≤mwe have,

a(T_m+1(u_n), DT_m+1(u_n))χ_{|un|<k}=a(T_k(u_n), DT_k(u_n))χ_{|un|<k} a.e. in Q, which implies that, passing to the limit as n→+∞,

(5.50) h_m+1χ_{|un|<k} =h_kχ_{|un|<k} a.e. in Q− {|u|=k} for k≤m.

As a consequence of (5.50) we have for k≤m, (5.51) hm+1DTk(u) =hkDTk(u) a.e. in Q.

Recalling (5.48), (5.49), (5.51) we conclude that (5.37) holds true and the proof of Lemma 5.8 is complete.

In this Lemma we prove the following monotonicity estimate:

Lemma 5.9. The subsequence of u_n satises for any k≥0 (5.52)

n→+∞lim Z T

0

Z t 0

Z

Ω

[a(Tk(un),DTk(un))−a(T_k(un), DTk(u))][DTk(un)−DTk(u)]dxdsdt= 0.

Proof. Letk≥0be xed. The character (3.10) ofa(x, t, s, d)with respect to dimplies that

(5.53)

n→+∞lim Z _T

0

Z _t

0

Z

Ω

[a(T_k(u_n),DT_k(u_n))−a(T_k(u_n), DT_k(u))][DT_k(u_n)−DT_k(u)]dxdsdt≥0.

To pass to the limit-sup asn tends to+∞ in (5.53) imply that a(T_k(un), DT_k(u))→a(T_k(u), DT_k(u)) a.e. in Q, and that,

|a_i(T_k(u_n), DT_k(u))| ≤βw

1 p

i (x)



C_k(x, t) +

N

X

j=1

w

1 p0

j (x)

∂T_k(u)

∂x_j

p−1

 a.e. in Q, uniformly with respect to n. It follows that whenn tends to+∞

(5.54) a(T_k(u_n), DT_k(u))→a(T_k(u), DT_k(u)) strongly in

N

Y

i=1

L^p0(Q, w_i^∗).

Lemma 5.8, weak convergence ofDTk(un),a(Tk(un), DTk(un))and (5.54) make it possible to pass to the limit-sup as n→ +∞ in (5.53) and to obtain the result .

In this lemma we identify the weak limit hk and we prove the weak-L¹ convergence of the truncated energy a(T(un), DT_k(un))DT(un) as n tends to +∞.

Lemma 5.10. For xed k≥0, we have

(5.55) h_k=a(T(u), DT_k(u)) a.e. in Q, (5.56)

a(T(un), DT_k(un))DT(un)* a(T(u), DT_k(u))DT_k(u) weakly in L¹(Q).

Proof. The proof is standard once we remark that for any k ≥ 0, any n > k and any d∈R^N

a_n(T_k(u_n), d) =a(T_k(u_n), d) a.e. in Q

which together with weak convergence of(T_k(u_n))and a(DT_k(u_n))and (5.54) we obtain from (5.52)

(5.57)

n→+∞lim Z _T

0

Z _t

0

Z

Ω

a(T_k(u_n), DT_k(u_n))DT_k(u_n)dxdsdt= Z _T

0

Z _t

0

Z

Ω

h_kDT_k(u)dxdsdt.

The usual Minty's argument applies in view of weak convergence of(T_k(u_n)) and a(DT_k(u_n))and (5.57). It follows that (5.55) hold true.