Báo cáo hóa học: "PARABOLIC INEQUALITIES WITH NONSTANDARD GROWTHS AND L1 DAT"

PARABOLIC INEQUALITIES WITH NONSTANDARD GROWTHS AND L1 DATA

R. ABOULAICH, B. ACHCHAB, D. MESKINE, AND A. SOUISSI

Received 25 July 2005; Revised 13 December 2005; Accepted 19 December 2005

We prove an existence result for solutions of nonlinear parabolic inequalities with L1 data in Orlicz spaces.

Copyright © 2006 R. Aboulaich et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction Let Ω be an open bounded subset of RN , N ≥ 2, let Q be the cylinder Ω × (0,T) with some given T > 0. Consider the following nonlinear parabolic problem:

in Q,

∂u ∂t + A(u) = χ

(1.1)

u(x,t) = 0

on ∂Ω × (0,T), in Ω,

u(x,0) = u0

LM(Ω),

0

where A(u)=−div(a(x,t,u, ∇u)) is a Leray-Lions operator defined on D(A) ⊂W 1,x with M is an N-function, and χ is a given data.

In the variational case (i.e., where χ ∈ W −1,xEM(Ω)), it is well known that the solvabil- ity of (1.1) is done by Donaldson [2] and Robert [11] when the operator A is monotone, t2 (cid:6) M(t), and M satisfies a Δ2 condition, and by finally the recent work [3] for the gen- eral case.

In the L1 case, an existence theorem is given in [4]. However, the techniques used in [4] do not allow us to adapt it for parabolic inequalities. It is our purpose in this paper to solve the obstacle problem associated to (1.1) in the case where χ ∈ L1(Q) + W −1,xEM(Q) and without assuming any growth restriction on M. The existence of solutions is proved via a sequence of penalized problems, with solutions un. A priori estimates of the trun- cation of un are obtained in some suitable Orlicz space. For the passage to the limit, the

Hindawi Publishing Corporation Boundary Value Problems Volume 2006, Article ID 29286, Pages 1–18 DOI 10.1155/BVP/2006/29286

2

Parabolic inequalities in L1

almost everywhere convergence of ∇un is proved via new techniques. As operators mod- els, we can consider slow or fast growth:

(cid:2) (cid:5)

(cid:4)2∇u log (cid:3) 1 + |u|

,

A(u) = −div

(1.2)

(cid:4) (cid:3) 1 + |∇u| |∇u| (cid:4)(cid:4) . (cid:3) |∇u|

A(u) = −div(∇uexp

For some classical and recent results in the setting of Orlicz spaces dealing with elliptic and parabolic equations, the reader can be referred to [8, 10, 12–14].

2. Preliminaries 2.1. Let M : R+ → R+ be an N-function, that is, M is continous, convex, with M(t) > 0 for t > 0, M(t)/t → 0 as t → 0, and M(t)/t → ∞ as t → ∞. (cid:6) t Equivalently, M admits the representation M(t) = 0

a(s)ds, where a : R+ → R+ is non- decreasing, right continuous, with a(0) = 0, a(t) > 0 for t > 0, and a(t) tends to ∞ as t → ∞.

The N-function M conjugate to M is defined by M(t) =

(cid:6) t 0 ¯a(s)ds, where a : R+ → R+

is given by ¯a(t) = sup{s : a(s) ≤ t} (see [1]).

The N-function is said to satisfy the Δ2 condion if, for some k > 0,

M(2t) ≤ kM(t), ∀t ≥ 0,

(2.1)

when (2.1) holds only for t ≥ some t0 > 0, then M is said to satisfy the Δ2 condition near infinity.

We will extend these N-functions into even functions on all R. Let P and Q be two N-functions. P (cid:6) Q means that P grows essentially less rapidly than Q, that is, for each (cid:2) > 0, P(t)/Q((cid:2)t) → 0 as t → ∞. This is the case if and only if limt→∞(Q−1(t))/(P−1(t)) = 0.

2.2. Let Ω be an open subset of RN . The Orlicz class KM(Ω) (resp., the Orlicz space LM(Ω)) is defined as the set of (equivalence classes of) real-valued measurable functions u on Ω such that (cid:7)

(cid:7) (cid:8) (cid:8)

M

M

(cid:9) . (cid:3) u(x) (cid:4) dx < +∞

resp.,

(cid:9) dx < +∞ for some λ > 0

(2.2)

u(x) λ

Ω

Ω

LM(Ω) is a Banach space under the norm

(cid:7) (cid:10) (cid:8) (cid:11)

M

λ > 0 :

(cid:9) dx ≤ 1

(2.3)

(cid:11)u(cid:11)M,Ω = inf

u(x) λ

Ω

and KM(Ω) is a convex subset of LM(Ω).

The closure in LM(Ω) of the set of bounded measurable functions with compact sup-

port in Ω is denoted by EM(Ω).

The equality EM(Ω) = LM(Ω) holds if and only if M satisfies the Δ2 condition, for all t

or for t large, according to whether Ω has infinite measure or not.

3

The dual of EM(Ω) can be identified with LM(Ω) by means of the pairing

R. Aboulaich et al. (cid:6) Ω uv dx, and

the dual norm of LM(Ω) is equivalent to (cid:11) · (cid:11)M,Ω.

The space LM(Ω) is reflexive if and only if M and M satisfy the Δ2 condition, for all t

or for t large, according to whether Ω has infinite measure or not.

2.3. We now turn to the Orlicz-Sobolev space, W 1LM(Ω) (resp., W 1EM(Ω)) is the space of all functions u such that u and its distributional derivatives up to order 1 lie in LM(Ω) (resp., EM(Ω)). It is a Banach space under the norm

(cid:12) (cid:13) (cid:13) (cid:13) (cid:13)Dαu

(2.4)

M.

|α|≤1

(cid:11)u(cid:11)1,M =

(cid:14)

LM, we will use the weak topologies σ(

(cid:14)

Thus, W 1LM(Ω) and W 1EM(Ω) can be identified with subspaces of product of N +1 (cid:14) copies of LM(Ω). Denoting this product by LM, (cid:14) (cid:14)

LM).

EM) and σ( The space W 1 0

(cid:14)

LM(Ω) as the σ(

LM,

(cid:7) (cid:8)

M

LM, EM(Ω) is defined as the (norm) closure of the Schwartz space D(Ω) in (cid:14) EM) closure of D(Ω) in W 1LM(Ω). W 1EM(Ω) and the space W 1 0 We say that un converges to u for the modular convergence in W 1LM(Ω) if for some λ > 0, (cid:9) dx −→ 0, ∀|α| ≤ 1.

(2.5)

Dαun − Dαu λ

Ω

(cid:14) (cid:14)

LM,

LM). If M satisfies the Δ2 condition on R+, then

This implies convergence for σ( modular convergence coincides with norm convergence.

2.4. Let W −1LM(Ω) (resp., W −1EM(Ω)) denote the space of distributions on Ω which can be written as sums of derivatives of order ≤ 1 of functions in LM (resp., EM(Ω)). It is a Banach space under the usual quotient norm.

If the open set Ω has the segment property, then the space D(Ω) is dense in W 1 0

(cid:14) (cid:14)

LM,

LM(Ω) LM) (cf. [6, 7]). Con- LM(Ω) is well

for the modular convergence and thus for the topology σ( sequently, the action of a distribution in W −1LM(Ω) on an element of W 1 0 defined.

2.5. Let Ω be a bounded open subset of RN , T > 0, and set Q = Ω × (0,T). Let M be an N-function. For each α ∈ NN , denote by Dα x the distributional derivatives on Q of order α with respect to the variable x ∈ RN . The inhomogeneous Orlicz-Sobolev spaces of order 1 are defined as follows:

x u ∈ LM(Q), ∀|α| ≤ 1

(cid:16) , (cid:16)

(2.6)

.

W 1,xLM(Q) = W 1,xEM(Q) =

x u ∈ EM(Q), ∀|α| ≤ 1

(cid:15) u ∈ LM(Q) : Dα (cid:15) u ∈ EM(Q) : Dα

The latest space is a subset of the first one. They are Banach spaces under the norm (cid:12)

(cid:13) (cid:13) (cid:11)u(cid:11) = (cid:13) (cid:13)Dα x u

(2.7)

M,Q.

|α|=1

(cid:14)

We can easily show that they form a complementary system when Ω satisfies the seg- LM(Q) ment property.These spaces are considered as subspaces of the product spaces

4

Parabolic inequalities in L1

(cid:14) (cid:14)

LM,

(cid:14) (cid:14)

LM,

EM) and which has N + 1 copies. We will also consider the weak topologies σ( LM). If u ∈ W 1,xLM(Q), then the function t → u(t) = u(·,t) is defined on σ( (0,T) with values in W 1LM(Ω). If, further, u ∈ W 1,xEM(Q), then u(t) is W 1EM(Ω)- valued and is strongly measurable. Furthermore, the following continuous imbedding holds: W 1,xEM(Q) ⊂ L1(0,T;W 1EM(Ω)). The space W 1,xLM(Q) is not in general sepa- rable, if u ∈ W 1,xLM(Q), we cannot conclude that the function u(t) is measurable from (0,T) into W 1LM(Ω). However, the scalar function t → (cid:11)Dα x u(t)(cid:11)M,Ω is in L1(0,T) for all |α| ≤ 1.

0

(cid:14)

(cid:14)

σ(

EM )

(cid:14)

(cid:14)

LM ,

σ(

x u)/λ)dx → 0 when n → ∞. Consequently, D(Q)

(cid:14)

LM(Q). Furthermore, W 1,x

0

0

LM , (cid:6) Ω M((Dα LM ) LM , (cid:14)

LM(Q) ∩

0

2.6. The space W 1,x EM(Q) is defined as the (norm) closure in W 1,xEM(Q) of D(Q). We can easily show as in [7] that when Ω has the segment property, then for all u ∈ there exist some λ > 0 and a sequence (un) ⊂ D(Q) such that for all D(Q) EM ) = x un − Dα |α| ≤ 1, (cid:14) σ( , this space will be denoted by W 1,x D(Q) EM(Q) = W 1,x EM. Poincar´e’s inequality also holds in W 1,x LM(Q) and then there is a 0 constant C > 0 such that for all u ∈ W 1,x

(cid:12)

LM(Q), one has (cid:12)

0 (cid:13) (cid:13)

(cid:13) (cid:13)Dα x u (cid:13) (cid:13)Dα x u

(2.8)

M,Q ≤ C

|α|≤1

|α|=1

(cid:13) (cid:13) M,Q,

LM(Q). We have then

0

thus both sides of the last inequality are equivalent norms on W 1,x the following complementary system:

(cid:2) (cid:5)

,

(2.9)

LM(Q) F EM(Q) F0

W 1,x 0 W 1,x 0

F being the dual space of W 1,x (cid:14) LM by the polar set W 1,x

EM(Q). It is also, up to an isomorphism, the quotient of 0 EM(Q)⊥, and will be denoted by F = W −1,xLM(Q) and it is

0

shown that

(cid:10) (cid:11) (cid:12)

f =

Dα

.

W −1,xLM(Q) =

x fα : fα ∈ LM(Q)

(2.10)

|α|≤1

This space will be equipped with the usual quotient norm:

(cid:12) (cid:13) (cid:13) (cid:13) (cid:13) fα (cid:11) f (cid:11) = inf

(2.11)

M,Q,

|α|≤1

x fα,

|α|≤1

(cid:17) (cid:17)

Dα

Dα fα ∈ LM(Q). x fα : fα ∈ EM(Q)} and is denoted by

|α|≤1

where the inf is taken on all possible decompositions f = The space F0 is then given by F0 = { f = F0 = W −1,xEM(Q). Defintion 2.1. We say that un → u in W −1,xLM(Q) + L1(Q) for the modular convergence if we can write

(cid:12) (cid:12)

Dα

u =

Dα

un =

x uα

n + u0 n,

x uα + u0

(2.12)

|α|≤1

|α|≤1

R. Aboulaich et al.

5

n → uα in LM(Q) for the modular convergence for all |α| ≤ 1 and u0

n → u0 strongly

with uα in L1(Q).

We will give the following approximation theorem which plays a crucial role when

0

EM(Q) ∩ L∞(Q) and consider the convex set (cid:2)φ = {v ∈ LM(Q) : v ≥ φ a.e. in Q}. Then for every u ∈ (cid:2)φ ∩ L∞(Q) such that ∂u/∂t ∈

proving the existence result of solutions for parabolic inequalities. Theorem 2.2. Let φ ∈ W 1,x W 1,x 0 W −1,xLM(Q) + L1(Q), there exists v j ∈ (cid:2)φ ∩ D(Q) such that

v j −→ u

in W 1,xLM(Q),

(2.13)

−→

in W −1,xLM(Q) + L1(Q)

∂v j ∂t

∂u ∂t

for the modular convergence.

Proof. It is easily adapted from that given in [4, Theorem 3] and the approximation tech- (cid:2) niques of [9]. Remark 2.3. The result is still true for φ ∈ W 1,xEM(Q) ∩ L∞(Q), when Ω is more regular (see [9]).

In order to deal with the time derivative, we introduce a time mollification of a func-

tion v ∈ LM(Q). Thus, we define, for all μ > 0 and all (x,t) ∈ Q,

(cid:7) t

(2.14)

vμ(x,t) = μ

−∞

(cid:18)v(x,s)exp (cid:3) μ(s − t) (cid:4) ds,

where (cid:18)v(x,s) = v(x,s)χ(0,T)(s) is the zero extension of v. The following proposition is fun- damental in the sequel. Proposition 2.4 [5]. If v ∈ LM(Q), then vμ is measurable in Q, ∂vμ/∂t = μ(v − vμ) and

(cid:7) (cid:7) (cid:4) dx dt ≤

M

(cid:3) vμ

(2.15)

M(v)dx dt.

Q

Q

Recall now the following compactness result which is proved in [5].

LM(Q) such that ∂un/∂t

Proposition 2.5. Assume that (un)n is a bounded sequence in W 1 0 is bounded in W −1,xLM(Q) + L1(Q), then un is relatively compact in L1(Q).

0

3. The main result Let Ω be an open bounded subset of RN , N ≥ 2, with the segment property. Let P and M be two N-functions such that P (cid:6) M. Consider now the operator A : D(A) ⊂ W 1,x LM(Q) → W −1LM(Q) in divergence form A(u) = −div(a(x,t,u, ∇u)), where a : Ω × R × R × RN → RN is a Carath´eodory function satisfying for a.e. x ∈ Ω and for all ζ,ζ (cid:14) ∈ RN ,

6

Parabolic inequalities in L1

(ζ (cid:15)= ζ (cid:14)) and all s,t ∈ R:

(cid:4) (cid:4)

,

(cid:3) k4|ζ| (cid:3) k2|s| (cid:4)(cid:4)(cid:3)

ζ − ζ (cid:14)

(3.1)

+ k3M−1M (cid:4) > 0,

(cid:4) (cid:19) (cid:19) (cid:19) ≤ h(x,t) + k1P−1M (cid:19)a(x,t,s,ζ) (cid:3) (cid:3) x,t,s,ζ (cid:14) a(x,t,s,ζ) − a (cid:3) |ζ|

a(x,t,s,ζ)ζ ≥ αM

− d(x,t),

with d ∈ L1(Q), α,k1,k2,k3,k4 > 0, and h ∈ EM(Q). Let

EM(Ω) ∩ L∞(Ω).

(3.2)

ψ ∈ W 1 0

Finally, consider

f ∈ L1(Q).

(3.3)

We define for all t ∈ R, k ≥ 0, Tk(t) = max(−k,min(k,t)), and Sk(t) =

Tk(η)dη.

(cid:6) t 0

We will prove the following existence theorem.

Theorem 3.1. Let u0 ∈ L1(Ω) such that u0 ≥ 0. Assume that (3.1)–(3.3) hold true. Then there exists at least one solution u ∈ C([0,T];L1(Ω)) such that u(x,0) = u0 a.e. and for all τ ∈]0,T],

u ≥ ψ

a.e. in Q,

LM(Q), (cid:7)

(cid:7)

Tk(u) ∈ W 1,x 0 (cid:21)

(cid:20)

Sk

+

(cid:4) dx +

a(x,t,u, ∇u)∇Tk(u − v)dx dt

Ω

Qτ

∂v ∂t ,Tk(u − v) (cid:7)

(cid:3) u(τ) − v(τ) (cid:7) ≤

Sk

Qτ (cid:4) dx,

f Tk(u − v)dx dt +

Ω

Qτ

(cid:3) u0 − v(x,0)

∀k > 0 and ∀v ∈ (cid:2)ψ ∩ L∞(Q) such that ∈ W −1,xLM(Q) + L1(Q),

∂v ∂t

(pψ)

where Qτ = Ω×]0,τ[. Remark 3.2. Since {v ∈ (cid:2)ψ ∩ L∞(Q) : ∂v/∂t ∈ W −1,xLM(Q) + L1(Q)} ⊂ C([0,T],L1(Ω)), (see [4]), the first and the latest terms of problem (pψ) are well defined.

Proof Step 1. A priori estimates.

For the sake of simplicity, we assume that d(x,t) = 0. Consider the approximate equations

(cid:4)(cid:4) (cid:3) a − nTn − div (cid:3) x,t,un, ∇un (cid:3) un − ψ)− = fn,

∂un ∂t

(Pn)

LM(Q),

un ∈ W 1,x 0

un(x,0) = un 0,

R. Aboulaich et al.

7

where fn → f strongly in L1(Q) and un → u0 strongly in L1(Ω). Thanks to [3, Theo- 0 rem 3.1], there exists at least one solution un of problem (Pn). By choosing Tk(un − Th(un)),h ≥ (cid:11)ψ(cid:11)∞ as test function in (Pn), we get

(cid:7) (cid:20) (cid:21) (cid:4)(cid:4)

a

(cid:3) un (cid:4) ∇undx dt

+

(cid:3) un, ∇un

∂un ∂t ,Tk

(cid:7) (cid:3) un − Th (cid:7) (cid:4)(cid:4) (cid:4)(cid:4) −

dx dt =

dx dt.

(cid:3) un − ψ (cid:4)−Tk

nTn

h≤|un|≤h+k (cid:3) un − Th

(cid:3) un (cid:3) un − Th (cid:3) un

fnTk

Q

Q

(3.4)

On the one hand, we have

(cid:7) (cid:7) (cid:20) (cid:21) (cid:4)(cid:4) (cid:4) dx − = (cid:3) un − Th (cid:3) un (cid:3) u0 n (cid:3) un(T) (cid:4) dx,

(3.5)

Sh k

Sh k

∂un ∂t ,Tk

Ω

Ω

(cid:6)

Tk(q − Th(q))dq, and by using the fact that

k(un(T))dx ≥ 0 and |

Ω Sh k

n(cid:11)1, we get

(cid:6) Ω Sh (cid:6) s 0

where Sh k(s) = n)| ≤ k(cid:11)u0 (u0 (cid:7)

(cid:7) (cid:3) (cid:4)(cid:4) (cid:19) (cid:19)

α

(cid:4) dx dt −

M

(cid:3)(cid:19) (cid:19)∇un

nTn

(cid:3) un − Th (cid:3) un

un − ψ)−Tk

dx dt ≤ Ck, ∀n ∈ N,

Q

h≤|un|≤h+k

(3.6)

so that

(cid:4)(cid:4) (cid:7) (cid:4)− Tk (cid:3) un −

dx dt ≤ C.

(cid:3) un − ψ

nTn

(3.7)

Q

(cid:3) un − Th k

Since −nTn(un − ψ)−Tk(un − Th(un)) ≥ 0, for every h ≥ (cid:11)ψ(cid:11)∞, we deduce by Fatou’s lemma as k → 0 that

(cid:7) (cid:3) (cid:4)− ≤ C.

nTn

un − ψ

(3.8)

Q

Using in (Pn) the test function Tk(un)χ(0,τ), we get for every τ ∈ (0,T),

(cid:7) (cid:7) (cid:3) (cid:4)(cid:4) (cid:4)

a

dx dt

Sk

(cid:3) un (cid:3) un (cid:3) un ∇Tk (cid:4) dx +

x,t,Tk

Ω

Qτ

(cid:4) , ∇Tk (cid:3) un(τ) (cid:7)

(3.9)

(cid:3)(cid:3) (cid:4)−(cid:4) (cid:4) dx dt ≤ Ck

nTn

un − ψ

(cid:3) un

Tk

+

Qτ

which gives thanks to (3.8) (cid:7)

(cid:7) (cid:3) (cid:4)(cid:4)

a

Sk

un

(cid:3) un ∇Tk (cid:3) un (cid:3) un(τ) (cid:4) dx + (cid:3) x,t,Tk (cid:4) , ∇Tk (cid:4) dx dt ≤ Ck,

(3.10)

Ω

Qτ

(cid:7) (cid:4)(cid:19) (cid:19)

M

(cid:4) dx dt ≤ Ck. (cid:3)(cid:19) (cid:19)∇Tk (cid:3) un

(3.11)

Q

8

Parabolic inequalities in L1

On the other hand, by using [6, Lemma 5.7], there exist two positive constants μ1 and μ2 such that

(cid:5) (cid:2) (cid:4) (cid:7) (cid:7)

Tk

(cid:4)(cid:19) (cid:19)

M

(cid:4) dx dt

M

(cid:3)(cid:19) (cid:19)∇Tk (cid:3) un

(3.12)

dx dt ≤ μ2

Q

Q

(cid:3) un μ1

which implies, by using (3.11), that

(cid:16) (cid:4) (cid:19) (cid:19) > k ≤ (cid:15)(cid:19) (cid:19)un

(3.13)

meas

M

μ2Ck (cid:3) k/μ1

so that

(cid:16) (cid:19) (cid:19) > k (cid:15)(cid:19) (cid:19)un

meas

= 0

uniformly with respect to n.

(3.14)

lim k→∞

Take now a nondecreasing function θk ∈ C2(R) such that θk(s) = s for |s| ≤ k/2 and θk(s) = k sign(s) for |s| > k. By multiplying the approximate equation by θ(cid:14)

k(un), we get

(cid:4) (cid:4)(cid:4) (cid:4) (cid:3) un

∂θk

(cid:3) a

un

∇unθ(cid:14)(cid:14) (cid:3) (cid:4)

un

∂t

(3.15)

(cid:3)

θ(cid:14) (cid:3) (cid:4) (cid:4)

(cid:3) x,t,un, ∇un (cid:4)−θ(cid:14) − div (cid:3) un − ψ − nTn

un

k

0

(cid:3) + a x,t,un, ∇un (cid:4) (cid:3) un , = fnθ(cid:14) k

which implies that ∂θk(un)/∂t is bounded in W −1,xLM(Q) + L1(Q). Since θk(un) is bound- ed in W 1,x LM(Q), we have by Proposition 2.5 that θk(un) is relatively compact in L1(Q) and so that un → u a.e. in Q, and from (3.8) by using Fatou’s lemma, we get u ≥ ψ a.e. in Q. Consequently,

(cid:4)

Tk

−→ Tk(u) weakly in W 1,x

LM(Q)

(3.16)

0

(cid:3) un (cid:14) (cid:14)

EM).

LM,

for the topology σ( Step 2. Almost everywhere convergence of the gradients.

LM(Q), then there exists a sequence (αk

j ) ⊂ D(Q) such that αk

Since Tk(u) ∈ W 1,x

0

0

j → Tk(u) for the modular convergence in W 1,x LM(Q). In the sequel and throughout the pa- per, χ j,s and χs will denote, respectively, the characteristic functions of the sets Q j,s = {(x,t) ∈ Ω : |∇Tk(αk j )| ≤ s} and Qs = {(x,t) ∈ Ω : |∇Tk(u)| ≤ s}. For the sake of sim- plicity, we will write only (cid:2)(n, j,μ,s) to mean all quantities (possibly different) such that lims→∞ limμ→∞ lim j→∞ limn→∞ (cid:2)(n, j,μ,s) = 0.

j )μ), η > 0 as test function in (Pn), we get

Taking now Tη(un − Tk(αk

(cid:7) (cid:20) (cid:23)(cid:21) (cid:22) (cid:23)

a

(cid:4) ∇Tη

un − Tk

(cid:3) αk j (cid:3) αk j

+

(cid:3) x,un, ∇un (cid:4) μ (cid:4) μ

∂un ∂t ,Tη

Q

(cid:22) un − Tk (cid:7)

(3.17)

(cid:22) (cid:23) (cid:22)(cid:3) (cid:4)− −

nTn

un − ψ

(cid:23) Tη

un − Tk

(cid:3) αk j

dx dt ≤ Cη,

Q

(cid:4) μ

and by using (3.8), we get

(cid:7) (cid:20) (cid:22) (cid:23)(cid:21) (cid:22) (cid:23) (cid:3)

a

≤ Cη.

un − Tk

(cid:4) ∇Tη

un − Tk

(cid:3) αk j

αk j

+

(cid:3) un, ∇un

(3.18)

(cid:4) μ (cid:4) μ

∂un ∂t ,Tη

Q

R. Aboulaich et al.

9

The first term of the left-hand side of the last equality reads as

(cid:25) (cid:24) (cid:20) (cid:22) (cid:23)(cid:21) (cid:23)

∂Tk

(cid:4) μ − =

un − Tk

(cid:22) un − Tk (cid:3) αk j (cid:3) αk j

,Tη

(cid:4) μ (cid:4) μ

∂un ∂t ,Tη

(cid:3) αk j ∂t

(3.19)

∂un ∂t (cid:24)

(cid:25) (cid:22) (cid:23)

∂Tk

(cid:4) μ

.

un − Tk

(cid:3) αk j

+

,Tη

(cid:4) μ (cid:3) αk j ∂t

The second term of the last equality can be written as

(cid:24) (cid:25) (cid:23) (cid:22)

∂Tk

(cid:4) μ −

un − Tk

(cid:3) αk j

,Tη

(cid:4) μ

∂un ∂t

(cid:7) (cid:7) (cid:7) (cid:23)

dx −

(cid:4) dx ≥ −η =

Sη

Sη

Ω

Ω

Ω

(cid:3) αk j (cid:3) αk j ∂t (cid:22) un(T) − Tk (cid:19) (cid:19)dx ≥ −ηC, (cid:4) μ(T) (cid:3) un 0 (cid:19) (cid:19)un 0

(3.20)

the third term can be written as

(cid:4) (cid:24) (cid:25) (cid:7) (cid:23) (cid:22) (cid:23)(cid:22) (cid:22) (cid:23)(cid:23) (cid:4)

∂Tk

μ

= μ (cid:22) un − Tk

Tk

− Tk

Tη

un − Tk

(cid:3) αk j (cid:3) αk j (cid:3) αk j (cid:3) αk j

,

,Tη

Q

(cid:4) μ (cid:4) μ (cid:4) μ (cid:3) αk j ∂t

(3.21)

thus by letting n, j → ∞ and since αk

j → Tk(u) a.e. in Q and by using Lebesgue theorem,

(cid:7) (cid:22) (cid:23)(cid:22) (cid:22) (cid:23)(cid:23) (cid:3) (cid:4)

dx dt

Tk

− Tk

Tη

un − Tk

αk j

Q

(cid:3) αk j (cid:3) αk j (cid:4) μ (cid:4) μ (cid:7)

(3.22)

(cid:4)(cid:3) (cid:4)(cid:4) =

Tη

(cid:3) Tk(u) − Tk(u)μ (cid:3) u − Tk(u)μ

dx dt + (cid:2)(n, j).

Q

Consequently,

(cid:20) (cid:22) (cid:22) (cid:23)(cid:23)(cid:21)

Tk

un − Tk

(cid:3) αk j ≥ (cid:2)(n, j) − ηC.

(3.23)

(cid:4) μ

∂un ∂t ,Tη

On the other hand,

(cid:7) (cid:22) (cid:23)

a

dx dt

(cid:4) ∇Tη

un − Tk

Q

(cid:3) αk j (cid:4) μ (cid:3) un, ∇un (cid:7) (cid:4)(cid:4) (cid:4) (cid:4) =

a

μχ j,sdx dt

j )μ|<η}

{|Tk(un)−Tk(αk (cid:7)

(cid:3) Tk (cid:3) un (cid:3) un (cid:3) un ∇Tk − ∇Tk (cid:3) αk j (cid:4) , ∇Tk

(3.24)

a

(cid:4) ∇undx dt (cid:3) un, ∇un

+

j )μ|<η}

{k<|un|}∩{|un−Tk(αk (cid:7)

(cid:3) −

a

(cid:4) ∇Tk (cid:3) αk j

un, ∇un

j )|>s}dx dt

{k<|un|}∩{|un−Tk(αk

j )μ|<η}

(cid:4) μχ{|∇Tk(αk

10

Parabolic inequalities in L1

which implies, by using the fact that

j )μ|<η} a(un, ∇un)∇undx dt ≥ 0, that

(cid:6) {k<|un|}∩{|un−Tk(αk

(cid:7) (cid:4)

a

{|Tk(un)−Tk(αk

j )μ|<η} (cid:7)

(cid:4) ∇Tk (cid:3) un − ∇Tk (cid:3) αk j (cid:3) un, ∇un (cid:4) μχ j,sdx dt

(3.25)

a

j )|>s}dx dt.

{k<|un|}∩{|un−Tk(αk

j )μ|<η}

(cid:4) ∇Tk (cid:3) αk j ≤ Cη + (cid:3) un, ∇un (cid:4) μχ{|∇Tk(αk

Since a(Tk+η(un), ∇Tk+η(un)) is bounded in (LM(Q))N , there exists some hk+η ∈(LM(Q))N such that

(cid:26) (cid:22)(cid:26) (cid:4)(cid:4) (cid:4)N (cid:23) .

a

(cid:3) un (cid:3) un

EM

(cid:3) LM(Q)

LM,

for σ

(cid:3) Tk+η (cid:4) , ∇Tk+η

hk+η weakly in

(3.26)

Consequently,

(cid:7)

a

j )|>s}dx dt

j )μ|<η}

{k<|un|}∩{|un−Tk(αk (cid:7)

(cid:4) ∇Tk (cid:3) αk j (cid:3) un, ∇un (cid:4) μχ{|∇Tk(αk

(3.27)

= (cid:3) αk j

hk+η∇Tk

j )|>s}dx dt + (cid:2)(n),

{k<|u|}∩{|u−Tk(αk

j )μ|<η}

j )μχ{k<|un|}∩{|un−Tk(αk

j )μ|<η} tends strongly to

(cid:4) μχ{|∇Tk(αk

where we have used the fact that ∇Tk(αk ∇Tk(αk

j )μχ{k<|u|}∩{|u−Tk(αk

j )μ|<η} in (EM(Q))N . Letting j → ∞, we obtain

(cid:7)

a

j )|>s}dx dt

j )μ|<η}

{k<|un|}∩{|un−Tk(αk (cid:7)

(cid:4) ∇Tk (cid:3) αk j (cid:3) un, ∇un (cid:4) μχ{|∇Tk(αk

(3.28)

=

hk+η∇Tk(u)μχ{|∇Tk(u)|>s}dx dt + (cid:2)(n, j).

{k<|u|}∩{|u−Tk(u)μ|<η}

Thanks to Proposition 2.4, one easily has

(cid:7)

hk+η∇Tk(u)μχ{|∇Tk(u)|>s}dx dt

{k<|u|}∩{|u−Tk(u)μ|<η}

(cid:7)

(3.29)

=

hk+η∇Tk(u)χ{|∇Tk(u)|>s}dx dt + (cid:2)(μ) = (cid:2)(μ,s).

{k<|u|}∩{|u−Tk(u)|<η}

Hence

(cid:7) (cid:4)(cid:4) (cid:4) (cid:3)

a

(cid:3) Tk (cid:3) un (cid:3) un (cid:3) un ∇Tk − ∇Tk

αk j

{|Tk(un)−Tk(αk

j )μ|<η}

(cid:4) , ∇Tk (cid:4) μχ j,sdx dt ≤ Cη + (cid:2)(n, j,μ,s).

(3.30)

R. Aboulaich et al.

11

On the other hand, note that

(cid:7) (cid:4)(cid:4) (cid:4)

a

{|Tk(un)−Tk(αk

j )μ|<η}

(cid:3) Tk (cid:3) un (cid:3) un (cid:3) un ∇Tk − ∇Tk (cid:3) αk j (cid:4) , ∇Tk (cid:4) μχ j,sdx dt (cid:7) (cid:4)(cid:4) (cid:3) (cid:4) =

a

(cid:3) Tk (cid:3) un (cid:3) un ∇Tk

un

j )μ|<η}

{|Tk(un)−Tk(αk (cid:7)

− ∇Tk (cid:3) αk j (cid:4) , ∇Tk (cid:4) χ j,sdx dt

(cid:3) (cid:4)(cid:4)(cid:27)

a

(cid:28) dx dt. (cid:3) Tk (cid:3) un

un

∇Tk (cid:3) αk j (cid:3) αk j (cid:4) , ∇Tk

+

{|Tk(un)−Tk(αk

j )μ|<η}

(cid:4) χ j,s − ∇Tk (cid:4) μχ j,s

(3.31)

The latest integral tends to 0 as n and j go to ∞. Indeed, we have that

(cid:7) (cid:28) (cid:4)(cid:4)(cid:27)

a

dx dt

(cid:3) Tk (cid:3) un (cid:3) un ∇Tk (cid:3) αk j (cid:3) αk j (cid:4) , ∇Tk

(3.32)

{|Tk(un)−Tk(αk

j )μ|<η}

(cid:4) χ j,s − ∇Tk (cid:4) μχ j,s

tends to

(cid:7) (cid:28) dx dt (cid:27) ∇Tk

hk

(cid:3) αk j (cid:3) αk j

(3.33)

{|Tk(u)−Tk(αk

j )μ|<η}

(cid:4) χ j,s − ∇Tk (cid:4) μχ j,s

(cid:23) (cid:26) (cid:22)(cid:26) (cid:4)(cid:4) (cid:4)N

a

as n → ∞, since (cid:3) un

(cid:3) Tk (cid:3) un

EM

(cid:4) , ∇Tk

hk weakly in

(cid:3) LM(Q)

LM,

for σ

(3.34)

while ∇Tk(αk

j )χ j,s − ∇Tk(αk

j )μχ j,s ∈ (EM(Q))N . It is obvious that

(cid:7) (cid:28) dx dt (cid:27) ∇Tk

hk

(cid:3) αk j (cid:3) αk j

(3.35)

{Tk(u)−Tk(αk

j )μ|<η}

(cid:4) χ j,s − ∇Tk (cid:4) μχ j,s

goes to 0 as j → ∞ by using Lebesgue theorem. We deduce then that (cid:7)

(cid:4)(cid:4) (cid:4)

a

{|Tk(un)−Tk(αk

j )μ|<η}

(cid:3) Tk (cid:3) un (cid:3) un (cid:3) un ∇Tk − ∇Tk (cid:3) αk j (cid:4) , ∇Tk (cid:4) χ j,sdx dt ≤ Cη + (cid:2)(n, j,μ,s).

(3.36)

(cid:7) (cid:28)δ (cid:4)(cid:4) (cid:4)(cid:28)(cid:27) (cid:4) (cid:27) a

dx dt

− a

Let now 0 < δ < 1. We have (cid:3) un

Qr

(cid:3) Tk (cid:3) un (cid:3) Tk (cid:3) un (cid:3) un ∇Tk (cid:4) , ∇Tk (cid:4) , ∇Tk(u) − ∇Tk(u)

(cid:30)δ (cid:4) (cid:19) (cid:19) (cid:19) > η (cid:29)(cid:19) (cid:19) (cid:19)Tk (cid:3) un − Tk (cid:3) αk j (cid:4) μ ≤ Cmeas (cid:31) (cid:7) (cid:4)(cid:4) (cid:4)! a − a (cid:3) Tk (cid:3) un (cid:3) un (cid:3) Tk (cid:3) un

+ C

{|Tk(un)−Tk(αk

j )μ|<η}∩Qr

(cid:4) , ∇Tk (cid:4) , ∇Tk(u)

"δ (cid:4) × ! dx dt

.

∇Tk (cid:3) un − ∇Tk(u)

(3.37)

12

Parabolic inequalities in L1

On the other hand, we have for every s ≥ r, r > 0,

{|Tk(un)−Tk(αk

j )μ|<η∩Qr }

(cid:7) (cid:4)(cid:4) (cid:4)! (cid:4) a ! dx dt −a (cid:3) (cid:3) un Tk (cid:3) un (cid:3) Tk (cid:3) un ∇Tk (cid:3) un (cid:4) , ∇Tk (cid:4) , ∇Tk(u) −∇Tk(u)

(cid:7) (cid:4)(cid:4) (cid:4) (cid:4)! ≤ a − a (cid:3) Tk (cid:3) un (cid:3) un (cid:3) Tk (cid:3) un (cid:4) , ∇Tk

, ∇Tk(u)χs

{|Tk(un)−Tk(αk

j )μ|<η}

(cid:4) × ! dx dt ∇Tk (cid:3) un − ∇Tk(u)χs (cid:7) (cid:4)(cid:4) (cid:4) (cid:4)! ≤ a − a (cid:3) Tk (cid:3) un (cid:3) un (cid:3) Tk (cid:3) un (cid:4) , ∇Tk

, ∇Tk(αk

j )χ j,s

{|Tk(un)−Tk(αk

j )μ|<η}

(cid:4) ! ×

dx dt

∇Tk (cid:3) un − ∇Tk (cid:3) αk j (cid:4) χ j,s (cid:7) (cid:3) (cid:4)(cid:4) !

a

dx dt

(cid:3) Tk (cid:3) un

un

∇Tk (cid:3) αk j (cid:4) , ∇Tk

+

{|Tk(un)−Tk(αk

j )μ|<η}

(cid:4) χ j,s − ∇Tk(u)χs (cid:7) (cid:3) (cid:4) (cid:4)!

a

(cid:4) dx dt −a (cid:3) Tk

un

(cid:3) Tk (cid:3) un ∇Tk (cid:3) un (cid:3) αk j (cid:4) , ∇Tk (cid:4) , ∇Tk(u)χs

+

j )μ|<η}

{|Tk(un)−Tk(αk (cid:7)

(cid:4) χ j,s

−

a

{|Tk(un)−Tk(αk

j )μ|<η}

(cid:3) Tk (cid:3) un (cid:4) ∇Tk (cid:3) αk j (cid:3) αk j (cid:4) , ∇Tk (cid:4) χ j,s (cid:4) χ j,sdx dt (cid:7)

a

(cid:3) Tk (cid:3) un (cid:4) , ∇Tk(u)χs (cid:4) ∇Tk(u)χsdx dt

+

{|Tk(un)−Tk(αk

j )μ|<η}

≤ I1(n, j,μ,s) + I2(n, j,μ,s) + I3(n, j,μ,s) + I4(n, j,μ,s) + I5(n, j,μ,s).

(3.38)

We will go to the limit as n, j, μ, and s → ∞ in the last fifth integrals of the last side. Starting with I1, we have

I1(n, j,μ,s) ≤ Cη + (cid:2)(n, j,μ,s)

(cid:7) (cid:3) (cid:4) (cid:4) −

a

(cid:3) Tk

un

{|Tk(un)−Tk(αk

j )μ|<η}

(cid:3) un ∇Tk − ∇Tk (cid:3) αk j (cid:3) αk j (cid:4) , ∇Tk (cid:4) χ j,s (cid:4) χ j,sdx dt

(3.39)

since

a

j )μ|<η}

(cid:3) Tk (cid:3) un (cid:3) αk j (cid:4) χ j,s

(3.40)

(cid:4)N −→ a (cid:4) , ∇Tk (cid:3) Tk(u), ∇Tk (cid:3) EM(Q)

,

in

j )μ|<η}

(cid:4) χ j,s (cid:4) χ{|Tk(u)−Tk(αk (cid:4) (cid:3) αk χ{|Tk(u)−Tk(αk j

while

(cid:4) (cid:3) (cid:4)N . (cid:3) un ∇Tk ∇Tk(u) weakly in

LM(Ω)

(3.41)

R. Aboulaich et al.

13

We deduce then that

(cid:7) (cid:3) (cid:4)

a

(cid:3) Tk (cid:3) un (cid:4) ∇Tk (cid:3) un − ∇Tk

αk j

{|Tk(un)−Tk(αk

j )μ|<η}

(cid:3) αk j (cid:4) , ∇Tk (cid:4) χ j,s (cid:4) χ j,sdx dt (cid:7) (cid:4) =

a

(cid:3) αk j (cid:3) αk j (cid:3) Tk(u), ∇Tk (cid:4) ∇Tk(u) − ∇Tk (cid:4) χ j,sdx dt + (cid:2)(n)

χ j,s

{|Tk(u)−Tk(αk

j )μ|<η}

(3.42)

which gives by letting j → ∞ and using the modular convergence of ∇Tk(αk

j ), that

(cid:7) (cid:4)

a

(cid:3) αk j (cid:3) αk j (cid:3) Tk(u), ∇Tk (cid:4) ∇Tk(u) − ∇Tk (cid:4) χ j,sdx dt

χ j,s

j )μ|<η}

(cid:7)

(3.43)

=

a

{|Tk(u)−Tk(αk (cid:3) Tk(u), ∇Tk(u)χs

Q

(cid:4) ∇Tk(u) − ∇Tk(u)χsdx dt + (cid:2)( j) = (cid:2)( j).

Finally,

(3.44)

I1(n, j,μ,s) ≤ Cη + (cid:2)(n, j,μ,s) + (cid:2)(n, j) = (cid:2)(n, j,μ,s,η).

For what concerns I2, by letting n → ∞, we have

(cid:7) !

hk

∇Tk (cid:3) αk j

dx dt + (cid:2)(n)

(3.45)

I2(n, j,μ,s) =

{|Tk(u)−Tk(αk

j )μ|<η}

(cid:4) χ j,s − ∇Tk(u)χs

(cid:26) (cid:22)(cid:26) (cid:4)(cid:4) (cid:4)N

since (cid:3) (cid:3) un Tk a

(cid:3) (cid:4) un , ∇Tk

hk weakly in

(cid:3) LM(Q)

for σ

LM,

χ{|Tk(un)−Tk(αk

j )μ|<η}

(cid:23) EM , (3.46)

while

! ∇Tk (cid:3) αk j (cid:3) αk j (cid:4) χ j,s −∇Tk(u)χs

χ{|Tk(un)−Tk(αk

j )μ|<η}

j )μ|<η}∇Tk

−→χ{|Tk(u)−Tk(αk (cid:4) χ j,s −∇Tk(u)χs (3.47)

strongly in (EM(Q))N . By letting now j → ∞, and using Lebesgue theorem, we deduce then that

(3.48)

I2(n, j,μ,s) = (cid:2)(n, j).

Similar tools as above give

(cid:7)

a

I3(n, j,μ,s) = (cid:2)(n, j), (cid:3) Tk(u), ∇Tk(u)

(cid:4) ∇Tk(u) + (cid:2)(n, j,μ,s),

I4(n, j,μ,s) =

Q

(3.49)

(cid:7)

a

(cid:3) Tk(u), ∇Tk(u) (cid:4) ∇Tk(u) + (cid:2)(n, j,μ,s).

I5(n, j,μ,s) =

Q

14

Parabolic inequalities in L1

Combining (3.37)–(3.48) and (3.49), we get

Qr

(cid:7) (cid:4)(cid:4) (cid:4)! (cid:4) a !δdx dt − a (cid:3) Tk (cid:3) un (cid:3) un (cid:3) Tk (cid:3) un (cid:3) un ∇Tk (cid:4) , ∇Tk (cid:4) , ∇Tk(u) − ∇Tk(u)

(3.50)

(cid:30)δ (cid:4) (cid:3) (cid:4)1−δ (cid:19) (cid:19) (cid:19) < η (cid:29)(cid:19) (cid:19) (cid:19)Tk (cid:3) un − Tk (cid:3) αk j ≤ Cmeas

+ C

(cid:2)(n, j,s,μ,η)

,

(cid:4) μ

and by passing to the limit sup over n, j, μ, s, and, η

(cid:7) (cid:4)(cid:4) (cid:4)! (cid:4)

a

− a (cid:3) Tk (cid:3) un (cid:3) un (cid:3) Tk (cid:3) un (cid:3) un ∇Tk (cid:4) , ∇Tk (cid:4) , ∇Tk(u) − ∇Tk(u)

limn→∞

Qr

!δdx dt = 0, (3.51)

and thus there exists a subsequence also denoted by (un) such that

∇un −→ ∇u

a.e. in Qr,

(3.52)

and since r is arbitrary, we obtain

∇un −→ ∇u

a.e. in Q.

(3.53)

Step 3. Passage to the limit.

Let φ ∈ (cid:2)ψ ∩ D(Q). Choosing now Tk(un − φ)χ(0,τ) as test function in (Pn), we get

(cid:7) (cid:20) (cid:21) (cid:4) (cid:3)

a

(cid:4) dx dt (cid:3) un − φ (cid:4) ∇Tk (cid:3) un − φ

+

x,t,un, ∇un

Qτ

Qτ

(cid:7)

∂un ∂t ,Tk (cid:7)

(3.54)

− (cid:4) dx dt = (cid:4) dx dt (cid:3) un − ψ (cid:4)−Tk (cid:3) un − φ

nTn

(cid:3) un − φ

fnTk

Qτ

nTn(un − ψ)−Tk(un − φ)dx dt ≥ 0,

Qτ (cid:6) Qτ

which gives, by − (cid:7)

(cid:20) (cid:21) (cid:4)

Sk

Ω

Qτ

(cid:3) un − φ (cid:3) un(τ) − φ(τ) (cid:4) dx + (cid:7)

a

(cid:4) dx dt

(3.55)

Qτ

(cid:3) x,t,un, ∇un

∂φ ∂t ,Tk (cid:3) (cid:4) un − φ ∇Tk (cid:7)

+ (cid:7)

≤ (cid:4) dx.

fnTk

(cid:3) un − φ

Sk

Ω

Qτ

(cid:4) dx dt + (cid:3) un(0) − φ(0)

We will show that

(cid:3) (cid:4) .

un −→ u

in C

[0,T],L1(Ω)

(3.56)

Since Tk(u) ∈ (cid:2)ψ, for every k ≥ (cid:11)ψ(cid:11)∞, there exists a sequence (w j) in D(Q) ∩ (cid:2)φ such that

w j −→ Tk(u)

in W 1,x

LM(Q)

(3.57)

0

R. Aboulaich et al.

15

j,μ = Tl(w j)μ + e−μtTl(ηi), with ηi ≥ 0 con-

for the modular convergence. Choosing now Φi,l verges to u0 in L1(Ω), as test function in (3.55),

(cid:7) (cid:20) (cid:21) (cid:4)

a

(cid:4) dx dt (cid:4) ∇Tk

+

Qτ

Qτ

(cid:3) x,t,un, ∇un (cid:3) un − Φi,l j,μ (cid:3) un − Φi,l j,μ (cid:7)

∂un ∂t ,Tk (cid:7)

(3.58)

(cid:3) − (cid:4) dx dt = (cid:4) dx dt.

nTn

fnTk

(cid:3) un − ψ)−Tk (cid:3) un − Φi,l j,μ

un − Φi,l j,μ

Qτ

Qτ

On the one hand, we have

(cid:7) (cid:20) (cid:4)(cid:14) (cid:21) (cid:4) (cid:4) = μ (cid:3) Tl (cid:3) w j (cid:4) Tk

,Tk

Qτ

Qτ

(cid:4) dx dt ≥ (cid:2)(n, j,μ,l); (cid:3) Φi,l j,μ (cid:3) un − Φi,l j,μ − Φi,l j,μ (cid:3) un − Φi,l j,μ

(3.59)

nTn(un −

(cid:6) Qτ

(cid:7)

on the other hand, by using the monotonicity of a and the fact that − ψ)−Tk(un − Φi,l (cid:20)

j,μ)dx dt ≥ 0, we deduce that (cid:21) (cid:4)

(cid:3)

a

(cid:4) dx dt (cid:4) ∇Tk

+

(cid:3) x,t,un, ∇Φi,l j,μ (cid:3) un − Φi,l j,μ

un − Φi,l j,μ

Qτ

Qτ

∂un ∂t ,Tk (cid:7)

(3.60)

≤ (cid:4) dx dt.

fnTk

Qτ

(cid:3) un − Φi,l j,μ

Since, for every (cid:2) > 0,

(cid:4)(cid:19) (cid:19) (cid:4) ∇Tk (cid:19) (cid:19)χQτ a (cid:3) x,t,un, ∇Φi,l j,μ (cid:3) un − Φi,l j,μ (cid:5) (cid:2) (cid:19) (cid:4)(cid:19) (cid:19)

(3.61)

(cid:4)(cid:4) (cid:19)∇Tk (cid:3) a ≤ (cid:2)M (cid:3) un

+ M

,

(cid:3) x,t,Tk+(cid:11)l(cid:11)∞ (cid:4) , ∇Φi,l j,μ (cid:3) un − Φi,l j,μ (cid:2)

we have by using Vitali’s theorem

(cid:20) (cid:21) (cid:4) ≤ 0

(3.62)

(cid:3) un − Φi,l j,μ

∂un ∂t ,Tk

limsup μ→∞

limsup n→∞

limsup i→∞

limsup j→∞

limsup l→∞

Qτ

uniformly on τ. Therefore, by writing

(cid:7) (cid:20) (cid:20) (cid:3) (cid:21) (cid:4) (cid:4)(cid:14) (cid:21) (cid:4) (cid:4) dx = −

Sk

,Tk

(cid:3) un(τ) − Φi,l j,μ (cid:3) un − Φi,l j,μ

Φi,l j,μ

Ω

Qτ

Qτ

(cid:3) un − Φi,l j,μ

(cid:4)(cid:4)

dx

Sk

(cid:3) ηi

+

∂un ∂t ,Tk (cid:7) (cid:3) u0 − Tl

Ω

(3.63)

and using (3.55) and (3.59), we see that

(cid:7)

Sk

(cid:4) dx ≤ (cid:2)(n, j,μ,i,l)

(3.64)

Ω

(cid:3) un(τ) − Φi,l j,μ

16

Parabolic inequalities in L1

which implies, by writing

(cid:7) (cid:7) (cid:8) (cid:8) (cid:7) (cid:9) (cid:4) dx

Sk

Sk

Sk

(cid:4) dx +

,

Ω

Ω

Ω

(cid:3) un(τ) − Φi,l j,μ (cid:3) um(τ) − Φi,l j,μ

un(τ) − um(τ) 2

(cid:9) dx ≤ 1 2

(3.65)

that

(cid:7) (cid:8)

Sk

(3.66)

Ω

(cid:9) dx ≤ (cid:2)1(n,m),

un(τ) − um(τ) 2

we deduce then that

(cid:7)

(cid:19) (cid:19)un(τ) − um(τ)

not depending on τ,

(3.67)

Ω

(cid:19) (cid:19)dx ≤ (cid:2)2(n,m),

and thus (un) is a Cauchy sequence in C([0,T],L1(Ω)), and since un → u, a.e. in Q, we deduce that

(cid:3) (cid:4) .

un −→ u

in C

[0,T],L1(Ω)

(3.68)

Go back now to (3.48) and pass to the limit to obtain

(cid:7) (cid:7) (cid:20) (cid:21) (cid:4)

a

Sk

(cid:3) u(τ) − φ(τ) (cid:4) dx +

+

(cid:3) x,t,u, ∇u ∇Tk(u − φ)dx dt

∂φ ∂t ,Tk(u − φ)

Ω

Qτ

Qτ

(cid:7) (cid:7) ≤ (cid:4) dx

Sk

f Tk(u − φ)dx dt +

Ω

Qτ

(cid:3) u(0) − φ(0)

(3.69)

since for every v ∈ (cid:2)ψ ∩ L∞(Q), there exists v j ∈ (cid:2)ψ ∩ D(Q) such that

v j −→ v

for the modular convergence in W 1,x

LM(Q),

0

(3.70)

−→

for the modular in W −1,xLM(Q) + L1(Q),

∂v j ∂t

∂v ∂t

we deduce then that

(cid:7) (cid:7) (cid:20) (cid:21) (cid:4)

a

Sk

(cid:3) u(τ) − v(τ) (cid:4) dx +

+

Ω

Qτ

Qτ

(cid:3) x,t,u, ∇u ∇Tk(u − v)dx dt

∂v ∂t ,Tk(u − v) (cid:7)

(cid:7) (cid:4) dx ≤

Sk

(cid:3) u(0) − v(0)

f Tk(u − v)dx dt +

Ω

Qτ

(3.71)

(cid:2)

which completes the proof.

R. Aboulaich et al.

17

Remark 3.3. A similar result can be proved when dealing with the right-hand side in L1(Q) + W −1,xEM(Q) or replacing the assumption (3.1) by the general one:

(cid:4)(cid:3) (cid:3) (cid:19) (cid:19) ≤ b |s| (cid:19) (cid:19)a(x,t,s,ζ)

h(x,t) + M−1M

(cid:4)(cid:4) ,

(3.72)

(cid:3) k4|ζ|

where b : R+ → R+ is an increasing continuous function. Indeed, we consider the follow- ing approximate problems:

(cid:4)(cid:4) (cid:3) (cid:3) a (cid:3) un − nTn

un − ψ

− div (cid:3) x,t,Tn (cid:4) , ∇un (cid:4)− = fn,

∂un ∂t

(Pn)

LM(Q),

un ∈ W 1,x 0

un(x,0) = un 0,

and we conclude by adapting the same steps.

Acknowledgments

The authors would like to thank the anonymous referees for interesting remarks. This work was supported in part by the Volkswagen Foundation, Grant number I/79315 and in the other part by the Moroccan-Tunisian Project 04/TM/19.

References

[1] R. A. Adams, Sobolev Spaces, Pure and Applied Mathematics, vol. 65, Academic Press, New York,

1975.

[2] T. Donaldson, Inhomogeneous Orlicz-Sobolev spaces and nonlinear parabolic initial value prob-

lems, Journal of Differential Equations 16 (1974), no. 2, 201–256.

[3] A. Elmahi and D. Meskine, Parabolic equations in Orlicz spaces, Journal of the London Mathe-

matical Society. Second Series 72 (2005), no. 2, 410–428.

, Strongly nonlinear parabolic equations with natural growth terms and L1 data in Orlicz

[4]

spaces, Portugaliae Mathematica. Nova S´erie 62 (2005), no. 2, 143–183.

, Strongly nonlinear parabolic equations with natural growth terms in Orlicz spaces, Non-

[5]

linear Analysis. Theory, Methods & Applications 60 (2005), no. 1, 1–35.

[6] J.-P. Gossez, Nonlinear elliptic boundary value problems for equations with rapidly (or slowly) increasing coefficients, Transactions of the American Mathematical Society 190 (1974), 163–205. , Some approximation properties in Orlicz-Sobolev spaces, Studia Mathematica 74 (1982),

[7]

no. 1, 17–24.

[8]

, A strongly nonlinear elliptic problem in Orlicz-Sobolev spaces, Nonlinear Functional Analysis and Its Applications, Part 1 (Berkeley, Calif, 1983), Proc. Sympos. Pure Math., vol. 45, American Mathematical Society, Rhode Island, 1986, pp. 455–462.

[9] J.-P. Gossez and V. Mustonen, Variational inequalities in Orlicz-Sobolev spaces, Nonlinear Anal-

ysis. Theory, Methods & Applications 11 (1987), no. 3, 379–392.

[10] V. K. Le and K. Schmitt, Quasilinear elliptic equations and inequalities with rapidly growing coef-

ficients, Journal of the London Mathematical Society. Second Series 62 (2000), no. 3, 852–872.

[11] J. Robert,

Journal de

In´equations variationnelles paraboliques

fortement non lin´eaires,

Math´ematiques Pures et Appliqu´ees. Neuvi`eme S´erie 53 (1974), 299–320.

[12] M. Rudd, Nonlinear constrained evolution in Banach spaces, Ph.D. thesis, University of Utah,

Utah, 2003.

18

Parabolic inequalities in L1

[13]

, Weak and strong solvability of parabolic variational inequalities in Banach spaces, Journal

of Evolution Equations 4 (2004), no. 4, 497–517.

[14] M. Rudd and K. Schmitt, Variational inequalities of elliptic and parabolic type, Taiwanese Journal

of Mathematics 6 (2002), no. 3, 287–322.

R. Aboulaich: LERMA, ´Ecole Mohammadia d’Ing´enieurs, Universit´e Mahammed V-Agdal, Avenue Ibn Sina, BP 765, Rabat-Agdal, Morocco E-mail address: aboul@emi.ac.ma

B. Achchab: LERMA, ´Ecole Mohammadia d’Ing´enieurs, Universit´e Mahammed V-Agdal, Avenue Ibn Sina, BP 765, Rabat-Agdal, Morocco Current address: Facult´e des Sciences Juridiques, ´Economiques et Sociales, Universit´e Hassan 1er, BP 784, Settat, Morocco E-mail address: achchab@yahoo.fr

D. Meskine: LERMA, ´Ecole Mohammadia d’Ing´enieurs, Universit´e Mahammed V-Agdal, Avenue Ibn Sina, BP 765, Rabat-Agdal, Morocco Current address: GAN, D´epartement de Math´ematiques et d’Informatiques, Facult´e des Sciences, Universit´e Mahammed V-Agdal, Avenue Ibn Battouta, BP 1014, Rabat, Morocco E-mail address: driss.meskine@laposte.net

A. Souissi: LERMA, ´Ecole Mohammadia d’Ing´enieurs, Universit´e Mahammed V-Agdal, Avenue Ibn Sina, BP 765, Rabat-Agdal, Morocco Current address: GAN, D´epartement de Math´ematiques et d’Informatiques, Facult´e des Sciences, Universit´e Mahammed V-Agdal, Avenue Ibn Battouta, BP 1014, Rabat, Morocco E-mail address: souissi@fsr.ac.ma