Nonlinear parabolic inequalities on a general convex

Nonlinear parabolic inequalities on a general convex domain B. Achchab, A. Agouzal, Na¨ıma Debit, M. Kbiri Alaoui, A. Soussi

To cite this version: B. Achchab, A. Agouzal, Na¨ıma Debit, M. Kbiri Alaoui, A. Soussi. Nonlinear parabolic inequalities on a general convex domain. Journal of Mathematical Inequalities, 2010, 4 (2), pp.271-284.

HAL Id: hal-00360256 https://hal.archives-ouvertes.fr/hal-00360256 Submitted on 10 Feb 2009

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Nonlinear parabolic inequalities on a general convex B. Achchab1 , A. Agouzal2 , N. Debit3 , M. Kbiri Alaoui4 , and A. Souissi5 1

LM2CE,Universit´e Hassan1, Facult´e des sciences ´economiques, juridiques et sociales Settat, Maroc. [email protected] 2 Universit´e de Lyon ; Universit´e Lyon 1 ; CNRS, UMR5208, Institut Camille Jordan, 43 blvd du 11 novembre 1918, F-69622 Villeurbanne-Cedex, France. [email protected] 3 Universit´e de Lyon ; Universit´e Lyon 1 ; CNRS, UMR5208, Bt ISTIL, 15 Bd Latarjet, F-69622 Villeurbanne-Cedex, France. [email protected] 4 LERMA, Ecole Mohammadia d’Ing´enieurs, avenue Ibn Sina B.P 765, Agdal, Rabat, Maroc and College of Sciences, Department of Mathematics, King Khalid University, PO.Box 9004, Abha KSA, mka [email protected] 5 GAN, D´epartement de Math´ematiques et d’Informatique Facult´e des Sciences, avenue Ibn Battouta, B.P 1014 Rabat, Maroc. [email protected]

Abstract The paper deals with the existence and uniqueness of solutions of some non linear parabolic inequalities in the Orlicz-Sobolev spaces framework.

1

Introduction

We consider boundary value problems of type  u∈K     ∂u + A(u) = f ∂t   u=0   u(x, 0) = u0 (x)

in Q,

(P )

on ∂Q, in Ω.

where

A(u) = −div(a(., t, ∇u)), Q = Ω × [0, T ], T > 0 and Ω is a bounded domain of RN , with the segment property. a : Ω × R × RN → RN is a Carath´eodory function (measurable with respect to x in Ω for every (t, ξ) in R × R × RN , and continuous with respect to ξ in R × RN for almost every x in Ω) such that for all ξ, ξ ∗ ∈ RN , ξ 6= ξ ∗ , a(x, t, ξ)ξ ≥ αM (|ξ|)

(1.1)

[a(x, t, ξ) − a(x, t, ξ ∗ )][ξ − ξ ∗ ] > 0,

(1.2)

|a(x, t, ξ)| ≤ c(x, t) + k1 M 1

−1

M (k2 |ξ|),

(1.3)

2

Nonlinear parabolic inequalities on a general convex

where c(x, t) ∈ EM (Q), c ≥ 0, ki ∈ R+ , for i = 1, 2 and α ∈ R+ ∗. f ∈ W −1,x EM (Q), f ≥ 0,

(1.4)

2

u0 ∈ L (Ω) ∩ K, u0 ≥ 0, where K is a given closed convex set of

W01 LM (Ω)

(1.5)

and K is defined by:

K := {v ∈ W01,x LM (Q) : v(t) ∈ K}. During the last decades, the theory of variational inequalities and complementarity problems have been applied in different fields such as mathematical programming, game theory, economics and Mathematical Finance. In the last case, the problem is used for the pricing of american options (see [1] and the references therein). One of the most interesting and important problems in the theory of variational inequalities is the development of efficient iterative algorithms to approximate their solutions. It is well known that (P ) admits at least one solution (see Lions [13] and Landes-Mustonen [15]. In the last papers, the function a(x, t, ξ) was assumed to satisfy a polynomial growth condition with respect to ∇u. When trying to relax this restriction on the function a(., ξ), we are led to replace the space Lp (0, T ; W 1,p (Ω)) by an inhomogeneous Sobolev space W 1,x LM built from an Orlicz space LM instead of Lp , where the N-function M which defines LM is related to the actual growth of the Carath´eodory’s function. It is our purpose in this paper, to prove existence results and uniqueness of the problem (P) in the setting of the inhomogeneous Sobolev space W 1,x LM . For the sake of simplicity, we suppose through this paper that a(x, t, ∇u) = m(|∇u|) ∇u, where Z |∇u| t m(s) ds. m is the derivative of the N-function M having the representation M (t) = 0

We refer the reader to [18, 19, 16, 6, 4] for additional recent and classical results for some parabolic inequalities problems.

2

Preliminaries

Let M : R+ → R+ be an N-function, i.e. M is continuous, convex, with M (t) > 0 for t > 0, M (t) → 0 as t → 0 and Mt(t) → ∞ as t → ∞. Equivalently, M admits the representation: t Z t M (t) = a(τ )dτ where a : R+ → R+ is non-decreasing, right continuous, with a(0) = 0, 0

a(t) > 0 for t > 0 and a(t) → ∞ as t → ∞. The N-function M conjugate to M is defined by Z t M (t) = a(τ )dτ , where a : R+ → R+ is given by a(t) = sup{s : a(s) ≤ t} (see [2],[11][12]). 0

The N-function M is said to satisfy the ∆2 condition if, for some k > 0: M (2t) ≤ k M (t) for all t ≥ 0,

(2.1)

In case this inequality holds only for t ≥ t0 > 0, M is said to satisfy the ∆2 condition near infinity. Let P and Q be two N-functions. P lQ means that P grows essentially less rapidly than Q, i.e., for each ε > 0, P (t) → 0 as t → ∞. Q(ε t)

3

Nonlinear parabolic inequalities on a general convex

This is the case if and only if lim

t→∞

Q−1 (t) = 0. P −1 (t)

Let Ω be an open subset of RN . The Orlicz class LM (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: Z Z u(x) )dx < +∞ for some λ > 0). M (u(x))dx < +∞ (resp. M ( λ Ω Ω Note that LM (Ω) is a Banach space with the norm Z n o u(x) M( kukM,Ω = inf λ > 0 : )dx ≤ 1 λ Ω

and LM (Ω) is a convex subset of LM (Ω). The closure in LM (Ω) of the set of bounded measurable functions with compact support in Ω is denoted by EM (Ω). The equality EM (Ω) = LM (Ω) holds if and only if M satisfies the ∆2 condition, for all t or for t large, according to whether Ω has infinite measure or not. Z The dual of EM (Ω) can be identified with LM (Ω) by means of the pairing u(x)v(x)dx, Ω

and the dual norm on LM (Ω) is equivalent to k.kM ,Ω . The space LM (Ω) is reflexive if and only if M and M satisfy the ∆2 condition, for all t or for t large, according to whether Ω has infinite measure or not. We now turn to the Orlicz-Sobolev space: W 1 LM (Ω) (resp. W 1 EM (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)). This is a Banach space with the norm X kuk1,M,Ω = kDα ukM,Ω . |α|≤1

Thus W 1 LM (Ω) and W 1 EM (Ω) can be identified by subspaces of the product of N + 1 copies of LM (Ω). Denoting this product by ΠLM , we will use the weak topologies σ(ΠLM , ΠEM ) and σ(ΠLM , ΠLM ). The space W01 EM (Ω) is defined as the (norm) closure of the Schwartz space D(Ω) in W 1 EM (Ω) and the space W01 LM (Ω) as the σ(ΠLM , ΠEM ) closure of D(Ω) in W 1 LM (Ω). We un converges to u for the modular convergence in W 1 LM (Ω) if for Z say that α D un − Dα u )dx → 0 for all |α| ≤ 1. This implies the σ(ΠLM , ΠLM ) M( some λ > 0, λ Ω convergence. If M satisfies the ∆2 condition on R+ (near infinity only when Ω has finite measure), then modular convergence coincides with norm convergence. Let W −1 LM (Ω) (resp. W −1 EM (Ω)) be the space of distributions on Ω which can be written as sums of derivatives of order ≤ 1 of functions in LM (Ω) (resp. EM (Ω)). It is a Banach space with the usual quotient norm. If the open set Ω has the segment property, then the space D(Ω) is dense in W01 LM (Ω) for the modular convergence and for the topology σ(ΠLM , ΠLM ) (cf. [8], [9]). Consequently, the action of a distribution in W −1 LM (Ω) on an element of W01 LM (Ω) is well defined. For k > 0, we define the truncation at height k, Tk : R → R by ½ s if |s| ≤ k, Tk (s) = k if |s| > k. The following abstract lemmas will be applied to the truncation operators.

4

Nonlinear parabolic inequalities on a general convex

Lemma 2.1 (see [3]) Let F : R → R be uniformly lipschitzian function such that F (0) = 0. Let M be an N-function and u ∈ W01 LM (Ω) (resp. W01 EM (Ω)). Then F (u) ∈ W01 LM (Ω) (resp. W01 EM (Ω) ). Moreover, if the set of discontinuity points of F ′ is finite then  ∂u  ′ ∂ a.e. in {x ∈ Ω : u(x) 6∈ D} F (u) F (u) = ∂xi  0 ∂xi a.e. in {x ∈ Ω : u(x) ∈ D}

Proof: By hypothesis, F (u) ∈ W 1 LM (Ω) for all u ∈ W 1 LM (Ω) and kF (u)k1,M,Ω ≤ C kuk1,M,Ω , which gives the result.

Let Ω be a bounded open subset of RN , T > 0 and set Q = Ω×]0, T [. Let m ≥ 1 be an integer and let M be an N-function. For each α ∈ NN , denote by Dxα the distributional derivative on Q of order α with respect to x ∈ RN . The inhomogeneous OrliczSobolev spaces are defined as follows:W m,x LM (Q) = {u ∈ LM (Q) : Dxα u ∈ LM (Q) ∀|α| ≤ m}, W m,x EM (Q) = {u ∈ EM (Q) : Dxα u ∈ EM (Q) ∀|α| ≤ m}. This second space is a subspace of the first one, and both are Banach spaces with the norm X kDxα ukM,Q . kuk = |α|≤m

These spaces constitute a complementary system since Ω satisfies the segment property. These spaces are considered as subspaces of the product space ΠLM (Q) which have as many copies as there is α-order derivatives, |α| ≤ m. We shall also consider the weak topologies σ(ΠLM , ΠEM ) and σ(ΠLM , ΠLM ). If u ∈ W m,x LM (Q) then the function : t 7−→ u(t) = u(t, .) is defined on [0, T ] with values in W m LM (Ω). If u ∈ W m,x EM (Q) the concerned function is a W m EM (Ω)-valued and is strongly measurable. Furthermore, the imbedding W m,x EM (Q) ⊂ L1 (0, T ; W m EM (Ω)) holds. The space W m,x LM (Q) is not in general separable; for u ∈ W m,x LM (Q) we can not conclude that the function u(t) is measurable on [0, T ]. However, the scalar function t 7→ ku(t)kM,Ω ∈ L1 (0, T ). The space W0m,x EM (Q) is defined as the (norm) closure in W m,x EM (Q) of D(Q). We can easily show as in [9] that when Ω has the segment property then each element u of the closure of D(Q) with respect to the weak * topology σ(ΠLM , ΠEM ) is limit in W m,x LM (Q) of some subsequence (ui ) ⊂ D(Q) for the modular convergence, i.e., there exists λ > 0 such that for all |α| ≤ m, Z Dα ui − Dxα u M( x ) dx dt → 0 as i → ∞, λ Q which gives that (ui ) converges to u in W m,x LM (Q) for the weak topology σ(ΠLM , ΠLM ). Consequently σ(ΠLM ,ΠEM ) σ(ΠLM ,ΠLM ) D(Q) = D(Q) ,

The space of functions satisfying such property will be denoted by W0m,x LM (Q). Furthermore, W0m,x EM (Q) = W0m,x LM (Q) ∩ ΠEM . Poincar´e’s inequality also holds in W0m,x LM (Q) i.e. there exists a constant C > 0 such that for all u ∈ W0m,x LM (Q), X X kDxα ukM,Q ≤ C kDxα ukM,Q . |α|≤m

|α|=m

Nonlinear parabolic inequalities on a general convex

5

Thus both sides of the last inequality are equivalent norms on W0m,x LM (Q). We have then the following complementary system µ ¶ W0m,x LM (Q) F . W0m,x EM (Q) F0

F states for the dual space of W0m,x EM (Q) and can be defined, except for an isomorphism, as the quotient of ΠLM by the polar set W0m,x EM (Q)⊥ . It will be denoted by F = W −m,x LM (Q) with n o X Dxα fα : fα ∈ LM (Q) . W −m,x LM (Q) = f = |α|≤m

This space will be equipped with the usual quotient norm X kfα kM ,Q kf kF = inf |α|≤m

where the infimum is taken over all possible decompositions X Dxα fα , fα ∈ LM (Q). f= |α|≤m

The space F0 is then given by n o X Dxα fα : fα ∈ EM (Q) F0 = f = |α|≤m

and is denoted by F0 = W −m,x EM (Q). Remark 2.1 Using lemma 4.4 of [9], we can check that each uniformly lipschitzian mapping F such that F (0) = 0, acts in inhomogeneous Orlicz-Sobolev spaces of order 1, W 1,x LM (Q) and W01,x LM (Q).

3

Main results

Theorem 3.1 Under the hypotheses (1.1)-(1.5), The problem (P ) has at least one solution in the following sense:   u ∈ K ∩ L2 (Q) Z ∂v a(., ∇u)(∇u − ∇v)dxdt ≤< f, u − v > ,u − v > +  < ∂t Q for all v ∈ K ∩ L∞ (Q) ∩ D,

where D := {v ∈ W01,x LM (Q) ∩ L2 (Q) :

∂v ∈ W −1,x LM (Q) + L2 (Q) and v(0) = u0 }. ∂t

Proof: In the sequel and throughout the paper, we will omit for simplicity the dependence on t in the function a(x, t, ξ) and denote ǫ(n, j, µ, i, s) all quantities (possibly different) such that lim lim lim lim lim ǫ(n, j, µ, i, s) = 0.

s→∞ i→∞ µ→∞ j→∞ n→∞

6

Nonlinear parabolic inequalities on a general convex

The order in which the parameters will tend to infinity, that is, first n, then j, µ, i and finaly s. Similarly, we will skip some parameters such as in ǫ(n) or ǫ(n, j), ...to mean that the limits are taken only on the specified parameters. Let us define the indicator functional: Φ : W01,x LM (Q) → R ∪ {+∞} such that: ½ 0 if v(t) ∈ K a.e. (almost everywhere), Φ(v) := +∞ otherwise. Φ is weakly lower semicontinuous. Z t Let us denote by Sk (t) := Tk (s)ds. 0

Step 1. Derivation of a priori estimate Let us consider the following approximate problem:  ∂un   + A(un ) + nTn (Φ(un )) = f ∂t   un (., 0) = u0n

in Q, (Pn ) in Ω.

where (u0n ) ⊂ D(Ω) such that u0n → u0 strongly in L2 (Ω). For the existence of a weak solution un ∈ W01,x LM (Q) ∩ L2 (Q), un ≥ 0 of the above problem, ∂un ∈ W −1,x LM (Q) + L2 (Q). see [7], also (un ) satisfies ∂t Let v = un be test function in (Pn ), then Z Z ∂un , un > + a(., ∇un )∇un dxdt + nTn (Φ(un ))un dxdt ≤< f, un > < ∂t Q Q We can deduce that:

(un ) is bounded in W01,x LM (Q), Z

a(., ∇un )∇un dxdt ≤ C,

Z

nTn (Φ(un ))un dxdt ≤ C.

Q

Q

There exists then a subsequence (also denoted (un )) and a measurable function u such that: un ⇀ u, weakly in

W01,x LM (Q)

for σ(ΠLM , ΠEM ),

strongly in EM (Q) and a.e in Q. Moreover there exists a measurable function h ∈ (LM (Q))N such that: a(., ∇un ) ⇀ h in (LM (Q))N weakly.

Nonlinear parabolic inequalities on a general convex

7

Let us consider now v = Tk (un ) ∈ W01,x LM (Q) as test function in (Pn ), which gives Z Z ∂un , Tk (un ) > + a(., ∇un )∇Tk (un )dxdt+ nTn (Φ(un ))Tk (un )dxdt ≤< f, Tk (un ) >≤ Ck < ∂t Q Q Z Z ∂un Since < Sk (un (T )) − Sk (un (0)), , Tk (un ) >= ∂t Ω Ω Z nTn (Φ(un ))Tk (un )dxdt ≤ Ck. Q

By letting k tend to infinity and using Fatou lemma, one has: Z nTn (Φ(un ))un dxdt ≤ Ck, Q

and since (Tn )n is a continuous increasing sequence , we can deduce Φ(u) = 0, which ensures that u ∈ K.

Step 2. Almost everywhere convergence of the gradients We intend to prove that Z lim (a(., ∇un ) − a(., ∇u)) (∇un − ∇u) dx dt = 0. n→∞ Q

Let us set i = (vj )µ + e−µt ψi , ωµ,j

where vj ∈ D(Q) such that vj → u with the modular convergence in W01,x LM (Q) , ψi a smooth function such that ψi → u0 strongly in L2 (Ω) and ωµ is the mollifier function defined i in [14], and the function ωµ,j have the following properties:  i ∂ωµ,j  i i   = µ(vj − ωµ,j ), ωµ,j (0) = ψi , ∂t ω i → uµ + e−µt ψi inW01,x LM (Q) for the modular convergence with respect to j,    µ,j −µt uµ + e ψi → u in W01,x LM (Q) for the modular convergence with respect to µ.

i as test function in (Pn ), Consider now v = un − ωµ,j Z ∂un i i < , un − ωµ,j > + a(., ∇un )(∇un − ∇ωµ,j )dxdt ∂t Q Z i nTn (Φ(un ))(un − ωµ,j )dxdt + Q

i > =< f, un − ωµ,j

Since un ∈ W01,x LM (Q), there exists a smooth function unσ (see [7]) such that: unσ → un for the modular convergence in W01,x LM (Q),

(3.1)

8

Nonlinear parabolic inequalities on a general convex

∂unσ ∂un → for the modular convergence in W −1,x LM (Q) + L2 (Q). ∂t ∂t Then, Z ∂un i i , un − ωµ,j >= lim (unσ )′ (unσ − ωµ,j )dxdt < ∂t µ σ→0+ Q ¶ Z Z i ′ i i ′ i (unσ − ωµ,j ) (unσ − ωµ,j )dxdt + (ωµ,j ) (unσ − ωµ,j )dxdt = lim σ→0+ Q ! ÷Q Z ¸T Z 1 i 2 i i = lim (unσ − ωµ,j ) + µ (vj − ωµ,j )(unσ − ωµ,j )dxdt 2 Ω σ→0+ Q 0 = lim (I1 (σ) + I2 (σ)). σ→0+

We have, 1 I1 (σ) = 2

Z

Ω

i (unσ − ωµ,j )2 (T )dx −

1 2

Z

Ω

i (unσ (0) − ωµ,j (0))2 dx

1 ≥− 2

Z

Ω

i (unσ (0) − ωµ,j (0))2 dx,

So, lim sup I1 (σ) ≥ ǫ(n, j, µ, i). σ→0+

Similarly, we have lim sup I2 (σ) = ǫ(n, j, µ, i), σ→0+

hence ∂un i , un − ωµ,j >≥ ǫ(n, j, µ, i). ∂t Now let us set for s > 0, Qs = {(x, t) ∈ Q : |∇u| ≤ s} and Qsj = {(x, t) ∈ Q : |∇vj | ≤ s}. Then Z i a(., ∇un )(∇un − ∇ωµ,j )dx dt QZ Z ¡ ¢ = a(., ∇un ) − a(., ∇vj χsj ) (∇un − ∇vj χsj )dx dt + a(., ∇vj χsj )(∇un − ∇vj χsj )dx dt Q Z ZQ s i a(., ∇un )∇vj χj dx dt − a(., ∇un )∇ωµ,j dx dt + <

Q

Q

=: J1 + J2 + J3 + J4 .

We consider first the term Z Z a(., ∇vj χsj )(∇un − ∇vj χsj )dx dt = a(., ∇vj χsj )(∇u − ∇vj χsj )dx dt + ǫ(n). J2 = Q

Q

Since a(., ∇vj χsj ) → a(., ∇uχs ) strongly in (EM (Q))N and ∇vj χsj → ∇uχs strongly in (LM (Q))N , one has J2 = ǫ(n, j). The same technique as in J2 gives, Analogously, we can derive that Z J3 = h∇udxdt + ǫ(n, j, s), Q

9

Nonlinear parabolic inequalities on a general convex

and J4 = −

Z

Q

Then, Z

Q

i h∇ωµ,j dxdt

i a(., ∇un )(∇un −∇ωµ,j )dxdt

Since terms

Z

Q

ǫ(n), we obtain: Z

Q

=

Z

Q

¡

+ ǫ(n) = −

Z

h∇udxdt + ǫ(n, j, µ, i).

Q

¢ a(., ∇un ) − a(., ∇vj χsj ) (∇un −∇vj χsj )dx dt+ǫ(n, j, µ, i, s).

i i > in (3.1) are of the form nTn (Φ(un ))(un − ωµ,j )dxdt and =< f, un − ωµ,j

¡ ¢ a(., ∇un ) − a(., ∇vj χsj ) (∇un − ∇vj χsj )dx dt ≤ ǫ(n, j, µ, i).

(3.2)

On the other hand, Z Z ¡ ¢ s s a(., ∇un ) − a(., ∇vj χsj ) (∇un − ∇vj χsj )dx dt (a(., ∇un ) − a(., ∇uχ )) (∇un − ∇uχ )dx dt − QZ Q Z s s a(., ∇un )(∇vj χj − ∇uχ )dx dt − a(., ∇u)(∇vj χsj − ∇un χs )dx dt = Q ZQ s s a(., ∇vj χj )(∇un − ∇vj χj )dx dt, + Q

Since all terms are of the last sum are ǫ(n, j, s), then Z (a(., ∇un ) − a(., ∇uχs )) (∇un − ∇uχs )dx dt QZ ¡ ¢ = a(., ∇un ) − a(., ∇vj χsj ) (∇un − ∇vj χsj )dx dt + ǫ(n, j, s) Q

Finally, for r < s, we get: Z lim

n→∞ Q r

(a(., ∇un ) − a(., ∇u)) (∇un − ∇u) dx dt = 0,

which gives by the same argument as in [3], ∇un → ∇u a.e. in Q. Step 3. The passage to the limit Let us consider v ∈ K ∩ L∞ (Q) ∩ D and 0 < θ < 1. Using un − θv as test function in (Pn ), the fact that

∂(un − θv) ∂v ∂un , un − θv >=< , un − θv > +θ < , un − θv > ∂t ∂t ∂t and letting n tend to infinity and θ to 1, we obtain Z ∂v < , u − θv > + a(., ∇u)(∇u − ∇v)dxdt ≤< f, u − v > . ∂t Q <

So u is a weak solution of the problem (P ).

(3.3)

10

Nonlinear parabolic inequalities on a general convex

Theorem 3.2 The solution u ∈ K ∩ L2 (Q) of the problem (P ) obtained as limit of approximations of solutions (un ) of the problem (Pn ) is unique. Proof: step1: The modular convergence of the gradients We have to prove that ∇un → ∇u in (LM (Q))N for the modular convergence. Let us recall that Z

Q

and Z

¡

¢ a(., ∇un ) − a(., ∇vj χsj ) (∇un − ∇vj χsj )dx dt ≤ ǫ(n, j, µ, i)

(a(., ∇un ) − a(., ∇uχs )) (∇un − ∇uχs )dx dt

QZ

=

Q

Then, Z

¡

¢ a(., ∇un ) − a(., ∇vj χsj ) (∇un − ∇vj χsj )dx dt + ǫ(n, j, s).

a(., ∇un )∇un dxdt

≤

Q

Z

a(., ∇un )∇uχs dxdt +

Q

Z

a(., ∇uχs )(∇un − ∇uχs )dx dt

Q

+ǫ(n, j, µ, i),

and, lim sup n

Z

a(., ∇un )∇un dxdt ≤

Q

Z

s

a(., ∇u)∇uχ dxdt ≤ lim inf n

Q

Z

a(., ∇un )∇un dxdt.

Q

Then, a(., ∇un )∇un → a(., ∇u)∇u.χs strongly in L1 (Q), and a(., ∇un )∇un → a(., ∇u)∇u strongly in L1 (Q). By a Vitali argument, we deduce ∇un → ∇u in (LM (Q))N for the modular convergence.

(3.4)

Step 2: Uniqueness Suppose there exist two solutions u1 , u2 of the problem (P ) obtained as limit of approximations of solutions of (Pn ) such that u1 (0) = u2 (0) = u0 . Let un1 and un2 be the sequences associated respectively to u1 and u2 . If we consider v = (un1 − un2 )χ(0,τ ) as test function in the approximate problem (where we omit the index τ ), we can deduce that:

< +

∂(un1 − un2 ) n , u1 − un2 > + ∂t Z Q

n(Tn (Φ(un1 ))

−

Z

(a(., ∇un1 ) − a(., ∇un2 ))(∇un1 − ∇un2 )dxdt

Q n Tn (Φ(u2 ))(un1

− un2 ) = 0.

(3.5)

11

Nonlinear parabolic inequalities on a general convex

Four situations may occur in the treatment of J(n) :=

Z

Q

n(Tn (Φ(un1 )) − Tn (Φ(un2 ))(un1 −

un2 ): i)- There exist two subsequences un1 , un2 belonging to K. ii)- all subsequences un1 , un2 are not in K. iii)-There exist two subsequences un1 , un2 such that un1 6∈ K and un2 ∈ K. iv)-There exist two subsequences un1 , un2 such that un1 ∈ K and un2 6∈ K. The cases i) and ii) are simple since J(n) = 0 and Z ∂(un1 − un2 ) n n < , u1 − u2 > + (a(., ∇u1n ) − a(., ∇un2 ))(∇un1 − ∇un2 )dxdt = 0. ∂t Q Then, Z Z Z 1 1 |un1 (τ )−un2 (τ )|2 dx+ (a(., ∇un1 )−a(., ∇un2 ))(∇un1 −∇un2 )dxdt = |un1 (0)−un2 (0)|2 dx. 2 Ω 2 Q Ω Letting n tend to infinity and using (3.4), we obtain: Z Z 1 |u1 (τ ) − u2 (τ )|2 dx + (a(., ∇u1 ) − a(., ∇u2 ))(∇u1 − ∇u2 )dxdt = 0, 2 Ω Q which gives u1 (τ ) = u2 (τ ) for all τ ∈ (0, T ), and using (1.2) , ∇u1 = ∇u2 a.e. in Q. Then u1 = u2 a.e. in Q. The cases iii) and iv) are similar. Let us consider case iii). We have ¸ · Z Z Z 1 2 2 n n (u1 − u2 ) = 0, |u1 (τ )−u2 (τ )| dx+ (a(., ∇u1 )−a(., ∇u2 ))(∇u1 −∇u2 )dxdt+ lim n n→∞ 2 Ω Q Q which gives as previously u1 = u2 a.e. in Q. Remark 3.1 The existence result of theorem 3.1 remains true if a depends on x, t, u, ∇u and condition (1.3) is replaced by the following one |a(x, t, s, ξ)| ≤ c(x, t) + k1 P

−1

M (k2 |s|) + k3 M

−1

M (k4 |ξ|),

where c(x, t) ∈ EM (Q), c ≥ 0 and ki ∈ R+ , i = 1, 2, 3, 4. Remark 3.2 The technique used in the proof of theorem 3.1 can be adapted to prove an existence result for solutions of the following parabolic inequalities:  u ∈ K ∩ L2 (Q),   Z Z T   ∂v   h , u − vidt + a(x, t, ∇u)∇(u − v) dx dt  0Z ∂t Q   + H(x, t, u, ∇u)(u − v) dx dt ≤ hf, u − vi,    Q   for all v ∈ K ∩ D ∩ L∞ (Q),

Nonlinear parabolic inequalities on a general convex

12

where H is a given Carath´eodory function satisfying, for all (s, ζ) ∈ R × RN and a.e. (x, t) ∈ Q, the following conditions |H(x, t, s, ζ)| ≤ λ(|s|)(δ(x, t) + |ζ|p ), and H(x, t, s, ζ)s ≥ 0; with λ : R+ → R+ is a continuous increasing function and δ(x, t) is a given positive function in L1 (Q) .

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