Nonlinear Parabolic Problems with Neumann-type Boundary Conditions and L 1-data

In this paper, we study existence, uniqueness and stability questions for the nonlinear parabolic equation: ∂u ∂t −� pu + α(u )= fi n]0 ,T [×Ω, with Neumann-type boundary conditions and initial data in L 1 .O ur approach is based essentially on the time discretization technique by Euler forward scheme. 2000 Subject Classifications: 35K05, 35J55, 35D65.


Introduction
In this work, we treat the nonlinear parabolic problem where ∆ p u = div |Du| p−2 Du , 1 < p < ∞, Ω is a connected open bounded set in R d , d ≥ 3, with a connected Lipschitz boundary ∂Ω, T is a fixed positive real number and α, γ are taken as continuous non decreasing real functions everywhere defined on R with α(0) = γ(0) = 0. We will have in mind especially the case when initial data in L 1 .
EJQTDE, 2007 No. 27, p. 1 The usual weak formulations of parabolic problems with initial data in L 1 do not ensure existence and uniqueness of solutions.There then arose formulations which were more suitable than that of weak solutions.Through that work it is hoped that we can arrive at a definition of solution so that we can prove existence and uniqueness.For that, three notions of solutions have been adopted: Solutions named SOLA ( Solution Obtained as the Limit of Approximations) defined by A. Dallaglio [6].Renormalized solutions defined by R. Diperna and P. L. Lions [7].Entropy solutions defined by Ph.Bénilan, L. Boccardo, T. Gallouet, R. Gariepy, M. Pierre, J.L. Vazquez in [4].We will have interested here at entropy formulation.Many authors are interested has this type of formulations, see for example [1,2,3,4,19,20,25,26].
We apply here a time discretization of given continuous problem by Euler forward scheme and we study existence, uniqueness and stability questions.We recall that the Euler forward scheme has been used by several authors while studying time discretization of nonlinear parabolic problems and we refer to the works [8,9,10,15] for some details.This scheme is usually used to prove existence of solutions as well as to compute the numerical approximations.
The problem (1), or some special cases of it, arises in many different physical contexts, for example: Heat equation, non Newtonian fluids, diffusion phenomena, etc.This paper is organized as follows: after some preliminary results in section 2, we discretize the problem (1) in section 3 by the Euler forward scheme and replace it by and show the existence and uniqueness of entropy solutions to the discretized problems.Section 4 is devoted to the analysis of stability of the discretized problem and in section 5 we study the asymptotic behavior of the solutions of the EJQTDE, 2007 No. 27, p. 2 discrete dynamical system associated with the discretized problems.We shall finish this paper by showing the existence and uniqueness of entropy solution to the parabolic problem (1).

Preliminaries and Notations
In this section we give some notations, definitions and useful results we shall need in this work.For a measurable set Ω of R d , |Ω| denotes its measure, the norm in L p (Ω) is denoted by .p and .1,p denotes the norm in the Sobolev space W 1,p (Ω), C i and C will denote various positive constants.For a Banach space X and a < b, L p (a, b; X) denotes the space of the measurable functions u : For a given constant k > 0 we define the cut function T k : R → R as For a function u = u(x), x ∈ Ω, we define the truncated function T k u pointwise, i.e., for every x ∈ Ω the value of (T k u) at x is just T k (u(x)).Let the function J k : R → R + such that what implies that For u ∈ W 1,p (Ω), we denote by τ u or u the trace of u on ∂Ω in the usual sense.
In ( [4]) the authors introduce the following spaces For bounded Ω s, we have Following [4], It is possible to give a sense to the derivative Du of a function u ∈ T We apply also the sets T 1,p tr (Ω) introduced in [2] as being the subset of functions in T 1,p (Ω) for which a generalized notion of trace may be defined.More precisely u ∈ T 1,p tr (Ω) if u ∈ T 1,p (Ω) and there exist a sequence (u n ) n∈N in W 1,p (Ω) and a measurable function The function v is the trace of u in the generalized sense introduced in [2].For u ∈ T 1,p tr (Ω), the trace of u on ∂Ω is denoted by tr(u) or u, the operator tr(.) satisfied the following properties In the case where u ∈ W 1,p (Ω), tr(u) coincides with τ u.Obviously, we have In [25], with Nonlinear Semigroup Theory, A. Siai demonstrated the following theorem EJQTDE, 2007 No. 27, p. 4 Theorem 2.1 ( [25]) If β, γ are non decreasing continuous functions on R such that β(0) = γ(0) = 0 and f ∈ L 1 (Ω), g ∈ L 1 (∂Ω), then there exists an entropy solution u ∈ T 1,p tr (Ω) to the problem u is unique, up to an additive constant.Furthermore, if β or γ is one-to-one, then the entropy solution is unique.Where a is an operator of Leray-Lions type defined as follows ) is a Carathéodory function in the sense that a is continuous in ξ for almost every x ∈ Ω, and measurable in x for every ξ ∈ R d .

The semi-discrete problem
By the Euler forward scheme, we consider the following system where )ds on ∂Ω.We assume the following hypotheses: EJQTDE, 2007 No. 27, p. 5 (H 1 ) α and γ are non decreasing continuous functions on R such that α(0 Recently, in [4], a new concept of solution has been introduced for the elliptic equation namely entropy solution.Following this idea we define the concept of entropy solution for the problems (Pn).
Definition 2 An entropy solution to the discretized problems (Pn), is a sequence (U n ) 0≤n≤N such that U 0 = u 0 and U n is defined by induction as an entropy solution of the problem Lemma 3 Let hypotheses (H 1 ) − (H 2 ) be satisfied, if (U n ) 0≤n≤N , N ∈ N is an entropy solution of problems (Pn), then ∀n = 1, ..., N, we have U n ∈ L 1 (Ω).
Proof.In inequality (6) we take ϕ = 0 as test function, we obtain By assumption (H 1 ) and the properties of T k , we get Now, since EJQTDE, 2007 No. 27, p. 6 and Thus, from inequality (7) we obtain, On the other hand, we have for each Then by Fatou's lemma, we deduce that U 1 ∈ L 1 (Ω) and By induction, we deduce in the same manner that U n ∈ L 1 (Ω), ∀n = 1, ..., N .
Lemma 4 If (U n ) 0≤n≤N , N ∈ N is an entropy solution of (Pn), then for all k > 0, for all n = 1, ..., N and for all h > 0, we have EJQTDE, 2007 No. 27, p. 7 Proof.Taking ϕ = T h (U n ) as test function in inequality (6), we have By using the definition of T k , we have where and Thus, we get In the same manner, using the hypothesis (H 1 ) we obtain EJQTDE, 2007 No. 27, p. 8 Now, let (U n ) 0≤n≤N and (V n ) 0≤n≤N , N ∈ N be two entropy solutions of problems (Pn) and let ϕ ∈ W 1,p (Ω) ∩ L ∞ (Ω) (for simplicity, we write u (13) and For the solution u, we take ϕ = T h (v) and for the solution v, we take ϕ = T h (u) as test functions and taking the limit as h → ∞, we get by applying Dominated Convergence Theorem that where and by applying hypothesis (H 1 ), we get Now, we show that lim h→∞ I k,h ≥ 0 .
To prove this, we pose EJQTDE, 2007 No. 27, p. 9 and we spilt , where and We have and on the other hand, from the Hölder's inequality, we have where Similarly, we have Finally It thus follows that Therefore, by inequalities ( 15), ( 16) and ( 17), we get Taking the limit as k → 0, by Dominated Convergence Theorem, we get By induction, we prove that ∀n = 1, ..., N, U n − V n 1 = 0.

Stability
Now we give some a priori estimates for the discrete entropy solution (U n ) 1≤n≤N which we use later to derive convergence results for the Euler forward scheme.
EJQTDE, 2007 No. 27, p. 11 Theorem 4.1 Let hypotheses (H 1 ) − (H 2 ) be satisfied and 1 < p < d.Then, there exists a positive constant C(u 0 , f, g) depending on the data but not on N such that for all n = 1, ..., N , we have ) ) Proof. 1) and 2): Let ϕ = 0 as test function in inequality ( 6) and dividing by k, we obtain Let k → 0, by the properties of T k and the Dominated Convergence Theorem we get, Summing (20) from i = 1 to n we obtain Then inequalities 1) and 2) are satisfied. 3) ) as test function in inequality (6) and using the fact that: where and we obtain Taking the limit as h → ∞ and using the Dominated Convergence Theorem, we get for k = 1 Summing ( 21) from i = 1 to n and applying the stability result 2) and inequality (9), we obtain

4)
Taking ϕ = 0 as test function in inequality ( 6), we deduce by ( 8) that Summing ( 22) from i = 1 to n and applying the stability result 3), we therefore get Hence, by using Sobolev's inequality we deduct the stability result 4).

The semi-discrete dynamical system
This section aims to study the discrete dynamical system.We show existence of absorbing sets in L 1 (Ω) and of the global attractor.(We refer to [27] for the definition of absorbing sets and global attractor).By the results of theorem (3.1), problems (Pn) generates a continuous semigroup S τ defined by Proposition 5 Let hypotheses (H 1 ) − (H 2 ) be satisfied and 1 < p < d.Then for τ small enough, there exists absorbing sets in L 1 (Ω).More precisely, there exists a positive integer n τ such that where C does not depend on τ.
On the other hand, according to the stability results of theorem 4.1, there exists n τ > 0 such that where C 6 does not depend on n 0 .By inequality (9), we have Now, applying the discrete Gronwall's lemma [8, lemma 7.5], we therefore get where C 8 is a constant not depending on τ.
Which implies the existence of absorbing sets in L 1 (Ω).Applying [27, theorem 1.1], we get the following result.
Corollary 6 Let hypotheses (H 1 ) − (H 2 ) be satisfied and 1 < p < d.Then for τ small enough, the semigroup associated with problems (Pn) possesses a compact attractor A τ which is bounded in L 1 (Ω).

Convergence and existence result
Definition 7 A function measurable u : for all k > 0, and and a piecewise constant function where t n := nτ.
As already shown, for any N ∈ N, the solution (U n ) 1≤n≤N of problems (Pn) is unique.Thus, u N and u N are uniquely defined and by construction, we have for any t ∈]t n−1 , t n ] and n = 1, ..., N, that 1) By using the stability results of theorem 4.1, we deduce the following a priori estimates concerning the Rothe function u N and the function u N .
Lemma 8 Let hypotheses (H 1 ) − (H 2 ) be satisfied and 1 < p < d.Then there exists a constant C(T, u 0 , f, g) not depending on N such that for all N ∈ N, we have Proof.We have EJQTDE, 2007 No. 27, p. 15 Using inequality 4) of theorem 4.1, we deduce that In the same manner, we prove the estimates (29), (30), ( 31) and (32).
Using estimates (29) and (31), we deduct that This implies the existence of a subsequence of (u N ) N ∈N converging to an element u in L 1 (Q T ).
And by estimate (28), we deduce hence that the sequence (u N ) N ∈N converges to u in L 1 (Q T ).
On the other hand, by (32) we have that Hence there exists a subsequence, still denoted by Hence, it follows that DT k (u N ) converges to DT k (u) weakly in L p (Q T ), and by (32) we conclude that T k (u) ∈ L p (0, T ; W 1,p (Ω)) f or all k > 0.
We follow the same technique used in [1] to show that u N converges to u on Σ T .

Lemma 9
The sequence (u N ) N ∈N converges to u in C 0, T ; L 1 (Ω) .