Electronic Journal of Qualitative Theory of Differential Equations 2007, No. 27, 1-22; http://www.math.u-szeged.hu/ejqtde/

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

Abderrahmane El Hachimi^{(1)} and Ahmed Jamea^{(2)}

UFR Mathématiques Appliquées et Industrielles Faculté des Sciences B. P. 20, El Jadida, Maroc (1)aelhachi@yahoo.fr; ^{(2)}a.jamea@yahoo.fr

Abstract In this paper, we study existence, uniqueness and stability questions

for the nonlinear parabolic equation:

\frac{∂u}{∂t} − △_{p}u + α(u) = f in ]0, T [×Ω,

with Neumann-type boundary conditions and initial data in L^{1}. Our approach is based essentially on the time discretization technique by Euler forward scheme.

2000 Subject Classifications: 35K05, 35J55, 35D65. Keywords: Entropy solution, Nonlinear parabolic problem, Neumann-type boundary conditions, P-Laplacian, Semi-discretization.

1 Introduction

In this work, we treat the nonlinear parabolic problem

\frac{∂u}{∂t} − △_{p}u + α(u) = f in Q_{T} :=]0, T [×Ω,

|Du|^{p−2} \frac{∂u}{∂η} + γ(u) = g on Σ_{T} :=]0, T [×∂Ω,

u(0, .) = u_{0} in Ω,

(1)

where Δ_{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}.

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].

The problem (1) is treated by F. Andereu, J. M. Mazón, S. Segura De león, J. Teledo [1] in the homogeneous case, i.e. f = 0, g = 0 and α = 0, with γ is a maximal monotone graph in ℝ × ℝ and 0 ∈ γ(0). Hulshof [12] considers the case where α is a uniformly Lipschitz continuous function, α(r) = 1 for r ∈ ℝ^{+}, α ∈ C^{1}(ℝ^{−}), α′ > 0 on ℝ^{−} and lim_{r→−∞} α(r) = 0 and some particular functions g. In [13], N. Igbida studies the case where α is the Heaviside maximal monotone graph. For p = 2, we obtain the heat equation, this equation is studied by many authors, see for example [14, 23] and the references therein. The elliptic case of problem (1) has been treated by many authors, see for example [3, 25, 26, 17], and the references therein.

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

U^{n} − τ△_{p}U^{n} + τα(U^{n}) = τf_{n} + U^{n−1} in Ω,

|DU^{n}|^{p−2} \frac{∂U^{n}}{∂η} + γ(U^{n}) = g_{n} on ∂Ω,

U^{0} = u_{0} in Ω,

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 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).

2 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 : [a, b] → X such that

(∫_{a}^{b} ∥u(t)∥_{X}^{p})^{\frac{1}{p}} := ∥u∥_{L^{p}(a,b;X)} < ∞.

For a given constant k > 0 we define the cut function T_{k} : ℝ → ℝ as

T_{k}(s) := \begin{cases}s & if |s| ≤ k, \\ k sign (s) & if |s| > k, \end{cases}

where

sign (s) := \begin{cases}1 & if s > 0, \\ 0 & if s = 0, \\ −1 & if s < 0. \end{cases}

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

J_{k}(x) = ∫_{0}^{x} T_{k}(s)ds

(J_{k} it is the primitive function of T_{k}). We have

< \frac{∂v}{∂t} , T_{k}(v) >= \frac{d}{dt} ∫_{Ω} J_{k}(v) in L^{1}(]0, T [),  (2)

what implies that

∫_{0}^{t} < \frac{∂v}{∂s} , T_{k}(v) >= ∫_{Ω} J_{k}(v(t)) − ∫_{Ω} J_{k}(v(0)).  (3)

For u ∈ W^{1,p}(Ω), we denote by τu or u the trace of u on ∂Ω in the usual sense. In ([4]) the authors introduce the following spaces

• T_{loc}^{1,1}(Ω) = {u : Ω → ℝ measurable : T_{k}(u) ∈ W_{loc}^{1,1}(Ω), for all k > 0} , • T_{loc}^{1,p}(Ω) = {u ∈ T_{loc}^{1,1}(Ω) : DT_{k}(u) ∈ L_{loc}^{p}(Ω), for all k > 0} ,

• T^{1,p}(Ω) = {u ∈ T_{loc}^{1,p}(Ω) : DT_{k}(u) ∈ L^{p}(Ω), for all k > 0} .

For bounded Ω′s, we have

T^{1,p}(Ω) = {u : Ω → ℝ measurable T_{k}(u) ∈ W^{1,p}(Ω), for all k > 0} .

Following [4], It is possible to give a sense to the derivative Du of a function u ∈ T_{loc}^{1,p}(Ω), generalizing the usual concept of weak derivative in W_{loc}^{1,1}(Ω), thanks to the following result

Lemma 1 ([4]) For every u ∈ T_{loc}^{1,p}(Ω) there exist a unique measurable function v : Ω → ℝ^{d} such that

DT_{k}(u) = v1_{{|v|<k}} a.e,

where 1_{B} is the characteristic function of the measurable set B ⊂ ℝ^{d}. Furthermore, u ∈ W_{loc}^{1,1}(Ω) if and only if v ∈ L_{loc}^{1}(Ω), and then v ≡ Du in the usual weak sense.

We apply also the sets T_{tr}^{1,p}(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_{tr}^{1,p}(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 v on ∂Ω such that

a) u_{n} → u a.e. in Ω,

b) DT_{k}(u_{n}) → DT_{k}(u) in L^{1}(Ω) for every k > 0,

c) u_{n} → v a.e. on ∂Ω.

The function v is the trace of u in the generalized sense introduced in [2]. For u ∈ T_{tr}^{1,p}(Ω), the trace of u on ∂Ω is denoted by tr(u) or u, the operator tr(.) satisfied the following properties

i) if u ∈ T_{tr}^{1,p}(Ω), then τT_{k}(u) = T_{k}(tr(u)), ∀k > 0,

ii) if φ ∈ W^{1,p}(Ω) ∩ L^{∞}(Ω), then ∀u ∈ T_{tr}^{1,p}(Ω), we have u − φ ∈ T_{tr}^{1,p}(Ω) and tr(u − φ) = tr(u) − τφ.

In the case where u ∈ W^{1,p}(Ω), tr(u) coincides with τu. Obviously, we have

W^{1,p}(Ω) ⊂ T_{tr}^{1,p}(Ω) ⊂ T^{1,p}(Ω).

In [25], with Nonlinear Semigroup Theory, A. Siai demonstrated the following theorem 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_{tr}^{1,p}(Ω) to the problem

− div[a(., Du)] + β(u) = f in Ω

\frac{∂u}{∂ν_{a}} + γ(τu) = g on ∂Ω (4)

i.e. ∀φ ∈ C_{0}^{∞}(ℝ^{N})

∫_{Ω} a(., Du)DT_{k}(u−φ)+∫_{Ω} α(u)T_{k}(u−φ)+∫_{∂Ω} γ(u)T_{k}(u−φ) ≤ ∫_{Ω} fT_{k}(u−φ)

+ ∫_{∂Ω} gT_{k}(u − φ),

with (β(u), γ(τu)) ∈ L^{1}(Ω) × L^{1}(∂Ω) and ∥(β(u), γ(τu))∥_{1} ≤ ∥(f, g)∥_{1} and 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

1) a : Ω × ℝ^{d} → ℝ^{d}, (x, ξ) ↦ a(x, ξ) 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}.

2) There exists p, 1 < p < d, and a constant A_{1} > 0, so that,

⟨a(x, ξ), ξ⟩ ≥ A_{1}|ξ|^{p}, for a.e. x ∈ Ω and every ξ ∈ ℝ^{d}.

3) ⟨a(x, ξ_{1}) − a(x, ξ_{2}), ξ_{1} − ξ_{2}⟩ > 0, if ξ_{1} ≠ ξ_{2}, for a.e. x ∈ Ω.

4) There exist some h_{0} ∈ L^{p′}(Ω), p′ = \frac{p}{p−1} and a constant A_{2} > 0, such that

|a(x, ξ)| ≤ A_{2} (h_{0}(x) + |ξ|^{p−1}) for a.e. x ∈ Ω and every ξ ∈ ℝ^{d}.

3 The semi-discrete problem

By the Euler forward scheme, we consider the following system

(P n) \{ \begin{matrix}U^{n} − τ△_{p}U^{n} + τα(U^{n}) = τf_{n} + U^{n−1} in Ω, \\ |DU^{n}|^{p−2} \frac{∂U^{n}}{∂η} + γ(U^{n}) = g_{n} on ∂Ω, \\ U^{0} = u_{0} in Ω, \end{matrix}

where Nτ = T, 1 ≤ n ≤ N and f_{n}(.) = \frac{1}{τ} ∫_{(n−1)τ}^{nτ} f(s, .)ds, in Ω, g_{n}(.) = \frac{1}{τ} ∫_{(n−1)τ}^{nτ} g(s, .)ds on ∂Ω. We assume the following hypotheses:

(H_{1}) α and γ are non decreasing continuous functions on ℝ such that α(0) = γ(0) = 0,

(H_{2}) u_{0} ∈ L^{1}(Ω), f ∈ L^{1}(Q_{T}) and g ∈ L^{1}(Σ_{T}).

Recently, in [4], a new concept of solution has been introduced for the elliptic equation

− div[a(x, Du)] = f(x) in Ω,

u = 0 on ∂Ω, (5)

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

u − τ△_{p}u + τα(u) = τf_{n} + U^{n−1} in Ω,

|Du|^{p−2} \frac{∂u}{∂η} + γ(u) = g_{n} on ∂Ω,

i.e. U^{n} ∈ T_{tr}^{1,p}(Ω) and ∀φ ∈ W^{1,p}(Ω) ∩ L^{∞}(Ω), ∀k > 0, we have

∫_{Ω} U^{n}T_{k}(U^{n} − φ) ∫_{Ω} |DU^{n}|^{p−2}DU^{n}DT_{k}(U^{n} − φ) + ∫_{Ω} τα(U^{n})T_{k}(U^{n} − φ)+

τ ∫_{∂Ω} γ(U^{n})T_{k}(U^{n} − φ) ≤ ∫_{Ω}(τf_{n} + U^{n−1})T_{k}(U^{n} − φ) + τ ∫_{∂Ω} g_{n}T_{k}(U^{n} − φ). (6)

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

τ ∫_{Ω} |DU^{1}|^{p−2}DU^{1}DT_{k}(U^{1}) + ∫_{Ω}(τα(U^{1}) + U^{1})T_{k}(U^{1}) + τ ∫_{∂Ω} γ(U^{1})T_{k}(U^{1})

≤ ∫_{Ω}(τf_{1} + u_{0})T_{k}(U^{1}) + τ ∫_{∂Ω} g_{1}T_{k}(U^{1}).  (7)

By assumption (H_{1}) and the properties of T_{k}, we get

∫_{Ω} τα(U^{1})T_{k}(U^{1}) + τ ∫_{∂Ω} γ(U^{1})T_{k}(U^{1}) ≥ 0.  (8)

Now, since

∑_{n=1}^{n=N} τ (∥f_{n}∥_{1} + ∥g_{n}∥_{L^{1}(∂Ω)}) ≤ ∥f∥_{L^{1}(Q_{T})} + ∥g∥_{L^{1}(Σ_{T})},  (9)

and

τ ∫_{Ω} |DU^{1}|^{p−2}DU^{1}DT_{k}(U^{1}) = τ ∫_{Ω} |DT_{k}(U^{1})|^{p} ≥ 0.

Thus, from inequality (7) we obtain,

0 ≤ ∫_{Ω} U^{1} \frac{T_{k}(U^{1})}{k} ≤ (∥f∥_{L^{1}(Q_{T})} + ∥g∥_{L^{1}(Σ_{T})} + ∥u_{0}∥_{1}) .  (10)

On the other hand, we have for each x ∈ Ω

lim_{k→0}U^{1}(x)\frac{T_{k}(U^{1}(x))}{k} = |U^{1}(x)|.

Then by Fatou’s lemma, we deduce that U^{1} ∈ L^{1}(Ω) and

∥U^{1}∥_{1} ≤ ∥f∥_{L^{1}(Q_{T})} + ∥g∥_{L^{1}(Σ_{T})} + ∥u_{0}∥_{1}.

By induction, we deduce in the same manner that U^{n} ∈ L^{1}(Ω), ∀n = 1, ..., N.

Theorem 3.1 Let hypotheses (H_{1}) − (H_{2}) be satisfied and 1 < p < d, then for all N ∈ ℕ the problems (Pn) has a unique entropy solution (U^{n})_{0≤n≤N}, such that for all n = 1, ..., N, U^{n} ∈ T_{tr}^{1,p}(Ω) ∩ L^{1}(Ω).

Proof. Existence. Let the problem

\begin{matrix}−τ△_{p}u + \overline{α}(u) = F in Ω, \\ |Du|^{p−2} \frac{∂u}{∂η} + γ(u) = G on ∂Ω, \end{matrix}  (11)

where u = U^{1}, F = τf_{1} + u_{0} and G = τg_{1}. According to inequality (9) and hypothesis (H_{2}), we have F ∈ L^{1}(Ω), G ∈ L^{1}(∂Ω) and, by hypothesis (H_{1}), the function defined by \overline{α}(s) = τα(s) + s is non decreasing, continuous and satisfies \overline{α}(0) = 0. Therefore, according to theorem 2.1, the problem (11) has an entropy solution U^{1} in T_{tr}^{1,p}(Ω). By induction, using Lemma 3, we deduct in the same manner that for n = 1, ..., N, the problem

u − τ△_{p}u + τα(u) = τf_{n} + U^{n−1} in Ω,

|Du|^{p−2} \frac{∂u}{∂η} + γ(u) = g_{n} on ∂Ω,

has an entropy solution U^{n} in T_{tr}^{1,p}(Ω) ∩ L^{1}(Ω).

Uniqueness. We firstly need the following lemma.

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

τ ∫_{{h<|U^{n}|<k+h}} |DU^{n}|^{p} ≤ k (∫_{{|U^{n}|>h}} τ|f_{n}| + ∫_{{|U^{n}|>h}} |U^{n−1}| + ∫_{∂Ω∩{|U^{n}|>h}} τ|g_{n}|) .

Proof. Taking φ = T_{h}(U^{n}) as test function in inequality (6), we have ∫_{Ω} U^{n}T_{k}(U^{n} − T_{h}(U^{n})) + τ ∫_{Ω} |DU^{n}|^{p−2}DU^{n}DT_{k}(U^{n} − T_{h}(U^{n}))

+τ ∫_{Ω} α(U^{n})T_{k}(U^{n} − T_{h}(U^{n})) + τ ∫_{∂Ω} γ(U^{n})T_{k}(U^{n} − T_{h}(U^{n}))
≤ ∫_{Ω}(τf_{n} + U^{n−1})T_{k}(U^{n} − T_{h}(U^{n})) + τ ∫_{∂Ω} g_{n}T_{k}(U^{n} − T_{h}(U^{n})).  (12)

By using the definition of T_{k}, we have

∫_{Ω} U^{n}T_{k}(U^{n} − T_{h}(U^{n})) = ∫_{\overline{Ω}_{h}} U^{n}T_{k}(U^{n} − hsign(U^{n}))

= ∫_{\overline{Ω}_{h}∩Ω_{(h,k)}} U^{n}(U^{n} − hsign(U^{n})) + ∫_{\overline{Ω}_{h}∩\overline{Ω}_{(h,k)}} U^{n}sign(U^{n} − hsign(U^{n})), where

\overline{Ω}_{h} = \{|U^{n}| > h\} , Ω_{(h,k)} = \{|U^{n} − hsign(U^{n})| ≤ k\} ,

and

\overline{Ω}_{(h,k)} = \{|U^{n} − hsign(U^{n})| > k\} .

However

sign(U^{n} − hsign(U^{n}))1_{\overline{Ω}_{h}} = sign(U^{n})1_{\overline{Ω}_{h}};

Thus, we get

∫_{Ω} U^{n}T_{k}(U^{n} − T_{h}(U^{n})) ≥ 0.

In the same manner, using the hypothesis (H_{1}) we obtain

τ ∫_{Ω} α(U^{n})T_{k}(U^{n} − T_{h}(U^{n})) + τ ∫_{∂Ω} γ(U^{n})T_{k}(U^{n} − T_{h}(U^{n})) ≥ 0. Now, we have

T_{k} (s − T_{h}(s)) = \begin{cases}s − hsign(s) & if h ≤ |s| < k + h, \\ k & if |s| ≥ k + h, \\ 0 & if |s| ≤ h, \end{cases}

then, it follows that τ ∫_{{h<|U^{n}|<k+h}} |DU^{n}|^{p} ≤ k (∫_{{|U^{n}|>h}} τ|f_{n}| + ∫_{{|U^{n}|>h}} |U^{n−1}| + ∫_{∂Ω∩{|U^{n}|>h}} τ|g_{n}|) .

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 = U^{1}, v = V^{1}), then we have

∫_{Ω} uT_{k}(u − φ) + τ ∫_{Ω} |Du|^{p−2}DuDT_{k}(u − φ) + τ ∫_{Ω} α(u)T_{k}(u − φ)

+τ ∫_{∂Ω} γ(u)T_{k}(u − φ) ≤ ∫_{Ω}(τf_{n} + U^{n−1})T_{k}(u − φ) + τ ∫_{∂Ω} g_{n}T_{k}(u − φ), (13)

and

∫_{Ω} vT_{k}(v − φ) + τ ∫_{Ω} |Dv|^{p−2}DvDT_{k}(v − φ) + τ ∫_{Ω} α(v)T_{k}(v − φ)

+τ ∫_{∂Ω} γ(v)T_{k}(v − φ) ≤ ∫_{Ω}(τf_{n} + U^{n−1})T_{k}(v − φ) + τ ∫_{∂Ω} g_{n}T_{k}(v − φ). (14)

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

∫_{Ω}(u − v)T_{k}(u − v) + τ lim_{h→∞}I_{k,h} + τ lim_{h→∞}J_{k,h} ≤ 0,  (15)

where

I_{k,h} := ∫_{Ω} |Du|^{p−2}DuDT_{k}(u − T_{h}(v)) + ∫_{Ω} |Dv|^{p−2}DvDT_{k}(v − T_{h}(u)),

and

J_{k,h} := ∫_{Ω} α(u)T_{k}(u − T_{h}(v)) + ∫_{Ω} α(v)T_{k}(v − T_{h}(u)) + ∫_{∂Ω} γ(u)T_{k}(u − T_{h}(v))
+ ∫_{∂Ω} γ(v)T_{k}(v − T_{h}(u)),

by applying hypothesis (H_{1}), we get

lim_{h→∞}J_{k,h} = ∫_{Ω}(α(u) − α(v))T_{k}(u − v) + ∫_{∂Ω}(γ(u) − γ(v))T_{k}(u − v) ≥ 0. (16)

Now, we show that lim_{h→∞}I_{k,h} ≥ 0 . To prove this, we pose

Ω_{1}(h) = {|u| < h, |v| < h}, Ω_{2}(h) = {|u| < h, |v| ≥ h},
Ω_{3}(h) = {|u| ≥ h, |v| < h} and Ω_{4}(h) = {|u| ≥ h, |v| ≥ h},

and we spilt

I_{k,h} = I_{k,h}^{1} + I_{k,h}^{2} + I_{k,h}^{3} + I_{k,h}^{4},

where

I_{k,h}^{1} = ∫_{Ω_{1}(h)} (|Du|^{p−2}DuDT_{k}(u − v) + |Dv|^{p−2}DvDT_{k}(v − u))
= ∫_{Ω_{1}(h)} (|Du|^{p−2}Du − |Dv|^{p−2}Dv) DT_{k}(u − v)
= ∫_{Ω_{1}(h)∩{|u−v|<k}} (|Du|^{p−2}Du − |Dv|^{p−2}Dv) (Du − Dv) ≥ 0,

and I_{k,h}^{2} = ∫_{Ω_{2}(h)} |Du|^{p−2}DuDT_{k}(u − hsign(v)) + ∫_{Ω_{2}(h)} |Dv|^{p−2}DvDT_{k}(v − u). We have

∫_{Ω_{2}(h)} |Du|^{p−2}DuDT_{k}(u − hsign(v)) = ∫_{Ω_{2}(h)∩{|u−hsign(v)|<k}} |Du|^{p} ≥ 0, and on the other hand, from the Hölder’s inequality, we have

|∫_{Ω_{2}(h)} |Dv|^{p−2}DvDT_{k}(v − u)|

≤ (∫_{Ω(k,h)} |Dv|^{p})^{\frac{1}{p′}} [(∫_{Ω(k,h)} |Dv|^{p})^{\frac{1}{p}} + (∫_{Ω(k,h)} |Du|^{p})^{\frac{1}{p}} ]
≤ (∫_{Ω_{1}(k,h)} |Dv|^{p})^{\frac{1}{p′}} [(∫_{Ω_{1}(k,h)} |Dv|^{p})^{\frac{1}{p}} + (∫_{Ω_{2}(k,h)} |Du|^{p})^{\frac{1}{p}} ] ,

where Ω(k, h) = Ω_{2}(h) ∩ {|u − v| < k}, Ω_{1}(k, h) = {h ≤ |v| ≤ h + k}, Ω_{2}(k, h) = \{h − k ≤ |u| ≤ h\} and \frac{1}{p} + \frac{1}{p′} = 1. By lemma 4, we have τ ∫_{{h−k<|u|<h}} |Du|^{p} ≤ k (∫_{{|u|>h−k}} τ|f_{n}| + ∫_{{|u|>h−k}} |U^{n−1}| + ∫_{∂Ω∩{|u|>h−k}} τ|g_{n}|) .

Now, τf_{n} ∈ L^{1}(Ω), τg_{n} ∈ L^{1}(∂Ω), U^{n−1} ∈ L^{1}(Ω) and lim_{h→∞} |{|u| ≥ h − k}| = 0,

then

lim_{h→∞} ∫_{{h−k<|u|<h}} |Du|^{p} = 0.

EJQTDE, 2007 No. 27, p. 10 In the same manner, we show that:

lim_{h→∞} ∫_{{h<|v|<h+k}} |Dv|^{p} = 0.

Hence

lim_{h→∞}I_{k,h}^{2} ≥ 0.

Similarly, we have

lim_{h→∞}I_{k,h}^{3} = lim_{h→∞} ∫_{Ω_{3}(h)} |Du|^{p−2}DuDT_{k}(u−v)+∫_{Ω_{3}(h)} |Du|^{p−2}DuDT_{k}(v−hsign(u)) ≥ 0.

Finally

I_{k,h}^{4} = ∫_{Ω_{4}(h)} |Du|^{p−2}DuDT_{k}(u − hsign(v)) + ∫_{Ω_{4}(h)} |Du|^{p−2}DuDT_{k}(v − hsign(u))
= ∫_{Ω_{4}(h)∩{|u−hsign(v)|<k}} |Du|^{p} + ∫_{Ω_{4}(h)∩{|v−hsign(u)|<k}} |Dv|^{p} ≥ 0.

It thus follows that

lim_{h→∞}I_{k,h} ≥ 0.  (17)

Therefore, by inequalities (15), (16) and (17), we get

∫_{Ω}(u − v)T_{k}(u − v) ≤ 0,

i.e.

∫_{Ω}(u − v)\frac{1}{k} T_{k}(u − v) ≤ 0.

Taking the limit as k → 0, by Dominated Convergence Theorem, we get

∥u − v∥_{1} ≤ 0.

By induction, we prove that ∀n = 1, ..., N, ∥U^{n} − V^{n}∥_{1} = 0.

4 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.

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

1) ∥U^{n}∥_{1} ≤ C(u_{0}, f, g),
2) τ ∑_{i=1}^{n} ∥α(U^{i})∥_{1} + τ ∑_{i=1}^{n} ∥γ(U^{i})∥_{L^{1}(∂Ω)} ≤ C(u_{0}, f, g),

3) ∑_{i=1}^{n} ∥U^{i} − U^{i−1}∥_{1} ≤ C(u_{0}, f, g),

4) ∑_{i=1}^{n} τ∥T_{k}(U^{i})∥_{1,p}^{p} ≤ k.C(u_{0}, f, g).

Proof. 1) and 2): Let φ = 0 as test function in inequality (6) and dividing by k, we obtain

τ \frac{1}{k} ∥DT_{k}(U^{i})∥_{p}^{p} + ∫_{Ω} U^{i} \frac{1}{k} T_{k}(U^{i}) + τ ∫_{Ω} α(U^{i})\frac{1}{k} T_{k}(U^{i}) + τ ∫_{∂Ω} γ(U^{i})\frac{1}{k} T_{k}(U^{i})

≤ τ (∥f_{i}∥_{1} + ∥g_{i}∥_{L^{1}(∂Ω)}) + ∥U^{i−1}∥_{1},  (18)

i.e.

∫_{Ω} (U^{i} + τα(U^{i})) \frac{T_{k}(U^{i})}{k}+τ ∫_{∂Ω} γ(U^{i})\frac{T_{k}(U^{i})}{k} ≤ τ (∥f_{i}∥_{1} + ∥g_{i}∥_{L^{1}(∂Ω)})+∥U^{i−1}∥_{1},

(19)

Let k → 0, by the properties of T_{k} and the Dominated Convergence Theorem we get,

τ∥α(U^{i})∥_{1} + ∥U^{i}∥_{1} + τ∥γ(U^{i})∥_{L^{1}(∂Ω)} ≤ τ (∥f_{i}∥_{1} + ∥g_{i}∥_{L^{1}(∂Ω)}) + ∥U^{i−1}∥_{1}. (20)

Summing (20) from i = 1 to n we obtain

∥U^{n}∥_{1} + τ ∑_{i=1}^{n} (∥α(U^{i})∥_{1} + ∥γ(U^{i})∥_{L^{1}(∂Ω)}) ≤ ∑_{i=1}^{n} τ (∥f_{i}∥_{1} + ∥g_{i}∥_{L^{1}(∂Ω)}) + ∥u_{0}∥_{1}
≤ ∥f∥_{L^{1}(Q_{T})} + ∥g∥_{L^{1}(Σ_{T})} + ∥u_{0}∥_{1}.

Then inequalities 1) and 2) are satisfied.

3) Taking φ = T_{h} (U^{i} − sign(U^{i} − U^{i−1})) as test function in inequality (6) and using the fact that:

∫_{Ω} |DU^{i}|^{p−2}DU^{i}DT_{k} (U^{i} − T_{h} (U^{i} − sign(U^{i} − U^{i−1}))) = ∫_{Ω_{k}∩\overline{Ω}_{h}} |DU^{i}|^{p} ≥ 0,

where

Ω_{3}(k, h) = {|U^{i} − T_{h}(U^{i} − sign(U^{i} − U^{i−1})| ≤ k}

and

\overline{Ω}_{h} = \{|U^{i} − sign(U^{i} − U^{i−1})| > h\} ,

we obtain

∫_{Ω} (U^{i} − U^{i−1}) T_{k} (U^{i} − T_{h} (U^{i} − sign(U^{i} − U^{i−1})))

≤ k (τ∥f_{i}∥_{1} + τ∥g_{i}∥_{L^{1}(∂Ω)} + τ∥α(U^{i})∥_{1} + τ∥γ(U^{i})∥_{L^{1}(∂Ω)}) .

Taking the limit as h → ∞ and using the Dominated Convergence Theorem, we get for k = 1

∥U^{i} − U^{i−1}∥_{1} ≤ τ∥f_{i}∥_{1} + τ∥g_{i}∥_{L^{1}(∂Ω)} + τ∥α(U^{i})∥_{1} + τ∥γ(U^{i})∥_{L^{1}(∂Ω)}. (21)

Summing (21) from i = 1 to n and applying the stability result 2) and inequality (9), we obtain

∑_{i=1}^{n} ∥U^{i} − U^{i−1}∥_{1} ≤ 2∥f∥_{L^{1}(Q_{T})} + 2∥g∥_{L^{1}(Σ_{T})} + ∥u_{0}∥_{1}.

4) Taking φ = 0 as test function in inequality (6), we deduce by (8) that

τ∥DT_{k}(U^{i})∥_{p}^{p} ≤ k (τ∥f_{i}∥_{1} + τ∥g_{i}∥_{L^{1}(∂Ω)} + ∥U^{i} − U^{i−1}∥_{1}) .  (22)

Summing (22) from i = 1 to n and applying the stability result 3), we therefore get

∑_{i=1}^{n} τ∥DT_{k}(U^{i})∥_{p}^{p} ≤ k.C(u_{0}, f, g), .

Hence, by using Sobolev’s inequality we deduct the stability result 4).

5 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

S_{τ} U^{n−1} = U^{n}.

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

∥U^{n}∥_{1} ≤ C, ∀n ≥ n_{τ} .  (23)

where C does not depend on τ.

Proof. By inequality (20), we have

y^{n} ≤ y^{n−1} + τh_{n},

where y^{n} = ∥U^{n}∥_{1} and h_{n} = ∥f_{n}∥_{1} + ∥g_{n}∥_{L^{1}(∂Ω)}. On the other hand, according to the stability results of theorem 4.1, there exists n_{τ} > 0 such that

τ ∑_{n=n_{0}}^{n=n_{0}+N} y_{n} ≤ C_{6} ∀n_{0} ≥ n_{τ} ,  (24)

where C_{6} does not depend on n_{0}. By inequality (9), we have

τ ∑_{n=n_{0}}^{n=n_{0}+N} ∥h_{n}∥_{1} ≤ C_{7} ∀n_{0} ≥ n_{τ} .

Now, applying the discrete Gronwall’s lemma [8, lemma 7.5], we therefore get

∥U^{n}∥_{1} ≤ C_{8} ∀n ≥ n_{τ} ,

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).

6 Convergence and existence result

Definition 7 A function measurable u : Q_{T} → ℝ is an entropy solution of parabolic problem (1) in Q_{T} if u ∈ C(0, T; L^{1}(Ω)), T_{k}(u) ∈ L^{p}(0, T; W^{1,p}(Ω)) for all k > 0, and

∫_{0}^{t} ∫_{Ω} |Du|^{p−2}DuDT_{k}(u − φ) + ∫_{0}^{t} ∫_{Ω} α(u)T_{k}(u − φ) + ∫_{0}^{t} ∫_{∂Ω} γ(u)T_{k}(u − φ)

≤ − ∫_{0}^{t} ⟨\frac{∂φ}{∂s}, T_{k}(u − φ)⟩ + ∫_{Ω} J_{k}(u(0) − φ(0)) − ∫_{Ω} J_{k}(u(t) − φ(t))

+ ∫_{0}^{t} ∫_{Ω} fT_{k}(u − φ) + ∫_{0}^{t} ∫_{∂Ω} gT_{k}(u − φ),  (25)

for all φ ∈ L^{∞}(Q_{T}) ∩ L^{p}(0, T; W^{1,p}(Ω)) ∩ W^{1,1}(0, T; L^{1}(Ω)) and t ∈ [0, T].

Now, we state our main result of this work.

Theorem 6.1 Let hypotheses (H_{1}) − (H_{2}) be satisfied and 1 < p < d. Then the nonlinear parabolic problem (1) admits a unique entropy solution.

Proof. Existence. Let us introduce a piecewise linear extension, called Rothe function, by

\{ \begin{matrix}u^{N}(0) := u_{0}, \\ u^{N}(t) := U^{n−1} + (U^{n} − U^{n−1})\frac{(t−t^{n−1})}{τ}, ∀ t ∈ ]t^{n−1}, t^{n}], n = 1, ..., N in Ω, \end{matrix}

(26)

and a piecewise constant function

\{ \begin{matrix}\overline{u}^{N}(0) := u_{0}, \\ \overline{u}^{N}(t) := U^{n} ∀ t ∈ ]t^{n−1}, t^{n}], n = 1, ..., N in Ω, \end{matrix}  (27)

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 \overline{u}^{N} are uniquely defined and by construction, we have for any t ∈]t^{n−1}, t^{n}] and n = 1, ..., N, that

1) \frac{∂u^{N}(t)}{∂t} = \frac{(U^{n} − U^{n−1})}{τ},
2) \overline{u}^{N}(t) − u^{N}(t) = (U^{n} − U^{n−1})\frac{t^{n} − t}{τ}.

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 \overline{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

∥\overline{u}^{N} − u^{N}∥_{L^{1}(Q_{T})} ≤ \frac{1}{N} C(T, u_{0}, f, g),  (28)
∥u^{N}∥_{L^{1}(Q_{T})} ≤ C(T, u_{0}, f, g),  (29)
∥\overline{u}^{N}∥_{L^{1}(Q_{T})} ≤ C(T, u_{0}, f, g),  (30)
∥\frac{∂u^{N}}{∂t} ∥_{L^{1}(Q_{T})} ≤ C(T, u_{0}, f, g),  (31)
∥T_{k}(\overline{u}^{N})∥_{L^{p}(0,T,W^{1,p}(Ω))} ≤ k.C(T, u_{0}, f, g).  (32)

Proof. We have

∥\overline{u}^{N} − u^{N}∥_{L^{1}(Q_{T})} = ∑_{n=1}^{N} ∫_{t^{n−1}}^{t^{n}} ∥U^{n} − U^{n−1}∥_{1} \frac{(t^{n} − t)}{τ}dt

= \frac{τ}{2} ∑_{n=1}^{N} ∥U^{n} − U^{n−1}∥_{1}.

Using inequality 4) of theorem 4.1, we deduce that

∥\overline{u}^{N} − u^{N}∥_{L^{1}(Q_{T})} ≤ \frac{1}{2N} T C(u_{0}, f, g).

In the same manner, we prove the estimates (29), (30), (31) and (32).

Using estimates (29) and (31), we deduct that

the sequence (u^{N})_{N∈ℕ} is relatively compact in L^{1}(Q_{T}).

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 (\overline{u}^{N})_{N∈ℕ} converges to u in L^{1}(Q_{T}).

On the other hand, by (32) we have that

(DT_{k}(\overline{u}^{N}))_{N∈ℕ} is uniformly bounded in L^{p}(Q_{T}).

Hence there exists a subsequence, still denoted by (DT_{k}(\overline{u}^{N}))_{N∈ℕ} such that

(DT_{k}(\overline{u}^{N}))_{N∈ℕ} converges to an element V in L^{p}(Q_{T}).

However

T_{k}(\overline{u}^{N}) converges to T_{k}(u) in L^{p}(Q_{T}).

Hence, it follows that

DT_{k}(\overline{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)) for all k > 0.

We follow the same technique used in [1] to show that

\overline{u}^{N} converges to u on Σ_{T} .

Lemma 9 The sequence (u^{N})_{N∈ℕ} converges to u in C (0, T; Ł^{1}(Ω)) .

Proof. Let φ ∈ L^{∞}(Q_{T}) ∩ L^{p}(0, T; W^{1,p}(Ω)) ∩ W^{1,1}(0, T; L^{1}(Ω)), we rewrite (6) in the form

∫_{0}^{t} ⟨\frac{∂u^{N}}{∂s} , T_{k}(\overline{u}^{N} − φ)⟩ + ∫_{0}^{t} ∫_{Ω} |D\overline{u}^{N}|^{p−2}D\overline{u}^{N}DT_{k}(\overline{u}^{N} − φ)

+ ∫_{0}^{t} ∫_{Ω} α(\overline{u}^{N})T_{k}(\overline{u}^{N} − φ) + ∫_{0}^{t} ∫_{∂Ω} γ(\overline{u}^{N})T_{k}(\overline{u}^{N} − φ)

≤ ∫_{0}^{t} ∫_{Ω} f_{N}T_{k}(\overline{u}^{N} − φ) + ∫_{0}^{t} ∫_{∂Ω} g_{N}T_{k}(\overline{u}^{N} − φ),  (33)

where f_{N}(t, x) = f_{n}(x), g_{N}(t, x) = g_{n}(x) ∀t ∈]t^{n−1}, t^{n}], n = 1, ..., N. Let (t^{n} = nτ_{N})_{n=1}^{N} and (t^{m} = mτ_{M})_{m=1}^{M} be two partitions of interval [0, T] and let (u^{N}(t), \overline{u}^{N}(t)) , (u^{M}(t), \overline{u}^{M}(t)) be the semi-discrete solutions defined by (26), (27) and corresponding to the partitions, respectively. The same method used in the proof of the uniqueness in the theorem 3.1, enables us to obtain for k = 1

∫_{0}^{t} ⟨\frac{∂(u^{N} − u^{M})}{∂s}, T_{1}(\overline{u}^{N} − \overline{u}^{M})⟩ ≤ ∫_{0}^{t} ∫_{Ω} |f_{N} − f_{M}| + ∫_{0}^{t} ∫_{∂Ω} |g_{N} − g_{M}| ,

that is

∫_{Ω} J_{1} (u^{N}(t) − u^{M}(t)) ≤ |∫_{0}^{t} ⟨\frac{∂(u^{N} − u^{M})}{∂s}, T_{1}(\overline{u}^{N} − \overline{u}^{M}) − T_{1}(u^{N} − u^{M})⟩|
+ ∥f_{N} − f_{M}∥_{L^{1}(Q_{T})} + ∥g_{N} − g_{M}∥_{L^{1}(Σ)} .

However,

|∫_{0}^{t} ⟨\frac{∂(u^{N} − u^{M})}{∂s}, T_{1}(\overline{u}^{N} − \overline{u}^{M}) − T_{1}(u^{N} − u^{M})⟩|

≤ ∥\frac{∂(u^{N} − u^{M})}{∂s}∥_{L^{1}(Q_{T})} ∥T_{1}(\overline{u}^{N} − \overline{u}^{M}) − T_{1}(u^{N} − u^{M})∥_{L^{∞}(Q_{T})}
≤ 2C(T, f, g, u_{0})∥T_{1}(\overline{u}^{N} − \overline{u}^{M}) − T_{1}(u^{N} − u^{M})∥_{L^{∞}(Q_{T})}.

Now, as

lim_{N,M→∞} ∥T_{1}(\overline{u}^{N} − \overline{u}^{M}) − T_{1}(u^{N} − u^{M})∥_{L^{∞}(Q_{T})} = 0,

we get

lim_{N,M→∞} |∫_{0}^{t} ⟨\frac{∂(u^{N} − u^{M})}{∂s}, T_{1}(\overline{u}^{N} − \overline{u}^{M}) − T_{1}(u^{N} − u^{M})⟩| = 0.  (34)

On the other hand, we have

lim_{N,M→∞} (∥f_{N} − f_{M}∥_{L^{1}(Q_{T})} + ∥g_{N} − g_{M}∥_{L^{1}(Σ)}) = 0,

then, we obtain

lim_{N,M→∞} ∫_{Ω} J_{1} (u^{N}(t) − u^{M}(t)) = 0.  (35)

Now, using the definition of J_{k} we have

∫_{\{|u^{N} −u^{M} |≤1\}} |u^{N}(t)−u^{M}(t)|^{2}+\frac{1}{2} ∫_{\{|u^{N} −u^{M} |≥1\}} |u^{N}(t)−u^{M}(t)| ≤ ∫_{Ω} J_{1} (u^{N}(t) − u^{M}(t)) .

Therefore, we obtain

∫_{Ω} |u^{N}(t) − u^{M}(t)| = ∫_{{|u^{N} −u^{M} |≤1}} |u^{N}(t) − u^{M}(t)| + ∫_{{|u^{N} −u^{M} |≥1}} |u^{N}(t) − u^{M}(t)|

≤ C(Ω) (∫_{\{|u^{N} −u^{M} |≤1\}} |u^{N}(t) − u^{M}(t)|^{2})^{\frac{1}{2}} + ∫_{\{|u^{N} −u^{M} |≥1\}} |u^{N}(t) − u^{M}(t)|

≤ C(Ω) (∫_{Ω} J_{1} (u^{N}(t) − u^{M}(t)))^{\frac{1}{2}} + 2 ∫_{Ω} J_{1} (u^{N}(t) − u^{M}(t)) .

Then by (35), we conclude that (u^{N})_{N∈ℕ} is a Cauchy sequence in C(0, T; Ł^{1}(Ω)); Which implies that

(u^{N})_{N∈ℕ} converges to u in C(0, T; Ł^{1}(Ω)).  (36)

It remains to prove that the limit function u is an entropy solution of the problem (1). Since u^{N}(0) = U^{0} = u_{0} for all N ∈ ℕ, then u(0, .) = u_{0}. By (33) we get

∫_{0}^{t} ⟨\frac{∂u^{N}}{∂s} , T_{k}(\overline{u}^{N} − φ) − T_{k}(u^{N} − φ)⟩ + ∫_{0}^{t} ∫_{Ω} |D\overline{u}^{N}|^{p−2}D\overline{u}^{N}DT_{k}(\overline{u}^{N} − φ)+

∫_{0}^{t} ∫_{Ω} α(\overline{u}^{N})T_{k}(\overline{u}^{N} − φ) + ∫_{0}^{t} ∫_{∂Ω} γ(\overline{u}^{N})T_{k}(\overline{u}^{N} − φ) ≤ − ∫_{0}^{t} ⟨\frac{∂φ}{∂s}, T_{k}(u^{N} − φ)⟩

+ ∫_{Ω} J_{k}(u^{N}(0) − φ(0)) − ∫_{Ω} J_{k}(u^{N}(t) − φ(t)) + ∫_{0}^{t} ∫_{Ω} f_{N}T_{k}(\overline{u}^{N} − φ)

+ ∫_{0}^{t} ∫_{∂Ω} g_{N}T_{k}(\overline{u}^{N} − φ).  (37)

By same manner, as used for the proof of the equality (34), we deduce that

lim_{N→∞} (∫_{0}^{t} ⟨\frac{∂u^{N}}{∂s} , T_{k}(\overline{u}^{N} − φ) − T_{k}(u^{N} − φ)⟩) = 0.  (38)

We follow the same technique used in [19], we show that

lim_{N→∞} ∫_{0}^{t} ∫_{Ω} |D\overline{u}^{N}|^{p−2}D\overline{u}^{N}DT_{k}(\overline{u}^{N} − φ) = ∫_{0}^{t} ∫_{Ω} |Du|^{p−2}DuDT_{k}(u − φ). (39)

And by Lemma 9, we deduce that u^{N}(t) → u(t) in L^{1}(Ω) for all t ∈ [0, T], which implies that

∫_{Ω} J_{k}(u^{N}(t) − φ(t)) → ∫_{Ω} J_{k}(u(t) − φ(t)) ∀t ∈ [0, T].  (40)

Finally, taking the limits as N → ∞, and using the above results, the continuities of α, γ and the facts that f_{N} → f in L^{1}(Q_{T}), g_{N} → g in L^{1}(Σ_{T}) and T_{k}(\overline{u}^{N} − φ) → T_{k}(u − φ) in L^{∞}(Q_{T}), we deduce that u is an entropy solution of the nonlinear parabolic problem (1). Uniqueness. Let v another entropy solution of the nonlinear parabolic problem (1). Taking φ = T_{h}(u^{N}) as test function in (25) and letting h → ∞, we get

∫_{Ω} J_{k}(v(t) − u^{N}(t)) + ∫_{0}^{t} ⟨\frac{∂u^{N}}{∂s} , T_{k}(v − u^{N})⟩ + lim_{h→∞} II_{1}^{N}(k, h)

+ ∫_{0}^{t} ∫_{Ω} α(v)T_{k}(v − u^{N}) + ∫_{0}^{t} ∫_{∂Ω} γ(v)T_{k}(v − u^{N})

≤ ∫_{0}^{t} ∫_{Ω} fT_{k}(v − u^{N}) + ∫_{0}^{t} ∫_{∂Ω} gT_{k}(v − u^{N});  (41)

where

II_{1}^{N}(k, h) = ∫_{0}^{t} ∫_{Ω} |Dv|^{p−2}DvDT_{k}(v − T_{h}(u^{N})).

On the other hand, taking φ = T_{h}(v) as a test function in the inequality (33) and taking the limit as h → ∞, we get

∫_{0}^{t} ⟨\frac{∂u^{N}}{∂s} , T_{k}(\overline{u}^{N} − v)⟩ + lim_{h→∞} II_{2}^{N}(k, h) + ∫_{0}^{t} ∫_{Ω} α(\overline{u}^{N})T_{k}(\overline{u}^{N} − v)+

∫_{0}^{t} ∫_{∂Ω} γ(\overline{u}^{N})T_{k}(\overline{u}^{N} − v) ≤ ∫_{0}^{t} ∫_{Ω} f_{N}T_{k}(\overline{u}^{N} − v) + ∫_{0}^{t} ∫_{∂Ω} g_{N}T_{k}(\overline{u}^{N} − v), (42)

where

II_{2}^{N}(k, h) = ∫_{0}^{t} ∫_{Ω} |D\overline{u}^{N}|^{p−2}D\overline{u}^{N}DT_{k}(\overline{u}^{N} − T_{h}(v)).

Adding (41) and (42), we get

∫_{Ω} J_{k}(v(t) − u^{N}(t)) + ∫_{0}^{t} ⟨\frac{∂u^{N}}{∂s} , T_{k}(v − u^{N}) + T_{k}(\overline{u}^{N} − v)⟩

+ lim_{h→∞} II^{N}(k, h) + ∫_{0}^{t} ∫_{Ω} [α(v)T_{k}(v − u^{N}) + α(\overline{u}^{N})T_{k}(\overline{u}^{N} − v)]

+ ∫_{0}^{t} ∫_{∂Ω} [γ(v)T_{k}(v − u^{N}) + γ(\overline{u}^{N})T_{k}(\overline{u}^{N} − v)]

≤ ∫_{0}^{t} ∫_{Ω} [fT_{k}(v − u^{N}) + f_{N}T_{k}(\overline{u}^{N} − v)] + ∫_{0}^{t} ∫_{∂Ω} [gT_{k}(v − u^{N}) + g_{N}T_{k}(\overline{u}^{N} − v)] ,

where

II^{N}(k, h) = II_{1}^{N}(k, h) + II_{2}^{N}(k, h).

Taking the limit as N → ∞, using the above convergence results and the hypothesis (H_{1}), we get

∫_{Ω} J_{k}(v(t) − u(t)) + lim_{N→∞} lim_{h→∞} II^{N}(k, h) ≤ 0.  (43)

Applying the technique used in the proof of uniqueness in theorem 3.1, we deduce that

lim_{N→∞} lim_{h→∞} II^{N}(k, h) ≥ 0.  (44)

Therefore the inequality (43) becomes

∫_{Ω} J_{k}(v(t) − u(t)) ≤ 0.

i.e.

∫_{Ω} \frac{J_{k}(v(t) − u(t))}{k} ≤ 0.

However

lim_{k→0} \frac{J_{k}(x)}{k} = |x|.

Then, by Fatou’s lamma, we get

∥v(t) − u(t)∥_{1} ≤ 0, ∀t ∈ [0, T].

Remark 10 The above results can be generalized, for example if the p-Laplacian operator Δ_{p}u is replaced by the operator a(., Du) defined in the theorem 2.1.

References

[1] F. Andereu, J. M. Mazón, S. Segura De león, J. Teledo: Existence and uniqueness for a degenerate parabolic equation with L^{1}-data, Trans. Am. Math. Soc. Vol. 351, No. 1 (1999), pp. 285–306.

[2] F. Andereu, J. M. Mazón, S. Segura De león, J. Teledo: Quasi-linear elliptic and parabolic equations in L^{1} with non-linear boundary conditions, Adv. Math. Sci. Appl. 7 (1997), pp. 183–213.

[3] F. Andreu, N. Igbida, J. M. Mazón, J. Toledo: L^{1} Existence and uniqueness results for quasi-linear elliptic equations with nonlinear boundary condition, to appear Ann. Inst. Poincar.

[4] Bénilan, L. Boccardo, T. Gallouet, R. Gariepy, M. Pierre, J.L. Vazquez: An L^{1} theory of existence and uniqueness of solutions of nonlinear elliptic equations, Annali Sc. Norm. Sup. Pisa, 22 (1995), pp. 241-273.

[5] H. Brezis: Analyse fonctionnelle, théorie et application, Masson, Paris, 1983.

[6] A. Dallaglio: Aproximated solutions of equations with L^{1} data. Aplication to the H-convergence of quasi-linear equation, Ann. Math. Pura Appl. 170 (1996), pp. 207–240.

[7] R. Diperna and P.L. Lions: On the Cauchy problem for the Boltzman equation: Global existence and stability, Ann. Math. 130 (1989), No. 2, pp. 321–366.

[8] A. Eden, B. Michaux and J.M. Rakotoson: Semi-discretized nonlinear evolution equations as dynamical systems and error analysis, Indiana Univ. J. Vol. 39, No 3 (1990) pp. 737–783.

[9] F. Benzekri and A. El Hachimi: Doubly nonlinear parabolic equations related to the p-Laplacian operator: Semi-discretization, EJDE, Vol. 2002, No. 113 (2002), pp 1–14.

[10] A. El Hachimi and M. R. Sidi Ammi: Thermistor problem: A nonlocal parabolic problem, EJDE, 11 (2004), pp. 117–128.

[11] A. El Hachimi and H. El Ouardi: Existence and regularity of a global attractor for doubly nonlinear parabolic equations, EJDE, Vol. 2002 (2002) no 45, pp. 1–15.

[12] J. Hulshof: Bounded weak solutions of an elliptic-parabolic neumann problem, Trans. Amer. Math. Soc. 303 (1987), pp. 211–227.

[13] N. Igbida: The mesa-limit of the porous-medium equation and the Hele-Shaw problem, Diff. Integ. Eq. 15 (2002), pp. 129–146.

[14] M. A. Jendoubi: Exponential stability of positives solutions to some nonlinear heat equation, Por. Math. Vol. 55 (1989), pp. 401–409.

[15] J. Kacur: Method of Rothe in evolution. Teubner, Leipzig, 1985.

[16] D. Kinderlhrer and G. Stampacchia: An introduction to variational inequalities and their applications, Acadimic Press, 1980.

[17] J. Liang and J. F. Rodrigues: Quasilinear elliptic problems with nonmonotone discontinuites and measure data, Por. Math. 53 (1996), pp. 239–252.

[18] J.L. Lions: Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod, Paris, 1969.

[19] A. Pringet: Existence and uniqueness of ”entropy” solutions of parapolic problems with L^{1} data, Nonlin. Anal. TMA 28 (1997), pp. 1943–1954.

[20] A. Prignet: Problèmes elliptiques et paraboliques dans un cadre non variationnel, Thèse 1996.

[21] J. F. Rodrigues: The Stefan problem revisited. In: Math. Modeles for phaze change problems, (J. F. Rodrigues, ed.), ISNM 88, Birkhauser, Basel (1989), pp. 129–190.

[22] J. F. Rodrigues: Variational méthods in the Stefan problem: Phase transitions and Hysteresis, (A. Visitin, ed.), Springer, Berlin (1994), pp. 147–212.

[23] T. Roubicek: Nonlinear heat equation with L^{1}-data, Nodea, 5 (1998) pp. 517–527.

[24] T. Roubicek: The Stefan problem in heterogeneaus media, Ann. Inst. Henri Poincaré, Vol. 6, No. 6 (1989), pp. 481–501.

[25] A. Siai: Nonlinear Neumann problems on bounded Lipschitz domains, EJDE, Vol 2005, No. 09 (2005), pp. 1–16.

[26] A. Siai: A fully nonlinear non homogeneous Neumann problem, Pot. Anal. 24 (2006), pp. 15–45.

[27] R. Temam: Infinite dimensional dynamical systems in mechanics and physics, Applied Mathematical Sciences, No 68, Springer Verlag 1988.

(Received August 8, 2006; Revised version received October 10, 2007)