Nonlinear Parabolic Problems with

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

and Ahmed Jamea

UFR Math´ ematiques Appliqu´ ees et Industrielles Facult´ e des Sciences

B. P. 20, El Jadida, Maroc

(1)

aelhachi@yahoo.fr;

a.jamea@yahoo.fr

Abstract

In this paper, we study existence, uniqueness and stability questions for the nonlinear parabolic equation:

∂u

∂t − 4pu+α(u) =f in]0, T[×Ω,

with Neumann-type boundary conditions and initial data in L¹. 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 bound- ary conditions, P-Laplacian, Semi-discretization.

1 Introduction

In this work, we treat the nonlinear parabolic problem

∂u

∂t − 4pu+α(u) =f inQT :=]0, T[×Ω,

|Du|^p−2∂u

∂η +γ(u) =g on ΣT :=]0, T[×∂Ω, u(0, .) =u0 in Ω,

(1)

where ∆pu= div |Du|^p−2Du

, 1< p <∞, Ω is a connected open bounded set inR^d, d≥3, with a connected Lipschitz boundary∂Ω, T is a fixed positive real number andα, γ are taken as continuous non decreasing real functions every- where defined onRwithα(0) =γ(0) = 0. We will have in mind especially the case when initial data inL¹.

The usual weak formulations of parabolic problems with initial data inL¹do 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 exis- tence 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. B´enilan, L. Boccardo, T. Gal- louet, 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. Maz´on, S. Segura De le´on, J. Teledo [1] in the homogeneous case, i.e. f = 0, g = 0 and α= 0, with γ is a maximal monotone graph inR×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∈C¹(R⁻), α⁰>0 onR⁻ and limr→−∞α(r) = 0 and some particular functionsg.In [13], N. Igbida studies the case whereαis the Heaviside maximal monotone graph. Forp= 2,we obtain the heat equation, this equation is stud- ied 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 phenom- ena, 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

Uⁿ−τ4pUⁿ+τ α(Uⁿ) =τ fn+Uⁿ⁻¹ inΩ,

|DUⁿ|^p−2∂Uⁿ

∂η +γ(Uⁿ) =gn on ∂Ω, U⁰=u0 inΩ,

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 byk.kp andk.k1,p denotes the norm in the Sobolev spaceW^1,p(Ω), C_i andCwill denote various positive constants. For a Banach spaceX anda < b, L^p(a, b;X) denotes the space of the measurable functionsu: [a, b]→X such that

Z b a

ku(t)k^p_X

!¹_p

:=kuk_L^p(a,b;X)<∞.

For a given constantk >0 we define the cut functionTk:R→Ras Tk(s) :=

(s if|s| ≤k, k sign(s) if|s|> k, where

sign(s) :=







1 ifs >0, 0 ifs= 0,

−1 ifs <0.

For a functionu=u(x),x∈Ω, we define the truncated functionTkupointwise, i.e., for everyx∈Ω the value of (Tku) atxis justTk(u(x)).

Let the functionJk :R→R⁺ such that Jk(x) =

Z x 0

Tk(s)ds (Jk it is the primitive function ofTk). We have

< ∂v

∂t, Tk(v)>= d dt

Z

Ω

Jk(v) in L¹(]0, T[), (2) what implies that

Z t 0

< ∂v

∂s, Tk(v)>=

Z

Ω

Jk(v(t))− Z

Ω

Jk(v(0)). (3)

Foru∈W^1,p(Ω),we denote byτ uoruthe trace ofuon∂Ω in the usual sense.

In ([4]) the authors introduce the following spaces

• T_loc^1,1(Ω) =n

u: Ω→R measurable : T_k(u)∈W_loc^1,1(Ω), f or all k >0o ,

• T_loc^1,p(Ω) =n

u∈ T_loc^1,1(Ω) :DTk(u)∈L^p_loc(Ω), f or all k >0o ,

• T^1,p(Ω) =n

u∈ T_loc^1,p(Ω) : DT_k(u)∈L^p(Ω), f or all k >0o . For bounded Ω⁰s, we have

T^1,p(Ω) =

u: Ω→R measurable T_k(u)∈W^1,p(Ω), f or all k >0 . Following [4], It is possible to give a sense to the derivativeDu 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 everyu∈ T_loc^1,p(Ω)there exist a unique measurable function v: Ω→R^d such that

DTk(u) =v1{|v|<k} a.e,

where1B is the characteristic function of the measurable setB ⊂R^d.

Furthermore,u∈W_loc^1,1(Ω) if and only if v ∈L¹_loc(Ω), and then v ≡Du in the usual weak sense.

We apply also the setsT_tr^1,p(Ω) introduced in [2] as being the subset of functions inT^1,p(Ω) for which a generalized notion of trace may be defined. More precisely u∈ T_tr^1,p(Ω) ifu∈ T^1,p(Ω) and there exist a sequence (un)n∈N inW^1,p(Ω) and a measurable functionv on∂Ω such that

a) un→ua.e. in Ω,

b) DTk(un)→DTk(u) inL¹(Ω) for everyk >0, c) un→v a.e. on∂Ω.

The function v is the trace ofuin the generalized sense introduced in [2]. For u∈ T_tr^1,p(Ω), the trace of uon∂Ω is denoted bytr(u) or u,the operatortr(.) satisfied the following properties

i) if u∈ T_tr^1,p(Ω), thenτ T_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) andtr(u−ϕ) =tr(u)−τ ϕ.

In the case whereu∈W^1,p(Ω), tr(u) coincides withτ u.

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]) Ifβ, γare non decreasing continuous functions onRsuch thatβ(0) =γ(0) = 0 andf ∈L¹(Ω),g∈L¹(∂Ω), then there exists an entropy solutionu∈T_tr^1,p(Ω) to the problem

−div[a(., Du)] +β(u) =f inΩ

∂u

∂νa

+γ(τ u) =g on∂Ω (4)

i.e. ∀ϕ∈ C₀^∞(R^N) Z

Ω

a(., Du)DTk(u−ϕ) + Z

Ω

α(u)Tk(u−ϕ) + Z

∂Ω

γ(u)Tk(u−ϕ)≤ Z

Ω

f Tk(u−ϕ) +

Z

∂Ω

gTk(u−ϕ),

with (β(u), γ(τ u)) ∈ L¹(Ω)×L¹(∂Ω) and k(β(u), γ(τ u))k1 ≤ k(f, g)k1 and u is unique, up to an additive constant. Furthermore, if β or γ is one-to-one, then the entropy solution is unique. Whereais an operator of Leray-Lions type defined as follows

1) a: Ω×R^d→R^d,(x, ξ)7→a(x, ξ)is a Carath´eodory function in the sense thata is continuous inξfor almost every x∈Ω,and measurable inxfor every ξ∈R^d.

2) There existsp, 1< p < d, and a constant A1>0,so that, ha(x, ξ), ξi ≥A1|ξ|^p, f or a.e. x∈Ωand every ξ∈R^d. 3) ha(x, ξ1)−a(x, ξ2), ξ1−ξ2i>0,if ξ16=ξ2, for a.e. x∈Ω.

4) There exist someh0∈L^p0(Ω), p⁰= _p−1^p and a constantA2>0,such that

|a(x, ξ)| ≤A2 h0(x) +|ξ|^p−1

f or a.e. x∈Ωand every ξ∈R^d.

3 The semi-discrete problem

By the Euler forward scheme, we consider the following system

(P n)











Uⁿ−τ4pUⁿ+τ α(Uⁿ) =τ f_n+Uⁿ⁻¹ in Ω,

|DUⁿ|^p−2∂Uⁿ

∂η +γ(Uⁿ) =g_n on∂Ω, U⁰=u0 in Ω,

where N τ = T, 1 ≤ n ≤ N and f_n(.) = ¹_τRnτ

(n−1)τf(s, .)ds, in Ω, gn(.) =

1 τ

Rnτ

(n−1)τg(s, .)dson∂Ω.

We assume the following hypotheses:

(H1) αand γ are non decreasing continuous functions onRsuch that α(0) = γ(0) = 0,

(H2) u0∈L¹(Ω), f ∈L¹(QT) andg∈L¹(ΣT).

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 se- quence (Uⁿ)0≤n≤N such that U⁰ = u0 and Uⁿ is defined by induction as an entropy solution of the problem

u−τ4pu+τ α(u) =τ f_n+Uⁿ⁻¹ in Ω,

|Du|^p−2∂u

∂η +γ(u) =g_n on ∂Ω, i.e. Uⁿ∈ T_tr^1,p(Ω) and∀ϕ∈W^1,p(Ω)∩L^∞(Ω),∀k >0,we have

Z

Ω

UⁿTk(Uⁿ−ϕ) Z

Ω

|DUⁿ|^p−2DUⁿDTk(Uⁿ−ϕ) + Z

Ω

τ α(Uⁿ)Tk(Uⁿ−ϕ)+

τ Z

∂Ω

γ(Uⁿ)Tk(Uⁿ−ϕ)≤ Z

Ω

(τ fn+Uⁿ⁻¹)Tk(Uⁿ−ϕ) +τ Z

∂Ω

gnTk(Uⁿ−ϕ). (6) Lemma 3 Let hypotheses(H1)−(H2)be satisfied, if (Uⁿ)0≤n≤N, N ∈Nis an entropy solution of problems (Pn), then∀n= 1, ..., N,we have Uⁿ ∈L¹(Ω).

Proof. In inequality (6) we takeϕ= 0 as test function, we obtain τ

Z

Ω

|DU¹|^p−2DU¹DTk(U¹) + Z

Ω

(τ α(U¹) +U¹)Tk(U¹) +τ Z

∂Ω

γ(U¹)Tk(U¹)

≤ Z

Ω

(τ f1+u0)Tk(U¹) +τ Z

∂Ω

g1Tk(U¹). (7) By assumption (H1) and the properties ofTk,we get

Z

Ω

τ α(U¹)Tk(U¹) +τ Z

∂Ω

γ(U¹)Tk(U¹)≥0. (8) Now, since

n=N

X

n=1

τ kfnk1+kgnkL¹(∂Ω)

≤ kfkL¹(QT)+kgkL¹(ΣT), (9)

and

τ Z

Ω

|DU¹|^p−2DU¹DT_k(U¹) =τ Z

Ω

|DTk(U¹)|^p≥0.

Thus, from inequality (7) we obtain, 0≤

Z

Ω

U¹Tk(U¹)

k ≤ kfk_L¹(QT)+kgk_L¹(ΣT)+ku0k1

. (10)

On the other hand, we have for eachx∈Ω

k→0limU¹(x)T_k(U¹(x))

k =|U¹(x)|.

Then by Fatou’s lemma, we deduce thatU¹∈L¹(Ω) and kU¹k1≤ kfkL¹(QT)+kgkL¹(ΣT)+ku0k1.

By induction, we deduce in the same manner thatUⁿ ∈L¹(Ω),∀n= 1, ..., N.

Theorem 3.1 Let hypotheses (H1)−(H2)be satisfied and 1< p < d,then for all N ∈ N the problems (Pn) has a unique entropy solution (Uⁿ)0≤n≤N, such that for alln= 1, ..., N, Uⁿ∈ T_tr^1,p(Ω)∩L¹(Ω).

Proof. Existence. Let the problem

−τ4pu+α(u) =F inΩ,

|Du|^p−2∂u

∂η +γ(u) =G on ∂Ω, (11)

where u =U¹, F = τ f1+u0 and G = τ g1. According to inequality (9) and hypothesis (H2),we haveF ∈L¹(Ω), G∈L¹(∂Ω) and, by hypothesis (H1),the function defined byα(s) =τ α(s) +sis non decreasing, continuous and satisfies α(0) = 0.Therefore, according to theorem 2.1, the problem (11) has an entropy solutionU¹ inT_tr^1,p(Ω).

By induction, using Lemma 3, we deduct in the same manner that for n = 1, ..., N,the problem

u−τ4pu+τ α(u) =τ f_n+Uⁿ⁻¹ in Ω,

|Du|^p−2∂u

∂η +γ(u) =g_n on ∂Ω, has an entropy solutionUⁿ inT_tr^1,p(Ω)∩L¹(Ω).

Uniqueness. We firstly need the following lemma.

Lemma 4 If (Uⁿ)0≤n≤N, N ∈N is an entropy solution of (Pn), then for all k >0, for alln= 1, ..., N and for allh >0, we have

τ Z

{h<|Uⁿ|<k+h}

|DUⁿ|^p≤k Z

{|Uⁿ|>h}

τ|fn|+ Z

{|Uⁿ|>h}

|Uⁿ⁻¹|+ Z

∂Ω∩{|Uⁿ|>h}

τ|gn|

! .

Proof. Takingϕ=Th(Uⁿ) as test function in inequality (6), we have Z

Ω

UⁿTk(Uⁿ−Th(Uⁿ)) +τ Z

Ω

|DUⁿ|^p−2DUⁿDTk(Uⁿ−Th(Uⁿ))

+τ Z

Ω

α(Uⁿ)Tk(Uⁿ−T_h(Uⁿ)) +τ Z

∂Ω

γ(Uⁿ)Tk(Uⁿ−T_h(Uⁿ))

≤ Z

Ω

(τ fn+Uⁿ⁻¹)Tk(Uⁿ−Th(Uⁿ)) +τ Z

∂Ω

gnTk(Uⁿ−Th(Uⁿ)). (12) By using the definition ofT_k, we have

Z

Ω

UⁿTk(Uⁿ−Th(Uⁿ)) = Z

Ω_h

UⁿTk(Uⁿ−hsign(Uⁿ))

= Z

Ωh∩Ω_(h,k)

Uⁿ(Uⁿ−hsign(Uⁿ)) + Z

Ωh∩Ω_(h,k)

Uⁿsign(Uⁿ−hsign(Uⁿ)), where

Ωh={|Uⁿ|> h},Ω(h,k)={|Uⁿ−hsign(Uⁿ)| ≤k}, and

Ω(h,k)={|Uⁿ−hsign(Uⁿ)|> k}. However

sign(Uⁿ−hsign(Uⁿ))1_Ωh =sign(Uⁿ)1_Ωh; Thus, we get

Z

Ω

UⁿT_k(Uⁿ−T_h(Uⁿ))≥0.

In the same manner, using the hypothesis (H1) we obtain τ

Z

Ω

α(Uⁿ)Tk(Uⁿ−T_h(Uⁿ)) +τ Z

∂Ω

γ(Uⁿ)Tk(Uⁿ−T_h(Uⁿ))≥0.

Now, we have

Tk(s−Th(s)) =







s−hsign(s) ifh≤ |s|< k+h, k if|s| ≥k+h, 0 if|s| ≤h, then, it follows that

τ Z

{h<|Uⁿ|<k+h}

|DUⁿ|^p≤k Z

{|Uⁿ|>h}

τ|fn|+ Z

{|Uⁿ|>h}

|Uⁿ⁻¹|+ Z

∂Ω∩{|Uⁿ|>h}

τ|gn|

! .

Now, let (Uⁿ)0≤n≤N and (Vⁿ)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 = U¹, v=V¹), then we have

Z

Ω

uTk(u−ϕ) +τ Z

Ω

|Du|^p−2DuDTk(u−ϕ) +τ Z

Ω

α(u)Tk(u−ϕ)

+τ Z

∂Ω

γ(u)Tk(u−ϕ)≤ Z

Ω

(τ fn+Uⁿ⁻¹)Tk(u−ϕ) +τ Z

∂Ω

g_nT_k(u−ϕ), (13) and

Z

Ω

vTk(v−ϕ) +τ Z

Ω

|Dv|^p−2DvDTk(v−ϕ) +τ Z

Ω

α(v)Tk(v−ϕ)

+τ Z

∂Ω

γ(v)Tk(v−ϕ)≤ Z

Ω

(τ fn+Uⁿ⁻¹)Tk(v−ϕ) +τ Z

∂Ω

gnTk(v−ϕ). (14) For the solutionu,we takeϕ=T_h(v) and for the solutionv,we takeϕ=T_h(u) as test functions and taking the limit ash→ ∞, we get by applying Dominated Convergence Theorem that

Z

Ω

(u−v)Tk(u−v) +τ lim

h→∞Ik,h+τ lim

h→∞Jk,h ≤0, (15) where

Ik,h:=

Z

Ω

|Du|^p−2DuDTk(u−Th(v)) + Z

Ω

|Dv|^p−2DvDTk(v−Th(u)), and

J_k,h :=

Z

Ω

α(u)Tk(u−T_h(v)) + Z

Ω

α(v)Tk(v−T_h(u)) + Z

∂Ω

γ(u)Tk(u−T_h(v)) +

Z

∂Ω

γ(v)Tk(v−T_h(u)), by applying hypothesis (H1),we get

h→∞limJ_k,h= Z

Ω

(α(u)−α(v))Tk(u−v) + Z

∂Ω

(γ(u)−γ(v))Tk(u−v)≥0. (16) Now, we show that lim

h→∞Ik,h≥0 . To prove this, we pose

Ω1(h) ={|u|< h, |v|< h}, Ω2(h) ={|u|< h, |v| ≥h}, Ω3(h) ={|u| ≥h, |v|< h} and Ω4(h) ={|u| ≥h, |v| ≥h},

and we spilt

Ik,h=I_k,h¹ +I_k,h² +I_k,h³ +I_k,h⁴ , where

I_k,h¹ = Z

Ω1(h)

|Du|^p−2DuDT_k(u−v) +|Dv|^p−2DvDT_k(v−u)

= Z

Ω1(h)

|Du|^p−2Du− |Dv|^p−2Dv

DTk(u−v)

= Z

Ω1(h)∩{|u−v|<k}

|Du|^p−2Du− |Dv|^p−2Dv

(Du−Dv)≥0, and

I_k,h² = Z

Ω2(h)

|Du|^p−2DuDTk(u−hsign(v)) + Z

Ω2(h)

|Dv|^p−2DvDTk(v−u).

We have Z

Ω2(h)

|Du|^p−2DuDTk(u−hsign(v)) = Z

Ω2(h)∩{|u−hsign(v)|<k}

|Du|^p≥0, and on the other hand, from the H¨older’s inequality, we have

Z

Ω2(h)

|Dv|^p−2DvDT_k(v−u)

≤ Z

Ω(k,h)

|Dv|^p

!_p¹0 

 Z

Ω(k,h)

|Dv|^p

!¹_p +

Z

Ω(k,h)

|Du|^p

!¹_p



≤ Z

Ω1(k,h)

|Dv|^p

!_p¹0 

 Z

Ω1(k,h)

|Dv|^p

!¹_p +

Z

Ω2(k,h)

|Du|^p

!_p¹

,

where Ω(k, h) = Ω2(h)∩ {|u−v|< k},Ω1(k, h) ={h≤ |v| ≤h+k}, Ω2(k, h) = {h−k≤ |u| ≤h}and ¹_p+_p¹⁰ = 1.

By lemma 4, we have τ

Z

{h−k<|u|<h}

|Du|^p≤k Z

{|u|>h−k}

τ|fn|+ Z

{|u|>h−k}

|Uⁿ⁻¹|+ Z

∂Ω∩{|u|>h−k}

τ|gn|

! .

Now,τ fn ∈L¹(Ω), τ gn∈L¹(∂Ω),Uⁿ⁻¹∈L¹(Ω) and lim

h→∞|{|u| ≥h−k}|= 0, then

h→∞lim Z

{h−k<|u|<h}

|Du|^p= 0.

In the same manner, we show that:

h→∞lim Z

{h<|v|<h+k}

|Dv|^p= 0.

Hence

h→∞limI_k,h² ≥0.

Similarly, we have

h→∞limI_k,h³ = lim

h→∞

Z

Ω3(h)

|Du|^p−2DuDTk(u−v)+

Z

Ω3(h)

|Du|^p−2DuDTk(v−hsign(u))≥0.

Finally I_k,h⁴ =

Z

Ω4(h)

|Du|^p−2DuDTk(u−hsign(v)) + Z

Ω4(h)

|Du|^p−2DuDTk(v−hsign(u))

= Z

Ω4(h)∩{|u−hsign(v)|<k}

|Du|^p+ Z

Ω4(h)∩{|v−hsign(u)|<k}

|Dv|^p≥0.

It thus follows that

h→∞limIk,h≥0. (17)

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

Ω

(u−v)Tk(u−v)≤0,

i.e. Z

Ω

(u−v)1

kT_k(u−v)≤0.

Taking the limit ask→0,by Dominated Convergence Theorem, we get ku−vk1≤0.

By induction, we prove that

∀n= 1, ..., N, kUⁿ−Vⁿk1= 0.

4 Stability

Now we give some a priori estimates for the discrete entropy solution (Uⁿ)1≤n≤N

which we use later to derive convergence results for the Euler forward scheme.

Theorem 4.1 Let hypotheses (H1)−(H2) be satisfied and 1 < p < d. Then, there exists a positive constantC(u0, f, g)depending on the data but not on N such that for alln= 1, ..., N, we have

1) kUⁿk1≤C(u0, f, g), 2) τ

n

X

i=1

kα(Uⁱ)k1+τ

n

X

i=1

kγ(Uⁱ)k_L¹(∂Ω)≤C(u0, f, g),

3)

n

X

i=1

kUⁱ−Uⁱ⁻¹k1≤C(u0, f, g),

4)

n

X

i=1

τkTk(Uⁱ)k^p_1,p≤k.C(u0, f, g).

Proof. 1) and 2): Letϕ= 0 as test function in inequality (6) and dividing byk, we obtain

τ1

kkDTk(Uⁱ)k^p_p+ Z

Ω

Uⁱ1

kTk(Uⁱ) +τ Z

Ω

α(Uⁱ)1

kTk(Uⁱ) +τ Z

∂Ω

γ(Uⁱ)1 kTk(Uⁱ)

≤τ kfik1+kgik_L¹(∂Ω)

+kUⁱ⁻¹k1, (18)

i.e.

Z

Ω

Uⁱ+τ α(Uⁱ)Tk(Uⁱ) k +τ

Z

∂Ω

γ(Uⁱ)Tk(Uⁱ)

k ≤τ kfik1+kgik_L¹(∂Ω)

+kUⁱ⁻¹k1, (19) Letk →0, by the properties ofTk and the Dominated Convergence Theorem we get,

τkα(Uⁱ)k1+kUⁱk1+τkγ(Uⁱ)k_L¹(∂Ω)≤τ kfik1+kgik_L¹(∂Ω)

+kUⁱ⁻¹k1. (20) Summing (20) fromi= 1 to nwe obtain

kUⁿk1+τ

n

X

i=1

kα(Uⁱ)k1+kγ(Uⁱ)k_L¹(∂Ω)

≤

n

X

i=1

τ kfik1+kgik_L¹(∂Ω)

+ku0k1

≤ kfkL¹(QT)+kgkL¹(ΣT)+ku0k1. Then inequalities 1) and 2) are satisfied.

3)Takingϕ=Th Uⁱ−sign(Uⁱ−Uⁱ⁻¹)

as test function in inequality (6) and using the fact that:

Z

Ω

|DUⁱ|^p−2DUⁱDTk Uⁱ−Th Uⁱ−sign(Uⁱ−Uⁱ⁻¹)

= Z

Ωk∩Ωh

|DUⁱ|^p≥0, where

Ω3(k, h) =

|Uⁱ−Th(Uⁱ−sign(Uⁱ−Uⁱ⁻¹)| ≤k

and

Ωh=

|Uⁱ−sign(Uⁱ−Uⁱ⁻¹)|> h , we obtain

Z

Ω

Uⁱ−Uⁱ⁻¹

T_k Uⁱ−T_h Uⁱ−sign(Uⁱ−Uⁱ⁻¹)

≤k τkfik1+τkgik_L¹(∂Ω)+τkα(Uⁱ)k1+τkγ(Uⁱ)k_L¹(∂Ω) .

Taking the limit ash→ ∞and using the Dominated Convergence Theorem, we get fork= 1

kUⁱ−Uⁱ⁻¹k1≤τkfik1+τkgik_L¹(∂Ω)+τkα(Uⁱ)k1+τkγ(Uⁱ)k_L¹(∂Ω). (21) Summing (21) fromi= 1 tonand applying the stability result 2) and inequality (9), we obtain

n

X

i=1

kUⁱ−Uⁱ⁻¹k1≤2kfkL¹(QT)+ 2kgkL¹(ΣT)+ku0k1. 4)Takingϕ= 0 as test function in inequality (6), we deduce by (8) that

τkDTk(Uⁱ)k^p_p≤k τkfik1+τkgikL¹(∂Ω)+kUⁱ−Uⁱ⁻¹k1

. (22) Summing (22) fromi= 1 tonand applying the stability result 3), we therefore get

n

X

i=1

τkDTk(Uⁱ)k^p_p≤k.C(u0, 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 inL¹(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 Sτ defined by

SτUⁿ⁻¹=Uⁿ.

Proposition 5 Let hypotheses (H1)−(H2) be satisfied and 1 < p < d. Then for τ small enough, there exists absorbing sets inL¹(Ω). More precisely, there exists a positive integernτ such that

kUⁿk1≤C, ∀n≥n_τ. (23) whereC does not depend onτ.

Proof. By inequality (20), we have

yⁿ≤yⁿ⁻¹+τ hn, whereyⁿ=kUⁿk1and hn=kfnk1+kgnk_L¹(∂Ω).

On the other hand, according to the stability results of theorem 4.1, there exists n_τ>0 such that

τ

n=n0+N

X

n=n0

yn≤C6 ∀n0≥nτ, (24) whereC6does not depend onn0.

By inequality (9), we have τ

n=n0+N

X

n=n0

khnk1≤C7 ∀n0≥n_τ.

Now, applying the discrete Gronwall’s lemma [8, lemma 7.5], we therefore get kUⁿk1≤C8 ∀n≥nτ,

whereC8is a constant not depending onτ.

Which implies the existence of absorbing sets inL¹(Ω).

Applying [27, theorem 1.1], we get the following result.

Corollary 6 Let hypotheses(H1)−(H2)be satisfied and1< p < d.Then forτ small enough, the semigroup associated with problems (Pn) possesses a compact attractorAτ which is bounded inL¹(Ω).

6 Convergence and existence result

Definition 7 A function measurable u : QT → R is an entropy solution of parabolic problem (1) in QT if u∈ C(0, T;L¹(Ω)), Tk(u)∈ L^p(0, T;W^1,p(Ω)) for allk >0,and

Z t 0

Z

Ω

|Du|^p−2DuDTk(u−ϕ) + Z t

0

Z

Ω

α(u)Tk(u−ϕ) + Z t

0

Z

∂Ω

γ(u)Tk(u−ϕ)

≤ − Z t

0

∂ϕ

∂s, T_k(u−ϕ)

+ Z

Ω

J_k(u(0)−ϕ(0))− Z

Ω

J_k(u(t)−ϕ(t))

+ Z t

0

Z

Ω

f T_k(u−ϕ) + Z t

0

Z

∂Ω

gT_k(u−ϕ), (25) for allϕ∈L^∞(QT)∩L^p(0, T;W^1,p(Ω))∩W^1,1(0, T;L¹(Ω))andt∈[0, T].

Now, we state our main result of this work.

Theorem 6.1 Let hypotheses(H1)−(H2)be satisfied and1< 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

( u^N(0) :=u0,

u^N(t) :=Uⁿ⁻¹+ (Uⁿ−Uⁿ⁻¹)^(t−tn

−1)

τ , ∀ t∈]tⁿ⁻¹, tⁿ], n= 1, ..., N in Ω, (26) and a piecewise constant function

u^N(0) :=u0,

u^N(t) :=Uⁿ ∀t∈]tⁿ⁻¹, tⁿ], n= 1, ..., N inΩ, (27) wheretⁿ :=nτ.

As already shown, for anyN ∈N,the solution (Uⁿ)1≤n≤N of problems (Pn) is unique. Thus, u^N and u^N are uniquely defined and by construction, we have for anyt∈]tⁿ⁻¹, tⁿ] and n= 1, ..., N,that

1) ∂u^N(t)

∂t = (Uⁿ−Uⁿ⁻¹)

τ ,

2) u^N(t)−u^N(t) = (Uⁿ−Uⁿ⁻¹)tⁿ−t τ .

By using the stability results of theorem 4.1, we deduce the following a priori estimates concerning the Rothe functionu^N and the function u^N.

Lemma 8 Let hypotheses (H1)−(H2) be satisfied and1 < p < d. Then there exists a constantC(T, u0, f, g)not depending on N such that for allN ∈N,we have

ku^N −u^Nk_L¹(QT)≤ 1

NC(T, u0, f, g), (28) ku^NkL¹(QT)≤C(T, u0, f, g), (29) ku^Nk_L¹(Q_T)≤C(T, u0, f, g), (30) k∂u^N

∂t k_L¹(QT)≤C(T, u0, f, g), (31) T_k(u^N)

L^p(0,T,W^1,p(Ω)) ≤k.C(T, u0, f, g). (32) Proof. We have

ku^N −u^NkL¹(QT) =

N

X

n=1

Z tⁿ tⁿ⁻¹

kUⁿ−Uⁿ⁻¹k1

(tⁿ−t) τ dt

= τ

2

N

X

n=1

kUⁿ−Uⁿ⁻¹k1.

Using inequality 4) of theorem 4.1, we deduce that ku^N−u^NkL¹(QT)≤ 1

2NT C(u0, 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∈Nis relatively compact in L¹(QT).

This implies the existence of a subsequence of (u^N)N∈Nconverging to an element uin L¹(QT).

And by estimate (28), we deduce hence that

the sequence(u^N)N∈Nconverges to u in L¹(QT).

On the other hand, by (32) we have that DTk(u^N)

N∈N is unif ormly bounded in L^p(QT).

Hence there exists a subsequence, still denoted by DT_k(u^N)

N∈Nsuch that DTk(u^N)

N∈N converges to an element V in L^p(QT).

However

T_k(u^N)converges to T_k(u)in L^p(QT).

Hence, it follows that

DTk(u^N)converges to DTk(u)weakly in L^p(QT), and by (32) we conclude that

Tk(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 onΣT.

Lemma 9 The sequence(u^N)N∈N converges touin C 0, T; L¹(Ω) .

Proof. Letϕ ∈ L^∞(QT)∩L^p(0, T;W^1,p(Ω))∩W^1,1(0, T;L¹(Ω)), we rewrite (6) in the form

Z t 0

∂u^N

∂s , Tk(u^N −ϕ)

+ Z t

0

Z

Ω

|Du^N|^p−2Du^NDTk(u^N−ϕ)

+ Z t

0

Z

Ω

α(u^N)Tk(u^N −ϕ) + Z t

0

Z

∂Ω

γ(u^N)Tk(u^N −ϕ)

≤ Z t

0

Z

Ω

fNTk(u^N−ϕ) + Z t

0

Z

∂Ω

gNTk(u^N−ϕ), (33) wherefN(t, x) =fn(x),gN(t, x) =gn(x)∀t∈]tⁿ⁻¹, tⁿ], n= 1, ..., N.

Let (tⁿ=nτN)^N_n=1 and (t^m=mτM)^M_m=1 be two partitions of interval [0, T] and let u^N(t), u^N(t)

, u^M(t), 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

Z t 0

∂(u^N−u^M)

∂s , T1(u^N −u^M)

≤ Z t

0

Z

Ω

|fN−fM|+ Z t

0

Z

∂Ω

|gN −gM|,

that is Z

Ω

J1 u^N(t)−u^M(t)

≤

Z t 0

∂(u^N −u^M)

∂s , T1(u^N−u^M)−T1(u^N −u^M)

+ kfN −f_Mk_L1(QT)+kgN−g_Mk_L1(Σ).

However,

Z t 0

∂(u^N−u^M)

∂s , T1(u^N −u^M)−T1(u^N−u^M)

≤

∂(u^N −u^M)

∂s L¹(QT)

kT1(u^N−u^M)−T1(u^N −u^M)kL^∞(QT)

≤ 2C(T, f, g, u0)kT1(u^N−u^M)−T1(u^N −u^M)k_L^∞_(QT). Now, as

N,Mlim→∞kT1(u^N −u^M)−T₁(u^N −u^M)kL^∞(QT)= 0, we get

N,M→∞lim

Z t 0

∂(u^N −u^M)

∂s , T1(u^N −u^M)−T1(u^N−u^M)

= 0. (34) On the other hand, we have

N,M→∞lim

kfN −fMk_L1(QT)+kgN −gMk_L1(Σ)

= 0, then, we obtain

N,M→∞lim Z

Ω

J1 u^N(t)−u^M(t)

= 0. (35)

Now, using the definition ofJ_k we have Z

{|u^N−u^M|≤1}

|u^N(t)−u^M(t)|²+1 2

Z

{|u^N−u^M|≥1}

|u^N(t)−u^M(t)| ≤ Z

Ω

J1 u^N(t)−u^M(t) .

Therefore, we obtain Z

Ω

|u^N(t)−u^M(t)|= Z

{|u^N−u^M|≤1}

|u^N(t)−u^M(t)|+ Z

{|u^N−u^M|≥1}

|u^N(t)−u^M(t)|

≤C(Ω) Z

{|u^N−u^M|≤1}

|u^N(t)−u^M(t)|²

!¹2

+ Z

{|u^N−u^M|≥1}

|u^N(t)−u^M(t)|

≤C(Ω) Z

Ω

J1 u^N(t)−u^M(t) ¹₂

+ 2 Z

Ω

J1 u^N(t)−u^M(t) .

Then by (35), we conclude that (u^N)N∈Nis a Cauchy sequence inC(0, T; L¹(Ω));

Which implies that

(u^N)N∈Nconverges to u in C(0, T; L¹(Ω)). (36)

It remains to prove that the limit function uis an entropy solution of the problem (1). Sinceu^N(0) =U⁰=u0for allN ∈N, thenu(0, .) =u0.

By (33) we get Z t

0

∂u^N

∂s , Tk(u^N −ϕ)−Tk(u^N−ϕ)

+ Z t

0

Z

Ω

|Du^N|^p−2Du^NDTk(u^N −ϕ)+

Z t 0

Z

Ω

α(u^N)Tk(u^N −ϕ) + Z t

0

Z

∂Ω

γ(u^N)Tk(u^N−ϕ)≤ − Z t

0

∂ϕ

∂s, T_k(u^N −ϕ)

+ Z

Ω

Jk(u^N(0)−ϕ(0))− Z

Ω

Jk(u^N(t)−ϕ(t)) + Z t

0

Z

Ω

fNTk(u^N−ϕ)

+ Z t

0

Z

∂Ω

gNTk(u^N −ϕ). (37)

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

Nlim→∞

Z t 0

∂u^N

∂s , Tk(u^N−ϕ)−Tk(u^N−ϕ)

= 0. (38)

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

N→∞lim Z t

0

Z

Ω

|Du^N|^p−2Du^NDTk(u^N−ϕ) = Z t

0

Z

Ω

|Du|^p−2DuDTk(u−ϕ). (39)

And by Lemma 9, we deduce that u^N(t) → u(t) in L¹(Ω) for all t ∈ [0, T], which implies that

Z

Ω

J_k(u^N(t)−ϕ(t))→ Z

Ω

J_k(u(t)−ϕ(t)) ∀t∈[0, T]. (40) Finally, taking the limits asN → ∞, and using the above results, the conti- nuities ofα, γ and the facts thatf_N →f in L¹(QT), gN →g in L¹(ΣT) and T_k(u^N −ϕ)→T_k(u−ϕ) inL^∞(QT),we deduce thatuis an entropy solution of the nonlinear parabolic problem (1).

Uniqueness. Let v another entropy solution of the nonlinear parabolic prob- lem (1). Taking ϕ= T_h(u^N) as test function in (25) and letting h→ ∞, we get

Z

Ω

Jk(v(t)−u^N(t)) + Z t

0

∂u^N

∂s , Tk(v−u^N)

+ lim

h→∞II^N₁ (k, h)

+ Z t

0

Z

Ω

α(v)Tk(v−u^N) + Z t

0

Z

∂Ω

γ(v)Tk(v−u^N)

≤ Z t

0

Z

Ω

f Tk(v−u^N) + Z t

0

Z

∂Ω

gTk(v−u^N); (41) where

II^N₁(k, h) = Z t

0

Z

Ω

|Dv|^p−2DvDTk(v−Th(u^N)).

On the other hand, takingϕ=T_h(v) as a test function in the inequality (33) and taking the limit ash→ ∞,we get

Z t 0

∂u^N

∂s , Tk(u^N −v)

+ lim

h→∞II^N₂(k, h) + Z t

0

Z

Ω

α(u^N)Tk(u^N −v)+

Z t 0

Z

∂Ω

γ(u^N)Tk(u^N −v)≤ Z t

0

Z

Ω

f_NT_k(u^N −v) + Z t

0

Z

∂Ω

g_NT_k(u^N −v), (42) where

II^N₂(k, h) = Z t

0

Z

Ω

|Du^N|^p−2Du^NDTk(u^N −Th(v)).

Adding (41) and (42), we get Z

Ω

Jk(v(t)−u^N(t)) + Z t

0

∂u^N

∂s , Tk(v−u^N) +Tk(u^N −v)

+ lim

h→∞II^N(k, h) + Z t

0

Z

Ω

α(v)Tk(v−u^N) +α(u^N)Tk(u^N −v)

+ Z t

0

Z

∂Ω

γ(v)Tk(v−u^N) +γ(u^N)Tk(u^N −v)

≤ Z t

0

Z

Ω

f Tk(v−u^N) +fNTk(u^N −v) +

Z t 0

Z

∂Ω

gTk(v−u^N) +gNTk(u^N−v) ,

where

II^N(k, h) =II^N₁(k, h) +II^N₂ (k, h).

Taking the limit asN → ∞, using the above convergence results and the hy- pothesis (H1), we get

Z

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

Nlim→∞ lim

h→∞II^N(k, h)≥0. (44)

Therefore the inequality (43) becomes Z

Ω

J_k(v(t)−u(t))≤0.

i.e. Z

Ω

J_k(v(t)−u(t))

k ≤0.

However

k→0lim Jk(x)

k =|x|.

Then, by Fatou’s lamma, we get

kv(t)−u(t)k₁≤0, ∀t∈[0, T].

Remark 10 The above results can be generalized, for example if thep-Laplacian operator∆puis replaced by the operator a(., Du) defined in the theorem 2.1.

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