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 Math´
ematiques Appliqu´
ees et Industrielles
Facult´
e 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:
∂u
∂t − △pu+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 − △pu+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 inRd, 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
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−), α′>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−τ△pUn+τ α(Un) =τ fn+Un−1 inΩ,
|DUn|p−2∂Un ∂η +γ(U
n) =g
n on ∂Ω,
U0=u0 inΩ,
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 Rd, |Ω| denotes its measure, the norm in Lp(Ω) is denoted byk.kp andk.k1,p denotes the norm in the Sobolev spaceW1,p(Ω), Ci andCwill denote various positive constants. For a Banach spaceX anda < b, Lp(a, b;X) denotes the space of the measurable functionsu: [a, b]→X such that
Z b
a
ku(t)kpX
!1p
:=kukLp(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∈W1,p(Ω),we denote byτ uoruthe trace ofuon∂Ω in the usual sense. In ([4]) the authors introduce the following spaces
• Tloc1,1(Ω) =nu: Ω→R measurable : Tk(u)∈Wloc1,1(Ω), f or all k >0
o
• Tloc1,p(Ω) =nu∈ Tloc1,1(Ω) :DTk(u)∈Lploc(Ω), f or all k >0
o
,
• T1,p(Ω) =nu∈ T1,p
loc(Ω) : DTk(u)∈Lp(Ω), f or all k >0
o
.
For bounded Ω′s, we have
T1,p(Ω) =
u: Ω→R measurable Tk(u)∈W1,p(Ω), f or all k >0 .
Following [4], It is possible to give a sense to the derivativeDu of a function
u ∈ Tloc1,p(Ω), generalizing the usual concept of weak derivative in Wloc1,1(Ω), thanks to the following result
Lemma 1 ([4]) For everyu∈ Tloc1,p(Ω)there exist a unique measurable function
v: Ω→Rd such that
DTk(u) =v1{|v|<k} a.e,
where1B is the characteristic function of the measurable setB ⊂Rd.
Furthermore,u∈Wloc1,1(Ω) if and only if v ∈L1
loc(Ω), and then v ≡Du in the
usual weak sense.
We apply also the setsTtr1,p(Ω) introduced in [2] as being the subset of functions inT1,p(Ω) for which a generalized notion of trace may be defined. More precisely
u∈ Ttr1,p(Ω) ifu∈ T1,p(Ω) and there exist a sequence (u
n)n∈N inW1,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∈ Ttr1,p(Ω), the trace of uon∂Ω is denoted bytr(u) or u,the operatortr(.) satisfied the following properties
i) if u∈ Ttr1,p(Ω), thenτ Tk(u) =Tk(tr(u)),∀k >0,
ii) if ϕ ∈ W1,p(Ω)∩L∞(Ω), then ∀u∈ T1,p
tr (Ω), we have u−ϕ ∈ T
1,p tr (Ω)
andtr(u−ϕ) =tr(u)−τ ϕ.
In the case whereu∈W1,p(Ω), tr(u) coincides withτ u. Obviously, we have
W1,p(Ω)⊂ Ttr1,p(Ω)⊂ T1,p(Ω).
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∈Ttr1,p(Ω) to the problem
−div[a(., Du)] +β(u) =f inΩ
∂u
∂νa+γ(τ u) =g on∂Ω
(4)
i.e. ∀ϕ∈ C∞
0 (RN)
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×Rd→Rd,(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∈Rd.
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∈Rd.
3) ha(x, ξ1)−a(x, ξ2), ξ1−ξ2i>0,if ξ16=ξ2, for a.e. x∈Ω.
4) There exist someh0∈Lp′
(Ω), p′= p−p1 and a constantA2>0,such that
|a(x, ξ)| ≤A2 h0(x) +|ξ|p−1
f or a.e. x∈Ωand every ξ∈Rd.
3
The semi-discrete problem
By the Euler forward scheme, we consider the following system
(P n)
Un−τ△pUn+τ α(Un) =τ fn+Un−1 in Ω,
|DUn|p−2∂U n
∂η +γ(U
n) =g
n on∂Ω,
U0=u0 in Ω,
where N τ = T, 1 ≤ n ≤ N and fn(.) = 1τR nτ
(n−1)τf(s, .)ds, in Ω, gn(.) =
1
τ
Rnτ
(n−1)τg(s, .)dson∂Ω.
(H1) αand γ are non decreasing continuous functions onRsuch that α(0) = γ(0) = 0,
(H2) u0∈L1(Ω), f ∈L1(Q
T) 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−τ△pu+τ α(u) =τ fn+Un−1 in Ω,
|Du|p−2∂u
∂η +γ(u) =gn on ∂Ω,
i.e. Un∈ T1,p
tr (Ω) and∀ϕ∈W1,p(Ω)∩L∞(Ω),∀k >0,we have
Z
Ω
UnTk(Un−ϕ)
Z
Ω
|DUn|p−2
DUnDTk(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−2
DU1DTk(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(Q
and
τ
Z
Ω
|DU1|p−2DU1DTk(U1) =τ
Z
Ω
|DTk(U1)|p≥0.
Thus, from inequality (7) we obtain,
0≤
Z
Ω
U1Tk(U 1)
k ≤ kfkL1(QT)+kgkL1(ΣT)+ku0k1
. (10)
On the other hand, we have for eachx∈Ω
lim k→0U
1( x)Tk(U
1(x)) k =|U
1( x)|.
Then by Fatou’s lemma, we deduce thatU1∈L1(Ω) and
kU1k1≤ kfkL1(Q
T)+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∈ T1,p
tr (Ω)∩L1(Ω).
Proof. Existence. Let the problem
−τ△pu+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 inT1,p
tr (Ω).
By induction, using Lemma 3, we deduct in the same manner that for n = 1, ..., N,the problem
u−τ△pu+τ α(u) =τ fn+Un−1 in Ω,
|Du|p−2∂u
∂η +γ(u) =gn on ∂Ω,
has an entropy solutionUn inT1,p
tr (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)T
k(Un−Th(Un)) +τ
Z
∂Ω
γ(Un)T
k(Un−Th(Un))
≤
Z
Ω
(τ fn+Un−1)Tk(Un−Th(Un)) +τ
Z
∂Ω
gnTk(Un−Th(Un)). (12)
By using the definition ofTk, we have
Z
Ω
UnTk(Un−Th(Un)) =
Z
Ωh
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))1Ωh =sign(Un)1Ωh; Thus, we get
Z
Ω
UnTk(Un−Th(Un))≥0.
In the same manner, using the hypothesis (H1) we obtain
τ
Z
Ω
α(Un)Tk(Un−Th(Un)) +τ
Z
∂Ω
γ(Un)Tk(Un−Th(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 as test functions and taking the limit ash→ ∞, we get by applying Dominated Convergence Theorem that
by applying hypothesis (H1),we get
lim
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},
and we spilt
and on the other hand, from the H¨older’s inequality, we have
In the same manner, we show that:
lim h→∞
Z
{h<|v|<h+k}
|Dv|p= 0.
Hence
lim h→∞I
2
k,h≥0.
Similarly, we have
lim h→∞I
3
k,h= lim h→∞
Z
Ω3(h)
|Du|p−2DuDT
k(u−v)+
Z
Ω3(h)
|Du|p−2DuDT
k(v−hsign(u))≥0.
Finally
Ik,h4 =
Z
Ω4(h)
|Du|p−2
DuDTk(u−hsign(v)) +
Z
Ω4(h)
|Du|p−2
DuDTk(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
lim
h→∞Ik,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
kTk(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−Vnk
1= 0.
4
Stability
Now we give some a priori estimates for the discrete entropy solution (Un)
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
Proof. 1) and 2): Letϕ= 0 as test function in inequality (6) and dividing byk, we obtain Letk →0, by the properties ofTk and the Dominated Convergence Theorem we get, 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:
and
Ωh=
|Ui−sign(Ui−Ui−1)|> h ,
we obtain
Z
Ω
Ui−Ui−1
Tk Ui−Th Ui−sign(Ui−Ui−1)
≤k τkfik1+τkgikL1(∂Ω)+τkα(Ui)k1+τkγ(Ui)kL1(∂Ω).
Taking the limit ash→ ∞and using the Dominated Convergence Theorem, we get fork= 1
kUi−Ui−1k1≤τkfik1+τkgikL1(∂Ω)+τkα(Ui)k1+τkγ(Ui)kL1(∂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(Q
T)+ 2kgkL1(ΣT)+ku0k1.
4)Takingϕ= 0 as test function in inequality (6), we deduce by (8) that
τkDTk(Ui)kpp≤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)kpp≤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≥nτ. (23)
Proof. By inequality (20), we have
yn≤yn−1+τ hn,
whereyn=kUnk1and hn=kfnk1+kgnkL1(∂Ω).
On the other hand, according to the stability results of theorem 4.1, there exists
nτ>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≥nτ.
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)∈ Lp(0, T;W1,p(Ω))
for allk >0,and
Z t
0
Z
Ω
|Du|p−2DuDT
k(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, Tk(u−ϕ)
+
Z
Ω
Jk(u(0)−ϕ(0))−
Z
Ω
Jk(u(t)−ϕ(t))
+
Z t
0
Z
Ω
f Tk(u−ϕ) +
Z t
0
Z
∂Ω
gTk(u−ϕ), (25)
for allϕ∈L∞(QT)∩Lp(0, T;W1,p(Ω))∩W1,1(0, T;L1(Ω))andt∈[0, T].
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
(
uN(0) :=u0,
uN(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
uN(0) :=u0,
uN(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, uN and uN 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) uN(t)−uN(t) = (Un−Un−1)t n−t
τ .
By using the stability results of theorem 4.1, we deduce the following a priori estimates concerning the Rothe functionuN and the function uN.
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
kuN −uNkL1(Q
T)≤ 1
NC(T, u0, f, g), (28) kuNkL1(Q
T)≤C(T, u0, f, g), (29)
kuNkL1(Q
T)≤C(T, u0, f, g), (30)
k∂u N
∂t kL1(QT)≤C(T, u0, f, g), (31)
Tk(uN)
Lp(0,T,W1,p(Ω)) ≤k.C(T, u0, f, g). (32) Proof. We have
kuN −uNk L1(Q
T) =
N
X
n=1
Z tn
tn−1
kUn−Un−1k 1
(tn−t)
τ dt
= τ
2 N
X
n=1
Using inequality 4) of theorem 4.1, we deduce that
kuN−uNkL1(Q
T)≤ 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(uN)N∈Nis relatively compact in L1(QT).
This implies the existence of a subsequence of (uN)
N∈Nconverging to an element
uin L1(QT).
And by estimate (28), we deduce hence that
the sequence(uN)N∈Nconverges to u in L1(QT).
On the other hand, by (32) we have that
DTk(uN)N∈N is unif ormly bounded in L
p(Q T).
Hence there exists a subsequence, still denoted by DTk(uN)
N∈Nsuch that
DTk(uN)N∈N converges to an element V in L
p(Q T).
However
Tk(uN)converges to Tk(u)in Lp(QT).
Hence, it follows that
DTk(uN)converges to DTk(u)weakly in Lp(QT),
and by (32) we conclude that
Tk(u)∈Lp(0, T;W1,p(Ω)) f or all k >0.
We follow the same technique used in [1] to show that
uN converges to u onΣT.
Lemma 9 The sequence(uN)
N∈N converges touin C 0, T; L1(Ω)
.
Proof. Letϕ ∈ L∞(Q
T)∩Lp(0, T;W1,p(Ω))∩W1,1(0, T;L1(Ω)), we rewrite (6) in the form
Z t
0
∂uN
∂s , Tk(u
N −ϕ)
+
Z t
0
Z
Ω
|DuN|p−2DuNDT
k(uN−ϕ)
+
Z t
0
Z
Ω
α(uN)T
k(uN −ϕ) +
Z t
0
Z
∂Ω
γ(uN)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
On the other hand, we have
lim
Now, using the definition ofJk we have
Therefore, we obtain
It remains to prove that the limit function uis an entropy solution of the problem (1). SinceuN(0) =U0=u0for allN ∈N, thenu(0, .) =u0.
By same manner, as used for the proof of the equality (34), we deduce that
lim
We follow the same technique used in [19], we show that
And by Lemma 9, we deduce that uN(t) → u(t) in L1(Ω) for all t ∈ [0, T],
Finally, taking the limits asN → ∞, and using the above results, the conti-nuities ofα, γ and the facts thatfN →f in L1(QT), gN →g in L1(ΣT) and
Tk(uN −ϕ)→Tk(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= Th(uN) as test function in (25) and letting h→ ∞, we
Adding (41) and (42), we get
+
Z t
0
Z
∂Ω
γ(v)Tk(v−uN) +γ(uN)Tk(uN −v)
≤
Z t
0
Z
Ω
f Tk(v−uN) +fNTk(uN −v)+
Z t
0
Z
∂Ω
gTk(v−uN) +gNTk(uN−v),
where
IIN(k, h) =IIN1(k, h) +IIN2 (k, h).
Taking the limit asN → ∞, using the above convergence results and the hy-pothesis (H1), we get
Z
Ω
Jk(v(t)−u(t)) + lim
N→∞hlim→∞II
N(k, h)≤0. (43)
Applying the technique used in the proof of uniqueness in theorem 3.1, we deduce that
lim
N→∞hlim→∞II
N(k, h)≥0. (44)
Therefore the inequality (43) becomes
Z
Ω
Jk(v(t)−u(t))≤0.
i.e.
Z
Ω
Jk(v(t)−u(t))
k ≤0.
However
lim k→0
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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