Electronic Journal of Differential Equations, Vol. 2011(2011), No. 03, pp. 1–26.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu

SOLVABILITY OF DEGENERATED PARABOLIC EQUATIONS WITHOUT SIGN CONDITION AND THREE UNBOUNDED

NONLINEARITIES

YOUSSEF AKDIM, JAOUAD BENNOUNA, MOUNIR MEKKOUR

Abstract. In this article, we study the problem

∂

∂tb(x, u)−div(a(x, t, u, Du)) +H(x, t, u, Du) =f in Ω×]0, T[, b(x, u)(t= 0) =b(x, u0) in Ω,

u= 0 in∂Ω×]0, T[

in the framework of weighted Sobolev spaces, withb(x, u) unbounded function
onu. The main contribution of our work is to prove the existence of a renor-
malized solution without the sign condition and the coercivity condition on
H(x, t, u, Du). The critical growth condition onH is with respect toDuand
no growth condition with respect tou. The second termf belongs toL^{1}(Q),
andb(x, u0)∈L^{1}(Ω).

1. Introduction

Let Ω be a bounded open set ofR^{N}, pbe a real number such that 2< p <∞,
Q = Ω×[0, T] and w = {wi(x) : 0 ≤ i ≤ N} be a vector of weight functions
(i.e., every component w_{i}(x) is a measurable almost everywhere strictly positive
function on Ω), satisfying some integrability conditions (see Section 2). And let
Au = −div(a(x, t, u, Du)) be a Leray-Lions operator defined from the weighted
Sobolev spaceL^{p}(0, T;W_{0}^{1,p}(Ω, w)) into its dualL^{p}^{0}(0, T;W^{−1,p}^{0}(Ω, w^{∗})).

Now, we consider the degenerated parabolic problem associated for the differen- tial equation

∂b(x, u)

∂t +Au+H(x, t, u, Du) =f in Q, u= 0 on∂Ω×]0, T[,

b(x, u)(t= 0) =b(x, u_{0}) on Ω

(1.1)

where b(x, u) is a unbounded function on u, H is a nonlinear lower order term.

Problem (1.1) is studied in [2] with f ∈ L^{p}^{0}(0, T;W^{−1,p}^{0}(Ω, w^{∗})) and under the
strong hypothesis relatively to H, more precisely they supposed that b(x, u) =u

2000Mathematics Subject Classification. A7A15, A6A32, 47D20.

Key words and phrases. Weighted Sobolev spaces; truncations; time-regularization;

renormalized solutions.

c

2011 Texas State University - San Marcos.

Submitted June 28, 2010. Published January 4, 2011.

1

and the nonlinearityH satisfying the sign condition

H(x, t, s, ξ)s≥0 (1.2)

and the growth condition of the form

|H(x, t, s, ξ)| ≤b(s)X^{N}

i=1

w_{i}(x)|ξ_{i}|^{p}+c(x, t)

. (1.3)

In the case where the second membref ∈L^{1}(Q) , (1.1) is studied in [3].

It is our purpose to prove the existence of renormalized solution for (1.1) in the setting of the weighted Sobolev space without the sign condition (1.2), and without the following coercivity condition

|H(x, t, s, ξ)| ≥β

N

X

i=1

w_{i}(x)|ξ_{i}|^{p} for|s| ≥γ, (1.4)
our growth condition on H is simpler than (1.3) it is a growth with respect to
Du and no growth condition with respect to u(see assumption (H3) below), the
second term f belongs toL^{1}(Q). Note that our paper generalizes [2, 3]. The case
H(x, t, u, Du) = div(φ(u)) is studied by Redwane in the classical Sobolev spaces
W^{1,p}(Ω) and in Orlicz spaces; see [15, 16].

The notion of renormalized solution was introduced by Diperna and Lions [8] in
their study of the Boltzmann equation. This notion was then adapted to an elliptic
version of (1.1) by Boccardo et al [5] when the right hand side is in W^{−1,p}

0

(Ω),
by Rakotoson [14] when the right hand side is inL^{1}(Ω), and finally by Dal Maso,
Murat, Orsina and Prignet [7] for the case of right hand side is general measure
data.

Our article can be see as a continuation of [4] in the case where b(x, u) = u, a(x, t, s, ξ) is independent ofsandH = 0. The plan of the article is as follows. In Section 2 we give some preliminaries and the definition of weighted Sobolev spaces.

In Section 3 we make precise all the assumptions onb, a, H, f, b(x, u0). In section 4 we give some technical results. In Section 5 we give the definition of a renormalized solution of (1.1) and we establish the existence of such a solution (Theorem 5.3).

Section 6 is devoted to an example which illustrates our abstract result, and finally an appendix in section 7.

2. Preliminaries

Let Ω be a bounded open set of R^{N}, pbe a real number such that 2< p <∞
andw={w_{i}(x), 0≤i≤N}be a vector of weight functions; i.e., every component
wi(x) is a measurable function which is strictly positive a.e. in Ω. Further, we
suppose in all our considerations that , there exits

r_{0}>max(N, p) such thatw

−r0 r0−p

i ∈L^{1}_{loc}(Ω), (2.1)

w_{i} ∈L^{1}_{loc}(Ω), (2.2)

w

−1 p−1

i ∈L^{1}_{loc}(Ω), (2.3)

for any 0≤ i ≤N. We denote by W^{1,p}(Ω, w) the space of real-valued functions
u∈L^{p}(Ω, w0) such that the derivatives in the sense of distributions fulfill

∂u

∂xi

∈L^{p}(Ω, w_{i}) fori= 1, . . . , N.

Which is a Banach space under the norm
kuk_{1,p,w}=hZ

Ω

|u(x)|^{p}w_{0}(x)dx+

N

X

i=1

Z

Ω

|∂u(x)

∂xi

|^{p}w_{i}(x)dxi1/p

. (2.4)
Condition (2.2) implies thatC_{0}^{∞}(Ω) is a space ofW^{1,p}(Ω, w) and consequently, we
can introduce the subspaceV =W_{0}^{1,p}(Ω, w) ofW^{1,p}(Ω, w) as the closure ofC_{0}^{∞}(Ω)
with respect to the norm (2.4). Moreover, condition (2.3) implies thatW^{1,p}(Ω, w)
as well asW_{0}^{1,p}(Ω, w) are reflexive Banach spaces.

We recall that the dual space of weighted Sobolev spacesW_{0}^{1,p}(Ω, w) is equivalent
to W^{−1,p}^{0}(Ω, w^{∗}), where w^{∗} = {w_{i}^{∗} = w^{1−p}_{i} ^{0}, i = 0, . . . , N} and where p^{0} is the
conjugate ofp; i.e., p^{0}= _{p−1}^{p} , (see [11]).

3. Basic assumptions

Assumption (H1). For 2≤p <∞, we assume that the expression
k|u|kV =X^{N}

i=1

Z

Ω

|∂u(x)

∂xi

|^{p}wi(x)dx^{1/p}

(3.1) is a norm defined on V which is equivalent to the norm (2.4), and there exists a weight functionσon Ω such that,

σ∈L^{1}(Ω) andσ^{−1}∈L^{1}(Ω).

We assume also the Hardy inequality, Z

Ω

|u(x)|^{p}σ dx1/q

≤cX^{N}

i=1

Z

Ω

|∂u(x)

∂x_{i} |^{p}wi(x)dx1/p

, (3.2)

holds for every u∈V with a constantc >0 independent ofu, and moreover, the imbedding

W^{1,p}(Ω, w),→,→L^{p}(Ω, σ), (3.3)
expressed by the inequality (3.2) is compact. Notice that (V,k| · |k_{V}) is a uniformly
convex (and thus reflexive) Banach space.

Remark 3.1. If we assume thatw0(x)≡1 and in addition the integrability con-
dition: There existsν ∈]^{N}_{p},+∞[∩[_{p−1}^{1} ,+∞[ such that

w_{i}^{−ν} ∈L^{1}(Ω) and w

N N−1

i ∈L^{1}_{loc}(Ω) for alli= 1, . . . , N. (3.4)
Notice that the assumptions (2.2) and (3.4) imply

k|uk|=X^{N}

i=1

Z

Ω

|∂u

∂x_{i}|^{p}wi(x)dx1/p

, (3.5)

which is a norm defined on W_{0}^{1,p}(Ω, w) and its equivalent to (2.4) and that, the
imbedding

W_{0}^{1,p}(Ω, w),→L^{p}(Ω), (3.6)

is compact for all 1≤q≤p^{∗}_{1} ifpν < N(ν+ 1) and for allq≥1 ifpν ≥N(ν+ 1)
wherep1=_{ν+1}^{pν} andp^{∗}_{1} is the Sobolev conjugate ofp1; see [10, pp 30-31].

Assumption (H2).

b: Ω×R→R is a Carath´eodory function. (3.7)
such that for everyx∈Ω,b(x, .) is a strictly increasingC^{1}-function withb(x,0) = 0.

Next, for anyk >0, there existsλk >0 and functionsAk∈L^{1}(Ω) andBk∈L^{p}(Ω)
such that

λk≤ ∂b(x, s)

∂s ≤Ak(x) and Dx

∂b(x, s)

∂s

≤Bk(x) (3.8) for almost every x∈Ω, for every s such that |s| ≤ k , we denote byDx ∂b(x,s)

∂s

the gradient of ^{∂b(x,s)}_{∂s} defined in the sense of distributions. For i= 1, . . . , N,

|ai(x, t, s, ξ)| ≤βw^{1/p}_{i} (x)[k(x, t) +σ^{1/p}^{0}|s|^{q/p}^{0}+

N

X

j=1

w^{1/p}_{j} ^{0}(x)|ξj|^{p−1}], (3.9)
for a.e. (x, t)∈Q,all (s, ξ)∈R×R^{N}, some function k(x, t)∈L^{p}^{0}(Q) and β >0.

Hereσandq are as in (H1).

[a(x, t, s, ξ)−a(x, t, s, η)](ξ−η)>0 for all (ξ, η)∈R^{N} ×R^{N}, (3.10)
a(x, t, s, ξ).ξ≥α

N

X

i=1

w_{i}|ξ_{i}|^{p}, (3.11)

Whereαis a strictly positive constant.

Assumption (H3). Furthermore, letH(x, t, s, ξ) : Ω×[0, T]×R×R^{N} →Rbe a
Carath´eodory function such that for a.e (x, t)∈Q and for alls∈R, ξ ∈R^{N}, the
growth condition

|H(x, t, s, ξ)| ≤γ(x, t) +g(s)

N

X

i=1

w_{i}(x)|ξi|^{p} (3.12)
is satisfied, whereg:R→R^{+}is a continuous positive positive function that belongs
toL^{1}(R), whileγ(x, t) belongs toL^{1}(Q).

We recall that, fork >1 andsinR, the truncation is defined as Tk(s) =

(s if|s| ≤k
k_{|s|}^{s} if|s|> k.

4. Some technical results

Characterization of the time mollification of a function u. To deal with time derivative, we introduce a time mollification of a function u belonging to a some weighted Lebesgue space. Thus we define for allµ≥0 and all (x, t)∈Q,

uµ=µ Z t

∞

˜

u(x, s) exp(µ(s−t))ds
where ˜u(x, s) =u(x, s)χ_{(0,T)}(s).

Proposition 4.1 ([2]). (1) If u ∈ L^{p}(Q, wi) then uµ is measurable in Q and

∂uµ

∂t =µ(u−u_{µ})and,

kuµk_{L}p(Q,wi)≤ kuk_{L}p(Q,wi).
(2) Ifu∈W_{0}^{1,p}(Q, w), thenuµ→uin W_{0}^{1,p}(Q, w)asµ→ ∞.

(3) Ifu_{n} →uinW_{0}^{1,p}(Q, w), then(u_{n})_{µ}→u_{µ} in W_{0}^{1,p}(Q, w).

Some weighted embedding and compactness results. In this section we es-
tablish some embedding and compactness results in weighted Sobolev spaces, some
trace results, Aubin’s and Simon’s results [17]. LetV =W_{0}^{1,p}(Ω, w),H =L^{2}(Ω, σ)
and let V^{∗} = W^{−1,p}^{0}, with (2 ≤ p < ∞). Let X = L^{p}(0, T;W_{0}^{1,p}(Ω, w)). The
dual space of X is X^{∗} =L^{p}^{0}(0, T, V^{∗}) where ^{1}_{p} +_{p}^{1}0 = 1 and denoting the space
W_{p}^{1}(0, T, V, H) ={v∈X:v^{0} ∈X^{∗}} endowed with the norm

kuk_{W}1

p =kukX+ku^{0}kX^{∗},

which is a Banach space. Hereu^{0} stands for the generalized derivative ofu; i.e.,
Z T

0

u^{0}(t)ϕ(t)dt=−
Z T

0

u(t)ϕ^{0}(t)dt for allϕ∈C_{0}^{∞}(0, T).

Lemma 4.2 ([18]). (1) The evolution tripleV ⊆H ⊆V^{∗} is satisfied.

(2) The imbeddingW_{p}^{1}(0, T, V, H)⊆C(0, T, H)is continuous.

(3) The imbeddingW_{p}^{1}(0, T, V, H)⊆L^{p}(Q, σ)is compact.

Lemma 4.3 ([2]). Let g∈L^{r}(Q, γ)and let gn∈L^{r}(Q, γ), with kgnk_{L}r(Q,γ)≤C,
1< r <∞. If gn(x)→g(x)a.e in Q, thengn * g inL^{r}(Q, γ)wheren→ ∞.

Lemma 4.4 ([2]). Assume that

∂vn

∂t =αn+βn in D^{0}(Q)

whereαn andβn are bounded respectively inX^{∗} and inL^{1}(Q). Ifvn is bounded in
L^{p}(0, T;W_{0}^{1,p}(Ω, w)), thenv_{n}→uinL^{p}_{loc}(Q, σ). Furtherv_{n}→v strongly inL^{1}(Q)
wheren→ ∞.

Lemma 4.5 ([2]). Assume that (H1) and (H2) are satisfied and let (u_{n}) be a
sequence in L^{p}(0, T;W_{0}^{1,p}(Ω, w))such that un * u weakly in L^{p}(0, T;W_{0}^{1,p}(Ω, w))
and

Z

Q

[a(x, t, un, Dun)−a(x, t, u, Du)][Dun−Du]dx dt→0. (4.1)
Then, un→uin L^{p}(0, T;W_{0}^{1,p}(Ω, w)).

Definition 4.6. A monotone mapT :D(T)→X^{∗} is called maximal monotone if
its graph

G(T) ={(u, T(u))∈X×X^{∗} for allu∈D(T)}

is not a proper subset of any monotone set inX×X^{∗}. Let us consider the operator

∂

∂t which induces a linear mapLfrom the subsetD(L) ={v∈X :v^{0}∈X^{∗}, v(0) =
0} ofX into X^{∗} by

hLu, viX = Z T

0

hu^{0}(t), v(t)Vdti u∈D(L), v∈X
Lemma 4.7 ([18]). Lis a closed linear maximal monotone map.

In our study we deal with mappings of the formF =L+S whereL is a given
linear densely defined maximal monotone map from D(L)⊂X toX^{∗} andS is a
bounded demicontinuous map of monotone type fromX toX^{∗}.

Definition 4.8. A mappingSis called pseudo-monotone withun* u,Lun* Lu and limn→∞suphS(un), un−ui ≤0, we have

n→∞lim suphS(un), un−ui= 0
andS(u_{n})* S(u) asn→ ∞.

5. Main results Consider the problem

b(x, u_{0})∈L^{1}(Ω), f ∈L^{1}(Q)

∂b(x, u)

∂t −div(a(x, t, u, Du)) +H(x, t, u, Du) =f in Q u= 0 on∂Ω×]0, T[,

b(x, u)(t= 0) =b(x, u_{0}) on Ω.

(5.1)

Definition 5.1. Let f ∈ L^{1}(Q) and b(x, u0) ∈ L^{1}(Ω). A real-valued function u
defined onQis a renormalized solution of problem 5.1 if

Tk(u)∈L^{p}(0, T;W_{0}^{1,p}(Ω, w)) for allk≥0 andb(x, u)∈L^{∞}(0, T;L^{1}(Ω));, (5.2)
Z

{m≤|u|≤m+1}

a(x, t, u, Du)Du dx dt→0 asm→+∞; (5.3)

∂BS(x, u)

∂t −div (S^{0}(u)a(x, t, u, Du))

+S^{00}(u)a(x, t, u, Du)Du+H(x, t, u, Du)S^{0}(u)

=f S^{0}(u) in D^{0}(Q);

(5.4)

for all functionsS∈W^{2,∞}(R) which is piecewiseC^{1}and such thatS^{0}has a compact
support inR, whereBS(x, z) =Rz

0

∂b(x,r)

∂r S^{0}(r)drand

BS(x, u)(t= 0) =BS(x, u0) in Ω. (5.5)
Remark 5.2. Equation (5.4) is formally obtained through pointwise multiplication
of (5.1) by S^{0}(u). However, while a(x, t, u, Du) and H(x, t, u, Du) does not in
general make sense in (5.1), all the terms in (5.1) have a meaning inD^{0}(Q).

Indeed, ifM is such thatsuppS^{0} ⊂[−M, M], the following identifications are made
in (5.4):

• S(u) belongs toL^{∞}(Q) sinceS is a bounded function.

• S^{0}(u)a(x, t, u, Du) identifies with S^{0}(u)a(x, t, TM(u), DTM(u)) a.e. in Q.

Since |TM(u)| ≤ M a.e. in Q and S^{0}(u) ∈L^{∞}(Q), we obtain from (3.9)
and (5.2) that

S^{0}(u)a(x, t, TM(u), DTM(u))∈

N

Y

i=1

L^{p}^{0}(Q, w^{∗}_{i})

• S^{00}(u)a(x, t, u, Du)Duidentifies withS^{00}(u)a(x, t, TM(u), DTM(u))DTM(u)
and

S^{00}(u)a(x, t, TM(u), DTM(u))DTM(u)∈L^{1}(Q).

• S^{0}(u)H(x, t, u, Du) identifies with S^{0}(u)H(x, t, TM(u), DTM(u)) a.e in Q.

Since|TM(u)| ≤M a.e inQandS^{0}(u)∈L^{∞}(Q), we obtain from (3.9) and
(3.12) that

S^{0}(u)H(x, t, T_{M}(u), DT_{M}(u))∈L^{1}(Q).

• S^{0}(u)f belongs toL^{1}(Q).

The above considerations show that (5.4) holds inD^{0}(Q) and that

∂B_{S}(x, u)

∂t ∈L^{p}^{0}(0, T;W^{−1, p}^{0}(Ω, w^{∗}_{i})) +L^{1}(Q).

Due to the properties of S and (5.4), ^{∂S(u)}_{∂t} ∈ L^{p}^{0}(0, T;W^{−1, p}^{0}(Ω, w^{∗}_{i})) +L^{1}(Q),
which implies thatS(u)∈C^{0}([0, T];L^{1}(Ω)) so that the initial condition (5.5) makes
sense, since, due to the properties ofS (increasing) and (6.1), we have

B_{S}(x, r)−B_{S}(x, r^{0})

≤A_{k}(x)

S(r)−S(r^{0})

for allr, r^{0}∈R. (5.6)
Theorem 5.3. Letf ∈L^{1}(Q)andb(x, u0)∈L^{1}(Ω). Assume that(H1)–(H3)hold.

Then, there exists at least one renormalized solution u of problem (5.1) (in the sense of Definition 5.1).

The proof of this theorem is done in four steps.

Step 1: Approximate problem and a priori estimates. For n > 0, let us define the following approximation ofb, H, f andu0;

b_{n}(x, r) =b(x, T_{n}(r)) +1

nr forn >0, (5.7)
In view of (5.7),b_{n}is a Carath´eodory function and satisfies (6.1), there existλ_{n} >0
and functionsA_{n}∈L^{1}(Ω) andB_{n}∈L^{p}(Ω) such that

λ_{n} ≤∂b_{n}(x, s)

∂s ≤A_{n}(x) and

D_{x}∂b_{n}(x, s)

∂s

≤B_{n}(x)
a.e. in Ω,s∈R.

Hn(x, t, s, ξ) = H(x, t, s, ξ)

1 +_{n}^{1}|H(x, t, s, ξ)|χΩn.

Note that Ω_{n} is a sequence of compacts covering the bounded open set Ω andχ_{Ω}_{n}
is its characteristic function.

fn∈L^{p}^{0}(Q), and fn →f a.e. inQand strongly inL^{1}(Q) asn→+∞,
(5.8)
u0n∈D(Ω), kbn(x, u0n)k_{L}1 ≤ kb(x, u0)k_{L}1, (5.9)
bn(x, u0n)→b(x, u0) a.e. in Ω and strongly inL^{1}(Ω). (5.10)
Let us now consider the approximate problem:

∂b_{n}(x, u_{n})

∂t −div(a(x, t, u_{n}, Du_{n})) +H_{n}(x, t, u_{n}, Du_{n}) =f_{n} inD^{0}(Q),
un= 0 in (0, T)×∂Ω,

bn(x, un(t= 0)) =bn(x, u0n).

(5.11)

Note thatHn(x, t, s, ξ) satisfies the following conditions

|Hn(x, t, s, ξ)| ≤H(x, t, s, ξ) and |Hn(x, t, s, ξ)| ≤n.

For allu, v∈L^{p}(0, T;W_{0}^{1,p}(Ω, w)),

Z

Q

H_{n}(x, t, u, Du)v dx dt

≤Z

Q

|Hn(x, t, u, Du)|^{q}^{0}σ^{−}^{q}

0

q dx dt1/q^{0}Z

Q

|v|^{q}σ dx dt1/q

≤n Z T

0

Z

Ω_{n}

σ^{1−q}^{0}dx^{1/q}^{0}

dtkvkL^{q}(Q,σ)

≤Cnkvk_{L}p(0,T;W_{0}^{1,p}(Ω,w)).

Moreover, sincefn∈L^{p}^{0}(0, T;W^{−1,p}^{0}(Ω, w^{∗})), proving existence of a weak solution
u_{n}∈L^{p}(0, T;W_{0}^{1,p}(Ω, w)) of (5.11) is an easy task (see e.g. [13],[2]).

Letϕ∈ L^{p}(0, T;W_{0}^{1,p}(Ω, w))∩L^{∞}(Q) withϕ >0, choosing v = exp(G(un))ϕ
as test function in 5.11 whereG(s) =Rs

0 g(r)

α dr(the functiong appears in (3.12)).

We have Z

Q

∂bn(x, un)

∂t exp(G(un))ϕ dx dt+ Z

Q

a(x, t, un, Dun)D(exp(G(un))ϕ)dx dt

= Z

Q

Hn(x, t, un, Dun) exp(G(un))ϕ dx dt+ Z

Q

fnexp(G(un))ϕ dx dt.

In view of (3.12), we obtain Z

Q

∂bn(x, un)

∂t exp(G(u_{n}))ϕ dx dt
+

Z

Q

a(x, t, u_{n}, Du_{n})Du_{n}g(un)

α exp(G(u_{n}))ϕ dx dt
+

Z

Q

a(x, t, u_{n}, Du_{n}) exp(G(u_{n}))Dϕ dx dt

≤ Z

Q

γ(x, t) exp(G(un))ϕ dx dt+ Z

Q

g(un)

N

X

i=1

∂un

∂x_{i}

wiexp(G(un))ϕdxdt +

Z

Q

fnexp(G(un))ϕ dx dt.

By (3.11), we obtain Z

Q

∂bn(x, un)

∂t exp(G(un))ϕ dx dt+ Z

Q

a(x, t, un, Dun) exp(G(un))Dϕ dx dt

≤ Z

Q

γ(x, t) exp(G(un))ϕ dx dt+ Z

Q

fnexp(G(un))ϕ dx dt,

(5.12)
for all ϕ∈ L^{p}(0, T;W_{0}^{1,p}(Ω, w))∩L^{∞}(Q), ϕ >0. On the other hand, taking v =
exp(−G(un))ϕas test function in (5.11) we deduce, as in (5.12), that

Z

Q

∂bn(x, un)

∂t exp(−G(un))ϕ dx dt+ Z

Q

a(x, t, un, Dun) exp(−G(un))Dϕ dx dt +

Z

Q

γ(x, t) exp(−G(un))ϕ dx dt

≥ Z

Q

fnexp(−G(un))ϕ dx dt, (5.13)

for allϕ∈L^{p}(0, T;W_{0}^{1,p}(Ω, w))∩L^{∞}(Q), ϕ >0. Let ϕ=T_{k}(u_{n})^{+}χ_{(0,τ}_{)}, for every
τ∈[0, T], in (5.12) we have,

Z

Ω

B_{k}^{n}(x, u_{n}(τ)) exp(G(u_{n}))dx+
Z

Q_{τ}

a(x, t, u_{n}, Du_{n}) exp(G(u_{n}))DT_{k}(u_{n})^{+}dx dt

≤ Z

Q_{τ}

γ(x, t) exp(G(u_{n}))T_{k}(u_{n})^{+}dx dt+
Z

Q_{τ}

f_{n}exp(G(u_{n}))T_{k}(u_{n})^{+}dx dt
+

Z

Ω

B_{k}^{n}(x, u_{0n})dx,

(5.14)
whereB^{n}_{k}(x, r) =Rr

0 Tk(s)^{+}^{∂b}^{n}_{∂s}^{(x,s)}ds. Due to this definition, we have
0≤

Z

Ω

B_{k}^{n}(x, u_{0n})dx≤k
Z

Ω

|bn(x, u_{0n})|dx≤kkb(x, u0)kL^{1}(Ω). (5.15)
Using this inequality,B^{n}_{k}(x, un)≥0 andG(un)≤^{kgk}^{L1 (R)}_{α} , we deduce

Z

Qτ

a(x, t, un, DTk(un)^{+})DTk(un)^{+}exp(G(un))dx dt

≤kexpkgk_{L}1(R)

α

ku0nk_{L}1(Ω)+kfnk_{L}1(Q)+kγk_{L}1(Q)+kbn(x, u0n)k_{L}1(Ω)

≤c1k.

Thanks to (3.11), we have α

Z

Q_{τ}
N

X

i=1

wi(x)

∂T_{k}(u_{n})^{+}

∂xi

pexp(G(un))dx dt≤c1k. (5.16) We deduce that

α Z

Q N

X

i=1

wi(x)

∂Tk(un)^{+}

∂xi

pdx dt≤c1k. (5.17)
Similarly to (5.17), we takeϕ=Tk(un)^{−}χ(0,τ)in (5.13) we deduce that

α Z

Q N

X

i=1

wi(x)

∂Tk(un)^{−}

∂x_{i}

pdx dt≤c2k (5.18) wherec2is a positive constant. Combining (5.17) and (5.18) we conclude that

kTk(u_{n})k^{p}

L^{p}(0,T;W_{0}^{1,p}(Ω,w))≤ck. (5.19)
We deduce from the above inequality, (5.14) and (5.15), that

Z

Ω

B_{k}^{n}(x, un)dx≤k(kfk_{L}1(Q)+kb(x, u0)k_{L}1(Ω))≡Ck. (5.20)
Then, Tk(un) is bounded in L^{p}(0, T;W_{0}^{1,p}(Ω, w)), and Tk(un) * vk in the space
L^{p}(0, T;W_{0}^{1,p}(Ω, w)), and by the compact imbedding (3.6) gives

Tk(un)→vk strongly inL^{p}(Q, σ) and a.e. in Q.

Letk >0 be large enough andBR be a ball of Ω, we have
kmeas({|u_{n}|> k} ∩B_{R}×[0, T])

= Z T

0

Z

{|un|>k}∩BR

|Tk(u_{n})|dx dt

≤ Z T

0

Z

B_{R}

|Tk(un)|dx dt

≤Z

Q

|Tk(u_{n})|^{p}σ dx dt^{1/p}Z T
0

Z

B_{R}

σ^{1−p}^{0}dx dt^{1/p}^{0}

≤T c_{R}Z

Q N

X

i=1

w_{i}(x)

∂T_{k}(u_{n})

∂xi

p

dx dt^{1/p}

≤ck^{1/p},
which implies

meas({|un|> k} ∩BR×[0, T])≤ c1

k^{1−}^{p}^{1}

, ∀k≥1.

So, we have

k→+∞lim (meas({|un|> k} ∩BR×[0, T])) = 0.

Now we turn to prove the almost every convergence ofunandbn(x, un). Consider
now a function non decreasing gk ∈ C^{2}(R) such that gk(s) = s for |s| ≤ ^{k}_{2} and
gk(s) = kfor |s| ≥k. Multiplying the approximate equation byg_{k}^{0}(bn(x, un)), we
obtain

∂gk(bn(x, un))

∂t −div(a(x, t, un, Dun)g^{0}_{k}(bn(x, un)))
+a(x, t, un, Dun)g_{k}^{00}(bn(x, un))Dx

∂bn(x, un)

∂s

Dun

+Hn(x, t, un, Dun)g^{0}_{k}(bn(x, un))

=fng^{0}_{k}(bn(x, un))

(5.21)

in the sense of distributions, which implies that

gk(bn(x, un)) is bounded inL^{p}(0, T;W_{0}^{1,p}(Ω, w)), (5.22)

∂gk(bn(x, un))

∂t is bounded inX^{∗}+L^{1}(Q), (5.23)
independent of nas long as k < n. Due to Definition (3.7) and (5.7) ofbn, it is
clear that

{|bn(x, un)| ≤k} ⊂ {|un| ≤k^{∗}}

as long ask < n andk^{∗} is a constant independent ofn. As a first consequence we
have

Dgk(bn(x, un)) =g^{0}_{k}(x, bn(un))Dx

∂bn(x, Tk^{∗}(un))

∂s

DTk^{∗}(un) a.e inQ (5.24)
as long ask < n. Secondly, the following estimate holds

g_{k}^{0}(b_{n}(x, u_{n}))D_{x}∂b_{n}(x, T_{k}^{∗}(u_{n}))

∂s

L^{∞}(Q)

≤ kg^{0}_{k}k_{L}^{∞}_{(Q)}
max

|r|≤k^{∗}

Dx

∂bn(x, s)

∂s

+ 1 .

As a consequence of (5.19), (5.24) we then obtain (5.22). To show that (5.23) holds, due to (5.21) we obtain

∂gk(bn(x, un))

∂t = div(a(x, t, un, Dun)g^{0}_{k}(bn(x, un)))

−a(x, t, u_{n}, Du_{n})g_{k}^{00}(b_{n}(u_{n}))D_{x}∂b_{n}(x, u_{n})

∂s

+H_{n}(x, t, u_{n}, Du_{n})g_{k}^{0}(b_{n}(x, u_{n})) +f_{n}g_{k}^{0}(b_{n}(x, u_{n})).

(5.25)

Since support of g^{0}_{k} and support of g^{00}_{k} are both included in [−k, k], un may be
replaced by Tk^{∗}(un) in each of these terms. As a consequence, each term on the
right-hand side of (5.25) is bounded either inL^{p}^{0}(0, T;W^{−1,p}^{0}(Ω, w^{∗})) or inL^{1}(Q).

Hence lemma 4.4 allows us to conclude thatgk(bn(x, un)) is compact inL^{p}_{loc}(Q, σ).

Thus, for a subsequence, it also converges in measure and almost every where in Q, due to the choice ofgk, we conclude that for eachk, the sequenceTk(bn(x, un)) converges almost everywhere inQ(since we have, for every λ >0,)

meas({

b_{n}(x, u_{n})−b_{m}(x, u_{m})

> λ} ∩B_{R}×[0, T])

≤meas({|bn(x, un)|> k} ∩BR×[0, T]) + meas({|bm(x, um)|> k} ∩BR×[0, T]) + meas({

gk(bn(x, un))−gk(bm(x, um)) > λ}).

Letε >0, then there existk(ε)>0 such that meas({

bn(x, un)−bm(x, um)

> λ} ∩BR×[0, T])≤ε

for all n, m ≥ n0(k(ε), λ, R). This proves that (bn(x, un)) is a Cauchy sequence in measure inBR×[0, T], thus converges almost everywhere to some measurable functionv. Then for a subsequence denoted againun,

u_{n}→u a.e. inQ, (5.26)

bn(x, un)→b(x, u) a.e. inQ. (5.27) We can deduce from (5.19) that

Tk(un)* Tk(u) weakly inL^{p}(0, T;W_{0}^{1,p}(Ω, w)) (5.28)
and then, the compact imbedding (3.3) gives

Tk(un)→Tk(u) strongly inL^{q}(Q, σ) and a.e. inQ.

Which implies, by using (3.9), for all k > 0 that there exists a function hk ∈ QN

i=1L^{p}^{0}(Q, w^{∗}_{i}), such that

a(x, t, T_{k}(u_{n}), DT_{k}(u_{n}))* h_{k} weakly in

N

Y

i=1

L^{p}^{0}(Q, w^{∗}_{i}). (5.29)
We now establish that b(x, u) belongs to L^{∞}(0, T;L^{1}(Ω)). Using (5.26) and
passing to the limit-inf in (5.20) asntends to +∞, we obtain that

1 k

Z

Ω

Bk(x, u)(τ)dx≤[kfk_{L}1(Q)+ku0k_{L}1(Ω)]≡C,

for almost any τ in (0, T). Due to the definition of Bk(x, s) and the fact that

1

kBk(x, u) converges pointwise to b(x, u), as k tends to +∞, shows that b(x, u)
belong toL^{∞}(0, T;L^{1}(Ω)).

Lemma 5.4. Let un be a solution of the approximate problem (5.11). Then

m→∞lim lim sup

n→∞

Z

{m≤|un|≤m+1}

a(x, t, un, Dun)Dundx dt= 0 (5.30)
Proof. Considering the function ϕ=T_{1}(u_{n}−T_{m}(u_{n}))^{−} := α_{m}(u_{n}) in (5.13) this
function is admissible sinceϕ∈L^{p}(0, T;W_{0}^{1,p}(Ω, w)) andϕ≥0. Then, we have

Z

Q

∂bn(x, un)

∂t αm(un)dx dt+ Z

{−(m+1)≤un≤−m}

a(x, t, un, Dun)Dunα^{0}_{m}(un)dx dt
+

Z

Q

f_{n}exp(−G(un))α_{m}(u_{n})dx dt

≤ Z

Q

γ(x, t) exp(−G(un))α_{m}(u_{n})dx dt.

Which, by settingB^{m}_{n}(x, r) =Rr
0

∂b_{n}(x,s)

∂s αm(s)ds, gives Z

Ω

B^{m}_{n}(x, un)(T)dx+
Z

{−(m+1)≤un≤−m}

a(x, t, un, Dun)Dunα^{0}_{m}(un)dx dt
+

Z

Q

f_{n}exp(−G(u_{n}))α_{m}(u_{n})dx dt

≤ Z

Q

γ(x, t) exp(−G(u_{n}))α_{m}(u_{n})dx dt+
Z

Ω

B_{n}^{m}(x, u_{0n})dx.

SinceB_{n}^{m}(x, u_{n})(T)≥0 and by Lebesgue’s theorem, we have

m→∞lim lim

n→∞

Z

Q

f_{n}exp(−G(u_{n}))α_{m}(u_{n})dx dt= 0. (5.31)
Similarly, sinceγ∈L^{1}(Ω), we obtain

m→∞lim lim

n→∞

Z

Q

γexp(−G(u_{n}))α_{m}(u_{n})dx dt= 0. (5.32)
We conclude that

m→∞lim lim sup

n→∞

Z

{−(m+1)≤un≤−m}

a(x, t, un, Dun)Dundx dt= 0. (5.33)
On the other hand, let ϕ = T1(un −Tm(un))^{+} as test function in (5.12) and
reasoning as in the proof of (5.33) we deduce that

m→∞lim lim sup

n→∞

Z

{m)≤un≤m+1}

a(x, t, un, Dun)Dundx dt= 0. (5.34)

Thus (5.30) follows from (5.33) and (5.34).

Step 2: Almost everywhere convergence of the gradients. This step is devoted to introduce for k ≥ 0 fixed a time regularization of the function Tk(u) in order to perform the monotonicity method. This kind of regularization has been first introduced by R. Landes (see Lemma 6 and proposition 3, p.230, and proposition 4, p.231, in[12]).

Let ψi ∈ D(Ω) be a sequence which converge strongly to u0 in L^{1}(Ω). Set
w^{i}_{µ} = (Tk(u))µ+e^{−µt}Tk(ψi) where (Tk(u))µ is the mollification with respect to
time ofT_{k}(u). Note thatw_{µ}^{i} is a smooth function having the following properties:

∂w^{i}_{µ}

∂t =µ(T_{k}(u)−w_{µ}^{i}), w^{i}_{µ}(0) =T_{k}(ψ_{i}),
w_{µ}^{i}

≤k, (5.35)

w^{i}_{µ}→T_{k}(u) in L^{p}(0, T;W_{0}^{1,p}(Ω, w)), (5.36)
asµ→ ∞. We introduce the following function of one real:

hm(s) =

1 if|s| ≤m 0 if|s| ≥m+ 1 m+ 1−s ifm≤s≤m+ 1 m+ 1 +s if −(m+ 1)≤s≤ −m wherem > k.

Letϕ= (Tk(un)−w^{i}_{µ})^{+}hm(un)∈L^{p}(0, T;W_{0}^{1,p}(Ω, w))∩L^{∞}(Q) andϕ≥0, then
we take this function in (5.12), we obtain

Z

{Tk(u_{n})−w^{i}_{µ}≥0}

∂bn(x, un)

∂t exp(G(un))(Tk(un)−w^{i}_{µ})hm(un)dx dt
+

Z

{Tk(un)−w^{i}_{µ}≥0}

a(x, t, un, Dun)D(Tk(un)−w_{µ}^{i})hm(un)dx dt

− Z

{m≤un≤m+1}

exp(G(u_{n}))a(x, t, u_{n}, Du_{n})Du_{n}(T_{k}(u_{n})−w_{µ}^{i})^{+}dx dt

≤ Z

Q

γ(x, t) exp(G(un))(Tk(un)−w^{i}_{µ})^{+}hm(un)dx dt
+

Z

Q

fnexp(G(un))(Tk(un)−w^{i}_{µ})^{+}hm(un)dx dt.

(5.37)

Observe that Z

{m≤u_{n}≤m+1}

exp(G(u_{n}))a(x, t, u_{n}, Du_{n})Du_{n}(T_{k}(u_{n})−w^{i}_{µ})^{+}dx dt

≤2k Z

{m≤un≤m+1}

a(x, t, un, Dun)Dundx dt.

Thanks to (5.30) the third integral tend to zero as nand m tend to infinity, and by Lebesgue’s theorem, we deduce that the right hand side converge to zero asn, mandµtend to infinity. Since

(Tk(un)−w_{µ}^{i})^{+}hm(un)*(Tk(u)−w^{i}_{µ})^{+}hm(u) weakly* in L^{∞}(Q), as n→ ∞,
and (Tk(u)−w^{i}_{µ})^{+}hm(u)*0 weakly* in L^{∞}(Q) asµ→ ∞.

Letεl(n, m, µ, i)l= 1, . . . , n various functions tend to zero asn, m, iand µtend to infinity.

The definition of the sequence w^{i}_{µ} makes it possible to establish the following
lemma, which will be proved in the Appendix.

Lemma 5.5. [14] Fork≥0 we have Z

{Tk(u_{n})−w_{µ}^{i}≥0}

∂b_{n}(x, u_{n})

∂t exp(G(u_{n}))(T_{k}(u_{n})−w^{i}_{µ})h_{m}(u_{n})dx dt≥ε(n, m, µ, i)
(5.38)
On the other hand, the second term of left hand side of (5.37) reads as follows
Z

{T_{k}(un)−w_{µ}^{i}≥0}

a(x, t, u_{n}, Du_{n})D(T_{k}(u_{n})−w^{i}_{µ})h_{m}(u_{n})dx dt

= Z

{Tk(u_{n})−w^{i}_{µ}≥0,|un|≤k}

a(x, t, Tk(un), DTk(un))D(Tk(un)−w_{µ}^{i})hm(un)dx dt

− Z

{Tk(un)−w^{i}_{µ}≥0,|un|≥k}

a(x, t, un, Dun)Dw^{i}_{µ}hm(un)dx dt.

Sincem > k,hm(un) = 0 on{|un| ≥m+ 1}, One has Z

{Tk(u_{n})−w^{i}_{µ}≥0}

a(x, t, un, Dun)D(Tk(un)−w_{µ}^{i})hm(un)dx dt

= Z

{Tk(un)−w_{µ}^{i}≥0}

a(x, t, Tk(un), DTk(un))D(Tk(un)−w^{i}_{µ})hm(un)dx dt

− Z

{Tk(u_{n})−w^{i}_{µ}≥0,|un|≥k}

a(x, t, T_{m+1}(u_{n}), DT_{m+1}(u_{n}))Dw^{i}_{µ}h_{m}(u_{n})dx dt

=J1+J2

(5.39) In the following we pass to the limit in (5.39): first we let n tend to +∞, then µ and finally m, tend to +∞. Since a(x, t, Tm+1(un), DTm+1(un)) is bounded in QN

i=1L^{p}^{0}(Q, w^{∗}_{i}), we have that

a(x, t, Tm+1(un), DTm+1(un))hm(un)χ_{{|u}_{n}_{|>k}}→hmhm(u)χ_{{|u|>k}}

strongly inQN

i=1L^{p}^{0}(Q, w^{∗}_{i}) asntends to infinity, it follows that
J2=

Z

{Tk(un)−w_{µ}^{i}≥0}

hmDw^{i}_{µ}hm(u)χ_{{|u|>k}}dx dt+ε(n)

= Z

{T_{k}(u_{n})−w_{µ}^{i}≥0}

h_{m}(DT_{k}(u)_{µ}−e^{−µt}DT_{k}(ψ_{i}))h_{m}(u)χ_{{|u|>k}}dx dt+ε(n).

By lettingµ→+∞,
J_{2}=

Z

{T_{k}(u_{n})−w^{i}_{µ}≥0}

h_{m}DT_{k}(u)dx dt+ε(n, µ).

Using now the termJ1 of (5.39) one can easily show that Z

{T_{k}(u_{n})−w_{µ}^{i}≥0}

a(x, t, T_{k}(u_{n}), DT_{k}(u_{n}))D(T_{k}(u_{n})−w^{i}_{µ})h_{m}(u_{n})dx dt

= Z

{Tk(u_{n})−w^{i}_{µ}≥0}

[a(x, t, Tk(un), DTk(un))−a(x, t, Tk(un), DTk(u))]

×[DTk(un)−DTk(u)]hm(un)dx dt +

Z

{T_{k}(u_{n})−w^{i}_{µ}≥0}

a(x, t, T_{k}(u_{n}), DT_{k}(u))(DT_{k}(u_{n})−DT_{k}(u))h_{m}(u_{n})dx dt
+

Z

{Tk(u_{n})−w^{i}_{µ}≥0}

a(x, t, Tk(un), DTk(un))DTk(u)hm(un)dx dt

− Z

{Tk(un)−w^{i}_{µ}≥0}

a(x, t, Tk(un), DTk(un))Dw_{µ}^{i}hm(un)dx dt

=K_{1}+K_{2}+K_{3}+K_{4}.

(5.40) We shall go to the limit asnandµ→+∞in the three integrals of the right-hand side. Starting withK2, we have by lettingn→+∞,

K2=ε(n). (5.41)

AboutK3, we have by lettingn→+∞and using (5.29), K3=

Z

{Tk(u_{n})−w^{i}_{µ}≥0}

hkDTk(u)hm(u)χ_{{|u|>k}}dx dt+ε(n)
By lettingµ→+∞,

K_{3}=
Z

{T_{k}(u_{n})−w^{i}_{µ}≥0}

h_{k}DT_{k}(u)dx dt+ε(n, µ). (5.42)
ForK4we can write

K_{4}=−
Z

{T_{k}(un)−w^{i}_{µ}≥0}

h_{k}Dw^{i}_{µ}h_{m}(u)dx dt+ε(n),
By lettingµ→+∞,

K4=− Z

{Tk(un)−w^{i}_{µ}≥0}

hkDTk(u)dx dt+ε(n, µ). (5.43) We then conclude that

Z

{Tk(u_{n})−w_{µ}^{i}≥0}

a(x, t, Tk(un), DTk(un))D(Tk(un)−w^{i}_{µ})hm(un)dx dt

= Z

{Tk(un)−w^{i}_{µ}≥0}

[a(x, t, Tk(un), DTk(un))−a(x, t, Tk(un), DTk(u))]

×[DT_{k}(u_{n})−DT_{k}(u)]h_{m}(u_{n})dx dt+ε(n, µ).

On the other hand, we have Z

{T_{k}(u_{n})−w_{µ}^{i}≥0}

[a(x, t, T_{k}(u_{n}), DT_{k}(u_{n}))−a(x, t, T_{k}(u_{n}), DT_{k}(u))]

×[DTk(un)−DTk(u)]dx dt

= Z

{Tk(un)−w^{i}_{µ}≥0}

[a(x, t, Tk(un), DTk(un))−a(x, t, Tk(un), DTk(u))]

×[DT_{k}(u_{n})−DT_{k}(u)]h_{m}(u_{n})dx dt
+

Z

{Tk(u_{n})−w^{i}_{µ}≥0}

a(x, t, Tk(un), DTk(un))(DTk(un)−DTk(u))

×(1−hm(un))dx dt

− Z

{T_{k}(u_{n})−w^{i}_{µ}≥0}

a(x, t, T_{k}(u_{n}), DT_{k}(u))(DT_{k}(u_{n})−DT_{k}(u))

×(1−hm(un))dx dt.

(5.44)

Since h_{m}(u_{n}) = 1 in {|u_{n}| ≤ m} and {|u_{n}| ≤ k} ⊂ {|u_{n}| ≤ m} for m large
enough, we deduce from (5.44) that

Z

{Tk(u_{n})−w_{µ}^{i}≥0}

[a(x, t, Tk(un), DTk(un))−a(x, t, Tk(un), DTk(u))]

×[DTk(un)−DTk(u)]dx dt

= Z

{Tk(u_{n})−w^{i}_{µ}≥0}

[a(x, t, T_{k}(u_{n}), DT_{k}(u_{n}))−a(x, t, T_{k}(u_{n}), DT_{k}(u))]

×[DTk(un)−DTk(u)]hm(un)dx dt +

Z

{Tk(un)−w^{i}_{µ}≥0,|un|>k}

a(x, t, Tk(un), DTk(u))DTk(u)(1−hm(un))dx dt.

It is easy to see that the last terms of the last equality tend to zero asn→+∞, which implies

Z

{Tk(u_{n})−w_{µ}^{i}≥0}

[a(x, t, T_{k}(u_{n}), DT_{k}(u_{n}))−a(x, t, T_{k}(u_{n}), DT_{k}(u))]

×[DTk(un)−DTk(u)]dx dt

= Z

{T_{k}(un)−w^{i}_{µ}≥0}

[a(x, t, T_{k}(u_{n}), DT_{k}(u_{n}))−a(x, t, T_{k}(u_{n}), DT_{k}(u))]

×[DT_{k}(u_{n})−DT_{k}(u)]h_{m}(u_{n})dx dt+ε(n)

Combining (5.38), (5.40), (5.41), (5.42), (5.43) and (5.44), we obtain Z

{Tk(u_{n})−w^{i}_{µ}≥0}

[a(x, t, Tk(un), DTk(un))−a(x, t, Tk(un), DTk(u))]

×[DTk(un)−DTk(u)]dx dt≤ε(n, µ, m)

(5.45) Passing to the limit in (5.45) asnandmtend to infinity, we obtain

n→∞lim Z

{Tk(un)−w^{i}_{µ}≥0}

[a(x, t, Tk(un), DTk(un))−a(x, t, Tk(un), DTk(u))]

×[DT_{k}(u_{n})−DT_{k}(u)]dx dt= 0.

(5.46)