Article-IJEE

International Journal of Evolution Equations

Volume 5, Number 4

ISSN: 1549-2907

c© 2011 Nova Science Publishers, Inc.

RENORMALIZED SOLUTIONS OF NONLINEAR

DEGENERATED PARABOLIC EQUATIONS

WITHOUT SIGN CONDITION AND L1 DATA

Youssef Akdim∗, Jaouad Bennouna

†and Mounir Mekkour

‡

Department of Mathematics Faculty of Science

Dhar-Mahraz B.P 1796 Atlas Fez, Morocco

Abstract

In this paper, we study the problem

∂u

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

u(x,0) = u0 in Ω

u = 0 in ∂Ω×]0,T[

in the framework of weighted Sobolev space. The main contribution of our work

is to prove the existence of a renormalized solution without the sign condition and

the coercivity condition on H(x, t,u,Du), the critical growth condition on H is with

respect to Du and no growth with respect to u. The second term f belongs to L1(Q)and u0 ∈ L1(Ω).

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

renormalized solutions.

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

1. Introduction

Let Ω be a bounded open set of RN , p be 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 wi(x)is a measurable almost everywhere strictly positive function on Ω), satisfying some inte-

grability conditions (see Section 2). And let Au = −div(a(x, t,u,Du)) be a Leray-Lions

operator defined from the weighted Sobolev space Lp(0,T ;W1, p0 (Ω,w)) into it’s dual

Lp′(0,T ;W−1,p′(Ω,w∗)).

Received: July 15, 2009; Accepted: September 30, 2009∗E-mail address: [email protected]†E-mail address: [email protected]‡E-mail address: [email protected]

422 Youssef Akdim, Jaouad Bennouna and Mounir Mekkour

Now, we consider the degenerated parabolic problem associated to the following differ-

ential equation∂u

∂t+Au+H(x, t,u,Du) = f in Q,

u = 0 on ∂Ω×]0,T [,

u(x,0) = u0 on Ω

(1.1)

where H is a nonlinear lower order term. The problem (1.1) is studied in [3], [2] under

the strong hypothesis relatively to H, more precisely they supposed that the nonlinearity H

satisfies 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)

(

N

∑i=1

wi(x) |ξi|p +c(x, t)

)

. (1.3)

In the case where the second membre f ∈ L1(Q), they assume also that H satisfies the

coercivity condition

|H(x, t, s,ξ)| ≥ βN

∑i=1

wi(x) |ξi|p

for |s| ≥ γ. (1.4)

It is our purpose, in this paper, to prove the existence of a renormalized solution for the

problem (1.1) in the setting of the weighted Sobolev space without the sign condition (1.2),

the coercivity condition (1.4), our growth condition on H is more simpler than (1.3) it is

a growth with respect to Du and no growth with respect to u (see assumption (H3)), the

second term f belongs to L1(Q).

The notion of renormalized solution was introduced by J. DiPerna and P. Lions [11] in

their study of the Boltzmann equation. This notion was then adapted to an elliptic version of

(1.1) by L. Boccardo et al [6] when the right hand side is in W−1,p′

(Ω), by J.M. Rakotoson

[19] when the right hand side is in L1(Ω), and finally by G. Dal Maso, F. Murat, L. Orsina

and A. Prignet [10] for the case where the right hand side is a general measure data.

For the degenerated parabolic equations the existence of weak solutions have been

proved by L. Aharouch et al [3] in the case where a is strictly monotone, H verify (1.2)-

(1.3) and f ∈ Lp′(0,T,W−1, p′(Ω,w∗)). See also the existence of renormalized solution by

Y.Akdim et al [5] in the case where a(x, t, s,ξ) is independent of s and H = 0.

The plan of the paper 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

on a, H, f and 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.

2. Preliminaries

Let Ω be a bounded open set of RN , p be a real number such that 2 < p < ∞ and w = wi(x),

0 ≤ i ≤ N be a vector of weight functions, i.e., every component wi(x) is a measurable

Renormalized Solutions of Nonlinear Degenerated Parabolic Equations 423

function which is strictly positive a.e. in Ω. Further, we suppose in all our considerations

that , there exit

r0 > max(N, p) such that w

−r0r0−p

i ∈ L1loc(Ω), (2.1)

wi ∈ L1loc(Ω), (2.2)

w−1p−1

i ∈ L1loc(Ω), (2.3)

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

∂u

∂xi

∈ Lp(Ω,wi) for i = 1, . . .,N.

Which is a Banach space under the norm

‖u‖1,p,w =[

Z

Ω|u(x)|pw0(x)dx+

N

∑i=1

Z

Ω

∣

∣

∣

∂u(x)

∂xi

∣

∣

∣

p

wi(x)dx]1/p

. (2.4)

The condition (2.2) implies that C∞0 (Ω) is a subspace of W 1,p(Ω,w) and consequently, we

can introduce the subspace V = W1,p0 (Ω,w) of W 1,p(Ω,w) as the closure of C∞

0 (Ω) with

respect to the norm (2.4). Moreover, condition (2.3) implies that W 1,p(Ω,w) as well as

W1,p0 (Ω,w) are reflexive Banach spaces.

We recall that the dual space of weighted Sobolev spaces W1,p0 (Ω,w) is equivalent to

W−1,p′(Ω,w∗), where w∗ = w∗i = w

1−p′

i , i = 0, . . . ,N and where p′ is the conjugate of p

i.e. p′ = pp−1

, (see [14]).

3. Basic Assumptions

Assumption (H1)

For 2 ≤ p < ∞, we assume that the expression

‖|u|‖V =( N

∑i=1

Z

Ω

∣

∣

∣

∂u(x)

∂xi

∣

∣

∣

p

wi(x)dx

)1/p

(3.5)

is a norm defined on V which equivalent to the norm (2.4), and there exist a weight function

σ on Ω such that,

σ ∈ L1(Ω) and σ−1 ∈ L1(Ω).

We assume also the Hardy inequality,

(

Z

Ω|u(x)|qσdx

)1/q

≤ c( N

∑i=1

Z

Ω

∣

∣

∣

∂u(x)

∂xi

∣

∣

∣

p

wi(x)dx)1/p

, (3.6)

holds for every u ∈V with a constant c > 0 independent of u, and moreover, the imbedding

W 1, p(Ω,w) →→ Lq(Ω,σ), (3.7)

expressed by the inequality (3.6) is compact. Note that (V,‖|.|‖V) is a uniformly convex

(and thus reflexive) Banach space.


Remark 3.1. If we assume that w0(x)≡ 1 and in addition the integrability condition: There

exists ν ∈]Np,+∞ [∩[ 1

p−1,+∞[ such that

w−νi ∈ L1(Ω) and w

NN−1

i ∈ L1loc(Ω) for all i = 1, . . .,N. (3.8)

Notice that the assumptions (2.2) and (3.8) imply

‖|u‖|=( N

∑i=1

Z

Ω

∣

∣

∣

∂u

∂xi

∣

∣

∣

p

wi(x)dx

)1/p

, (3.9)

which is a norm defined on W1,p0 (Ω,w) and its equivalent to (2.4) and that, the imbedding

W1,p0 (Ω,w) → Lp(Ω), (3.10)

is compact for all 1 ≤ q ≤ p∗1 if pν < N(ν + 1) and for all q ≥ 1 if pν ≥ N(ν + 1) where

p1 = pνν+1

and p∗1 is the Sobolev conjugate of p1; see [13, pp 30-31].

Assumption (H2)

For i = 1, . . .,N

|ai(x, t, s,ξ)| ≤ βw1p

i (x)[k(x, t)+σ1p′ |s|

q

p′ +N

∑j=1

w1p′

j (x)|ξ j|p−1], (3.11)

for a.e. (x, t)∈ Q,all (s,ξ)∈ R×RN , some function k(x, t)∈ Lp′(Q) and β > 0. Here σ and

q are as in (H1).

[a(x, t, s,ξ)−a(x, t, s,η)](ξ−η) > 0 for all (ξ,η) ∈ RN ×R

N , (3.12)

a(x, t, s,ξ).ξ≥ αN

∑i=1

wi|ξi|p, (3.13)

Assumption (H3)

Furthermore, let H(x, t, s,ξ) : Ω× [0,T ]×R×RN → R be a Caratheodory function such

that for a.e (x, t) ∈ Q and for all s ∈ R,ξ ∈ RN , the growth condition

|H(x, t, s,ξ)| ≤ γ(x, t)+g(s)N

∑i=1

wi(x)|ξi|p, (3.14)

is satisfied, where g : R → R+ is a continuous positive positive function that belongs to

L1(R), while γ(x, t) belongs to L1(Q).

f is an element of L1(Q), (3.15)

u0 is an element of L1(Ω). (3.16)


Where α are strictly positive constant. We recall that, for k > 1 and s in R, the truncation is

defined as,

Tk(s) =

s if |s| ≤ k

k s|s| if |s|> k.

And we define the function that will be used later

ϕk(r) =

Z r

0Tk(s)ds =

r2

2if r ≤ k

k |r|− k2

2if |r|> k.

4. Some Technical Results

Characterization of the time mollification of a function u

In order to deal with time derivative, we introduce a time mollification of a function u be-

longing to a some weighted Lebesgue space. Thus we define for all µ ≥ 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 ([3]).

1) if u ∈ Lp(Q,wi) then uµ is measurable in Q and∂uµ

∂t= µ(u−uµ) and,

∥

∥uµ

∥

∥

Lp(Q,wi)≤ ‖u‖Lp(Q,wi)

.

2) If u ∈W1,p0 (Q,w), then uµ → u in W

1,p0 (Q,w) as µ → ∞.

3) If un → u in W1,p0 (Q,w) , then (un)µ → uµ in W

1,p0 (Q,w).

Some weighted embedding and compactness results

In this section we give some embedding and compactness results in weighted Sobolev

spaces, some trace results, Aubin’s and Simon’s results [21].

Let V = W1, p0 (Ω,w), H = L2(Ω,σ) and let V ∗ = W−1,p′ , with (2 ≤ p < ∞).

Let X = Lp(0,T ;W1, p0 (Ω,w)). The dual space of X is X∗ = Lp′(0,T,V∗) where 1

p+ 1

p′=

1 and denoting the space W 1p (0,T,V,H)= v ∈ X : v′ ∈ X∗ endowed with the norm

‖u‖W 1p

= ‖u‖X +∥

∥u′∥

∥

X∗ ,

which is a Banach space. Here u′ stands for the generalized derivative of u, i.e.,

Z T

0u′(t)ϕ(t)dt = −

Z T

0u(t)ϕ′(t)dt for all ϕ ∈ C∞

0 (0,T ).

Lemma 4.2 ([22]).

1) The evolution triple V ⊆ H ⊆V ∗ is verified.

2) The imbedding W 1p (0,T,V,H)⊆C(0,T,H) is continuous.

3) The imbedding W 1p (0,T,V,H)⊆ Lp(Q,σ) is compact.


Lemma 4.3 ([3]). Let g ∈ Lr(Q,γ) and let gn ∈ Lr(Q,γ), with ‖gn‖Lr(Q,γ) ≤ C, 1 < r < ∞.

If gn(x)→ g(x) a.e in Q, then gn g in Lr(Q,γ)

Lemma 4.4 ([3]). Assume that,

∂vn

∂t= αn +βn in D′(Q)

where αn and βn are bounded respectively in X∗ and in L1(Q). If vn is bounded in

Lp(0,T ;W1, p0 (Ω,w)), then vn → u in L

ploc(Q,σ).

Further vn → v strongly in L1(Q)

Lemma 4.5 ([3]). Assume that (H1) and (H2) are satisfied and let (un) be a sequence in

Lp(0,T ;W1, p0 (Ω,w)) such that un u weakly in Lp(0,T ;W

1, p0 (Ω,w)) and

Z

Q[a(x, t,un,Dun)−a(x, t,u,Du)][Dun−Du]dxdt → 0. (4.17)

Then, un → u in Lp(0,T ;W1, p0 (Ω,w)).

Definition 4.6. A monotone map T : D(T )→ X∗ is called maximal monotone if its graph

G(T) = (u,T(u)) ∈ X ×X∗ for all u ∈ D(T)

is not a proper subset of any monotone set in X ×X∗.

Let us consider the operator ∂∂t

which induces a linear map L from the subset D(L) =v ∈ X : v′ ∈ X∗,v(0) = 0 of X into X∗ by

〈Lu,v〉X =

Z T

0〈u′(t),v(t)Vdt〉 u ∈ D(L), v ∈ X

Lemma 4.7 ([22]). L is a closed linear maximal monotone map.

In our study we deal with mappings of the form F = L + S where L is a given linear

densely defined maximal monotone map from D(L) ⊂ X to X∗ and S is a bounded demi-

continuous map of monotone type from X to X∗.

Definition 4.8. A mapping S is called pseudo-monotone with un u , Lun Lu and

limn→∞

sup〈S(un),un −u〉 ≤ 0, that we have limn→∞

sup〈S(un),un −u〉 = 0 and S(un) S(u) as

n → ∞.

5. Main Results

Consider the problem

u0 ∈ L1(Ω), f ∈ L1(Q)

∂u

∂t−div(a(x, t,u,Du))+H(x, t,u,Du)= f in Q

u = 0 on ∂Ω×]0,T [,

u(x,0) = u0 on Ω.

(5.18)


Definition 5.1. Let f ∈ L1(Q) and u0 ∈ L1(Ω). A real-valued function u defined on Q is a

renormalized solution of problem 5.18 if

Tk(u) ∈ Lp(0,T ;W1, p0 (Ω,w)) for all (k ≥ 0) and u ∈C0(0,T ;L1(Ω)); (5.19)

Z

m≤|u|≤m+1a(x, t,u,Du)Dudxdt → 0 as m → +∞; (5.20)

∂S(u)

∂t−div

(

S′(u)a(x, t,u,Du))

+S′′(u)a(x, t,u,Du)Du

+H(x, t,u,Du)S′(u) = f S′(u) in D′(Q); (5.21)

for all functions S∈W 2, ∞(R) which is piecewise C1 and such that S′ has a compact support

in R,

S(u)(t = 0) = S(u0) (5.22)

Remark 5.2. Equation (5.21) is formally obtained through pointwise multiplication of equa-

tion (5.18) by S′(u). However, while a(x, t,u,Du) and H(x, t,u,Du) do not in general make

sense in (5.18), all the terms in (5.18) have a meaning in D′(Q).

Indeed, if M is such that suppS′ ⊂ [−M,M], the following identifications are made in

(5.21):

• S(u) belongs to L∞(Q) since S is a bounded function.

• S′(u)a(x, t,u,Du) is identifies with S′(u)a(x, t,TM(u),DTM(u)) a.e in Q.

Since |TM(u)| ≤ M a.e in Q and S′(u) ∈ L∞(Q), we obtain from (3.11) and (5.19) that

S′(u)a(x, t,TM(u),DTM(u)) ∈N

∏i=1

Lp′(Q,w∗i )

• S′′(u)a(x, t,u,Du)Du identifies with S′′(u)a(x, t,TM(u),DTM(u))DTM(u) and

S′′(u)a(x, t,TM(u),DTM(u))DTM(u) ∈ L1(Q).

• S′(u)H(x, t,u,Du) identifies with S′(u)H(x, t,TM(u),DTM(u)) a.e in Q. Since |TM(u)| ≤M a.e in Q and S′(u) ∈ L∞(Q), we obtain from (3.11) and (3.14) that

S′(u)H(x, t,TM(u),DTM(u)) ∈ L1(Q)

• S′(u) f belongs to L1(Q) .

The above considerations show that equation (5.21) holds in D′(Q) and∂S(u)

∂tbelongs to

Lp′(0,T ;W−1, p′(Ω,w∗i ))+ L1(Q). It follows that u belongs to C0(0,T ;L1(Ω)) so that the

initial condition (5.22) makes sense.

Theorem 5.3. Let f ∈ L1(Q) and u0 ∈ L1(Ω). Assume that (H1), (H2) and (H3), there

exists at least a renormalized solution u (in the sense of Definition 5.1).

Proof of Theorem 5.3

Step 1: Approximate problem and a priori estimates.

For n > 0, let us define the following respective approximations of H , f and u0;

Hn(x, t, s,ξ) =H(x, t, s,ξ)

1+ 1n|H(x, t, s,ξ)|

χΩn.


Note that Ωn is a sequence of compacts covering the bounded open set Ω and χΩnis its

characteristic function.

fn ∈ Lp′(Q) and fn → f a.e in Q and strongly in L1(Q) as n → +∞, (5.23)

u0n ∈ L2(Ω) and u0n → u0 a.e in Ω and strongly in L1(Ω) as n → +∞. (5.24)

Let us now consider the approximate problem:

∂un

∂t−div(a(x, t,un,Dun))+Hn(x, t,un,Dun) = fn in D′(Q),

un = 0 in (0,T )×∂Ω,

un(t = 0) = u0n.

(5.25)

Note that Hn(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 all u,v ∈ Lp(0,T ;W1, p0 (Ω,w))

∣

∣

∣

∣

Z

QHn(x, t,u,Du)vdxdt

∣

∣

∣

∣

≤

(

Z

Q|Hn(x, t,u,Du)|q

′

σ− q′

q dxdt

)1q′(

Z

Q|v|q σdxdt

)1q

≤ n

Z T

0

(

Z

Ωn

σ1−q′dx

)1q′

dt ‖v‖Lq(Q,σ)

≤ Cn‖v‖Lp(0,T ;W

1, p0 (Ω,w))

.

Moreover, since fn ∈ Lp′(0,T ;W−1,p′(Ω,w∗)) then, proving existence of a weak solution

un ∈ Lp(0,T ;W1, p0 (Ω,w)) of (5.25) is an easy task (see e.g. [16],[3]).

Let ϕ ∈ Lp(0,T ;W1, p0 (Ω,w))∩L∞(Q) with ϕ > 0, choosing v = exp(G(un))ϕ as test

function in 5.25 where G(s) =R s

0g(r)

α dr (the function g appears in (3.14)). We have,

Z

Q

∂un

∂texp(G(un))ϕdxdt +

Z

Qa(x, t,un,Dun)D(exp(G(un))ϕ)dxdt

=Z

QHn(x, t,un,Dun)exp(G(un))ϕdxdt +

Z

Qfn exp(G(un))ϕdxdt.

In view of (3.14) we obtain

Z

Q

∂un


Z

Qa(x, t,un,Dun)Dun

g(un)

αexp(G(un))ϕdxdt

+Z

Qa(x, t,un,Dun)exp(G(un))Dϕdxdt ≤

Z

Qγ(x, t)exp(G(un))ϕdxdt

+

Z

Qg(un)

N

∑i=1

∣

∣

∣

∣

∂un

∂xi

∣

∣

∣

∣

wi exp(G(un))ϕdxdt +

Z

Qfn exp(G(un))ϕdxdt.


By using (3.13) we obtain

Z

Q

∂un


Z

Qa(x, t,un,Dun)exp(G(un))Dϕdxdt

≤Z

Qγ(x, t)exp(G(un))ϕdxdt +

Z

Qfn exp(G(un))ϕdxdt, (5.26)

for all ϕ ∈ Lp(0,T ;W1, p0 (Ω,w))∩L∞(Q),ϕ > 0.

On the other hand, taking v = exp(−G(un))ϕ as test function in (5.25) we deduce as in

(5.26) that,

Z

Q

∂un

∂texp(−G(un))ϕdxdt +

Z

Qa(x, t,un,Dun)exp(−G(un))Dϕdxdt

+

Z

Qγ(x, t)exp(−G(un))ϕdxdt ≥

Z

Qfn exp(−G(un))ϕdxdt, (5.27)

for all ϕ ∈ Lp(0,T ;W1, p0 (Ω,w))∩L∞(Q),ϕ > 0.

Let ϕ = Tk(un)+χ(0,τ) , for every τ ∈ [0,T ], in (5.26) we have,

Z τ

0

Z

Ω

∂un

∂texp(G(un))Tk(un)

+dxdt +

Z

Qτ

a(x, t,un,Dun)exp(G(un))DTk(un)+dxdt

≤Z

Qτ

γ(x, t)exp(G(un))Tk(un)+dxdt +

Z

Qτ

fn exp(G(un))Tk(un)+dxdt,

which gives

Z τ

0

Z

Ω

∂un

∂texp(G(un))Tk(un)

+dxdt +

Z

Qτ

a(x, t,un,DTk(un)+)DTk(un)

+ exp(G(un))dxdt

≤

Z

Qτ

γ(x, t)exp(G(un))Tk(un)+dxdt +

Z

Qτ

fn exp(G(un))Tk(un)+dxdt.

Since G(un) ≤‖g‖

L1(R)

α we have∣

∣ϕ1k(r)

∣

∣ ≤ k exp

(

‖g‖L1(R)

α

)

|r| where ϕ1k(r) =

R r0 Tk(s)+ exp(G(r))ds then,

Z

Q

∂un

∂tTk(un)

+ exp(G(un))dx ≥

Z

Ωϕ1

k(un(τ))dx−k exp

(

‖g‖L1(R)

α

)

‖u0n‖L1(Ω) .

Then we deduce that,

Z

Ωϕ1

k(un(τ))dx+Z

Qτ

a(x, t,un,DTk(un)+)DTk(un)

+ exp(G(un))dxdt

≤ k exp

(

‖g‖L1(R)

α

)

(‖u0n‖L1(Ω) +‖ fn‖L1(Q) +‖γ‖L1(Q)) ≤ c1k.


Thanks to (3.13) we have

Z

Ωϕ1

k(un(τ))dx+α

Z

Qτ

N

∑i=1

wi(x)

∣

∣

∣

∣

∂Tk(un)+

∂xi

∣

∣

∣

∣

p

exp(G(un))dxdt ≤ c1k. (5.28)

And due to the fact ϕ1k(un(τ))≥ 0, we deduce that,

α

Z

Q

N

∑i=1

wi(x)

∣

∣

∣

∣

∂Tk(un)+

∂xi

∣

∣

∣

∣

p

dxdt ≤ c1k. (5.29)

Similarly to the (5.29) we take ϕ = Tk(un)−χ(0,τ) in (5.27) we deduce that

α

Z

Q

N

∑i=1

wi(x)

∣

∣

∣

∣

∂Tk(un)−

∂xi

∣

∣

∣

∣

p

dxdt ≤ c2k (5.30)

where c2 is a positive constant.

Combining (5.29) and (5.30) we conclude that,

‖Tk(un)‖p

Lp(0,T ;W1, p0 (Ω,w))

≤ ck. (5.31)

Then, Tk(un) is bounded in Lp(0,T ;W1, p0 (Ω,w)), Tk(un) vk in Lp(0,T ;W

1, p0 (Ω,w)),

and by the compact imbedding (3.10) gives,

Tk(un) → vk strongly in Lp(Q,σ) and a.e in Q.

Let k > 0 large enough and BR be a ball of Ω, we have,

k meas(|un| > k∩BR × [0,T ]) =

Z T

0

Z

|un|>k∩BR

|Tk(un)|dxdt

≤

Z T

0

Z

BR

|Tk(un)|dxdt

≤

(

Z

Q|Tk(un)|

p σdxdt

)1p(

Z T

0

Z

BR

σ1−p′dxdt

)1p′

≤ T cR

(

Z

Q

N

∑i=1

wi(x)

∣

∣

∣

∣

∂Tk(un)

∂xi

∣

∣

∣

∣

p

dxdt

)1p

≤ ck1p ,

which implies that,

meas(|un| > k∩BR × [0,T ])≤c1

k1− 1

p

, ∀k ≥ 1.

So, we have

limk→+∞

(meas(|un| > k∩BR × [0,T ])) = 0.

Consider now a function non decreasing gk ∈ C2(R) such that gk(s) = s for |s| ≤ k2 and

gk(s) = k for |s| ≥ k


Multiplying the approximate equation by g′k(un), we get

∂gk(un)

∂t−div(a(x, t,un,Dun)g′k(un))+a(x, t,un,Dun)g′′k(un)

+Hn(x, t,un,Dun)g′k(un) = fng′k(un)

in the sense of distributions. This implies, thanks to the fact g′k has compact support, that

gk(un) is bounded in Lp(0,T ;W1, p0 (Ω,w)), while it’s time derivative

∂gk(un)∂t

is bounded in

X∗ +L1(Q).

Hence lemma 4.4 allows us to conclude that gk(un) is compact in Lploc(Q,σ). Thus, for

a subsequence, it also converges in measure and almost every where in Q (since we have,

for every λ > 0, )

meas(|un −um|> λ∩BR × [0,T ]) ≤ meas(|un| > k∩BR × [0,T ])

+meas(|um| > k∩BR × [0,T ])+meas(|gk(un)−gk(um)|> λ).

Let ε > 0, then, there exist k(ε) > 0 such that,

meas(|un −um|> λ∩BR × [0,T ]) ≤ ε for all n,m ≥ n0(k(ε),λ,R).

This proves that (un) is a Cauchy sequence in measure in BR×[0,T ]), thus converges almost

everywhere to some measurable function u. Then for a subsequence denoted again un,

un → u a.e in Q, we can deduce from (5.31) that,

Tk(un) Tk(u) weakly in Lp(0,T ;W1, p0 (Ω,w)) (5.32)

and then, the compact imbedding (3.7) gives,

Tk(un) → Tk(u) strongly in Lq(Q,σ) and a.e in Q.

Which implies, by using (3.11), for all k > 0 that there exists a function Λk ∈N

∏i=1

Lp′(Q,w∗i ),

such that

a(x, t,Tk(un),DTk(un)) Λk weakly inN

∏i=1

Lp′(Q,w∗i ). (5.33)

Lemma 5.4. Let un be a solution of the approximate problem (5.25). Then

limm→∞

limsupn→∞

Z

m≤|un|≤m+1a(x, t,un,Dun)Dundxdt = 0 (5.34)

Proof. Considering the following function ϕ = T1(un −Tm(un))− := αm(un) in (5.27) this

function is admissible since ϕ ∈ Lp(0,T ;W1, p0 (Ω,w)) and ϕ ≥ 0. Then, we have,

Z

Q

(

−∂un

∂t

)

exp(−G(un))αm(un)dxdt

+

Z

−(m+1)≤un≤−ma(x, t,un,Dun)Dun exp(−G(un))dxdt


+

Z

Qfn exp(−G(un))αm(un)dxdt ≤

Z

Qγ(x, t)exp(−G(un))αm(un)dxdt.

Which gives, by setting

βm(s) =Z s

0T1(r−Tm(r))−exp(−G(r))dr

(observe that αm(r)r ≥ 0 and βm(s)≤ exp

(

‖g‖L1(R)

α

)

|s|)

Z

Ωβm(un(T ))dx+

Z

−(m+1)≤un≤−ma(x, t,un,Dun)Dun exp(−G(un))dxdt

≤ exp

(

‖g‖L1(R)

α

)

[

Z

|un|>m| fn|dxdt +

Z

|un|>m|γ|dxdt +

Z

|un0|>m|u0n|dx

]

.

Since βm(s) > 0 , γ ∈ L1(Ω), fn → f in L1(Q) and u0n → u0 in L1(Ω) then by Lebesgue’s

theorem, We conclude that

limm→∞

limsupn→∞

Z

−(m+1)≤un≤−ma(x, t,un,Dun)Dundxdt = 0. (5.35)

On the other hand, let ϕ = T1(un −Tm(un))+ as test function in (5.26) and reasoning as in

the proof of (5.35) we deduce that

limm→∞

limsupn→∞

Z

m≤un≤m+1a(x, t,un,Dun)Dundxdt = 0. (5.36)

Thus (5.34) follows from (5.35) and (5.36).

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.Lands (see Lemma 6 and proposition 3,p.230, and proposition 4, p.231,

in[15]).

Let ψi ∈ D(Ω) be a sequence which converge strongly to u0 in L1(Ω).

Set wiµ = (Tk(u))µ +e−µtTk(ψi) where (Tk(u))µ is the mollification with respect to time

of Tk(u). Note that wiµ is a smooth function having the following properties:

∂wiµ

∂t= µ(Tk(u)−wi

µ), wiµ(0) = Tk(ψi),

∣

∣wiµ

∣

∣≤ k, (5.37)

wiµ → Tk(u) in Lp(0,T ;W

1, p0 (Ω,w)), (5.38)

for the modular convergence as µ → ∞.We will introduce the following function of one real variable s, which is define as:

hm(s) =

1 if |s| ≤ m

0 if |s| ≥ m+1

m+1− s if m ≤ s ≤ m+1

m+1+ s if − (m+1) ≤ s ≤ −m


where m > k.

Let ϕ = (Tk(un)−wiµ)

+hm(un) ∈ Lp(0,T ;W1, p0 (Ω,w))∩L∞(Q) and ϕ ≥ 0, then we take

this function in (5.26), we obtain

Z

Tk(un)−wiµ≥0

∂un

∂texp(G(un))(Tk(un)−wi

µ)hm(un)dxdt

+

Z

Tk(un)−wiµ≥0

a(x, t,un,Dun)D(Tk(un)−wiµ)hm(un)dxdt

−

Z

m≤un≤m+1exp(G(un))a(x, t,un,Dun)Dun(Tk(un)−wi

µ)+dxdt

≤

Z

Qγ(x, t)exp(G(un))(Tk(un)−wi

µ)+hm(un)dxdt

+

Z

Qfn exp(G(un))(Tk(un)−wi

µ)+hm(un)dxdt. (5.39)

Observe thatZ

m≤un≤m+1exp(G(un))a(x, t,un,Dun)Dun(Tk(un)−wi

µ)+dxdt

≤ 2k exp

(

‖g‖L1(R)

α

)

Z

m≤un≤m+1a(x, t,un,Dun)Dundxdt.

Tanks to (5.34) the third integral tend to zero as n and m tend to infinity, and by Lebesgue’s

theorem, we deduce that the right hand side converge to zero as n, m and µ tend to infinity.

Since

(Tk(un)−wiµ)

+hm(un) (Tk(u)−wiµ)

+hm(u) weakly* in L∞(Q), as n → ∞ and

(Tk(u)−wiµ)

+hm(u) 0 weakly* in L∞(Q) as µ → ∞.

Let εl(n,m,µ, i) l = 1, . . . ,n various functions tend to zero as n, m, i and µ tend to infinity.

On the other hand, let Hm,k(r) =R r

0 hm(s)Tk(s)exp(G(s))ds, and Hm(r) =R r

0 hm(s)exp(G(s))ds , hence Integration by parts and the use of the properties of (w)iµ yield

Z T

0

Z

x∈Ω;Tk(un)−wiµ≥0

∂un

∂thm(un)exp(G(un))(Tk(un)−wi

µ)dxdt

=

Z T

0

Z


∂un

∂thm(un)Tk(un)exp(G(un)),dxdt

−

Z T

0

Z


∂un

∂thm(un)exp(G(un))wi

µdxdt

= In1 + I

n,µ2 . (5.40)


By a standard argument we can write the first term on the right-hand side of (5.40) as

In1 =

[

Z

x∈Ω; Tk(un)−wiµ≥0

Hm,k(un)dx

]T

0

=

Z

x∈Ω; Tk(un)(T)−wiµ(T)≥0

Hm,k(Tm(un)(T ))dx

−

Z

x∈Ω; Tk(un)(0)−wiµ(0)≥0

Hm,k(Tm(un)(0))dx. (5.41)

for n > m with supphm ⊂ [−m;m]. Passing to the limit in (5.41) as n →+∞, we deduce that

In1 =

Z

x∈Ω; Tk(u)(T)−wiµ(T )≥0

Hm,k(Tm(u(T)))dx

−

Z

x∈Ω; Tk(u)(0)−wiµ(0)≥0

Hm,k(Tm(u0))dx+ε(n). (5.42)

Passing to the limit in (5.42) as i → +∞ and µ → +∞, we have

In1 =

Z

Ω[Hm,k(u(T))−Hm,k(u0)]dx+ε(n,µ, i). (5.43)

The second term on the right-hand side of (5.40) can be written as

In,µ2 = −

Z T

0

Z

x∈Ω;/Tk(un)−wiµ≥0

∂un

∂thm(un)exp(G(un))wi

µdxdt

= −

[

Z


Hm(un)wiµdx

]T

0

+

Z T

0

Z


Hm(un)∂wi

µ

∂tdxdt

= −

Z

x∈Ω; Tk(un)(T)−wiµ(T)≥0

Hm(Tm(un(T)))wiµ(T )dx

+

Z

x∈Ω; Tk(un)(0)−wiµ(0)≥0

Hm(u0n)wiµ(0)dx

+µ

Z T

0

Z


Hm(un)(Tk(u)−wiµ)dxdt. (5.44)

By passing to the limit as n tends to infinity in (5.44), we obtain

In,µ2 = −

Z

x∈Ω; Tk(u)−wiµ≥0

[Hm(u(T))wiµ(T )−Hm(u0)wi

µ(0)dx

+µ

Z


Z T

0Hm(u)(Tk(u)−wi

µ)dxdt +ε(n),

in the first terms on the right-hand side of the last equality, we deduce thatZ


[Hm(u(T))wiµ(T )−Hm(u0)wi

µ(0)dx

=

Z

Ω[Hm(u(T))(Tk(u(T))−Hm(u0)Tk(u0))dx+ε(n,µ, i).

(5.45)


The second term on the right-hand side of (5.44) can be rewritten as

µ

Z T

0

Z


Hm(u)(Tk(u)−wiµ)dxdt

= µ

Z T

0

Z


(Hm(u)−Hm(Tk(u)))(Tk(u)−wiµ)dxdt

+µ

Z T

0

Z


(Hm(Tk(u))−Hm(wiµ)(Tk(u)−wi

µ)dxdt

+µ

Z T

0

Z


Hm(wiµ)(Tk(u)−wi

µ)dxdt

= J1 +J2 +J3, (5.46)

where

J1 = µ

Z T

0

Z

x∈Ω; k−wiµ≥0;u>k

(Hm(x,u)−Hm(k))(k−wiµ)dxdt ≥ 0. (5.47)

As Hm(z) is non-decreasing for z and −k ≤ wiµ ≤ k, Tk(u)−wi

µ ≥ 0;u <−k= /0 , it follows

that

J2 ≥ 0. (5.48)

Moreover,

J3 = µ

Z T

0

Z


Hm(wiµ)(Tk(u)−wi

µ)dxdt

=Z T

0

Z


Hm(wiµ)

∂(w)iµ

∂tdxdt

=

Z

x∈Ω; Tk(u)(T)−wiµ(T )≥0

H(wiµ(T))dx−

Z

x∈Ω; Tk(u)(0)−wiµ(0)≥0

H((wiµ(0))dx, (5.49)

where H(z) =R z

0 Hm(r)dr. Also wiµ → Tk(u) a.e. in Q as i and µ tends to +∞ and |wi

µ| ≤ k.

Then Lebegue’s convergence theorem shows that

J3 =

Z

Ω(H(Tk(u(T)))−H(Tk(u0)))dx+ε(n,µ, i). (5.50)

In view of (5.45)-(5.50), one has

In,µ2 ≥−

Z

Ω[Hm(u(T))Tk(u(T))−Hm(u0)Tk(u0)]dx

+

Z


As a consequence of (5.40), (5.43) and (5.51), we deduce thatZ

(x,t)∈Ω×(0,T); Tk(u)−wiµ≥0

∂un

∂thm(un)exp(G(un))(Tk(un)−wi

µ)dxdt

≥

Z

Ω[Hm,k(u(T))−Hm,k(u0)]dx−

Z

Ω[Hm(u(T))Tk(u(T))−Hm(u0)Tk(u0)]dx

+Z



Observe that for any z ∈ R , we have

H(Tk(z)) = Hm(z)Tk(z)−Hm,k(z).

Indeed,

H(Tk(z)) =

Z Tk(z)

0Hm(r)dr

=[

r

Z r

0hm(σ)exp(G(σ))dσ

]Tk(z)

0−

Z Tk(z)

0rhm(r)exp(G(r))dr

= Tk(z)

Z Tk(z)

0hm(r)exp(G(r))dr−

Z Tk(z)

0Tk(r)hm(r)exp(G(r))dr

= Tk(z)Hm(Tk(z))−Hm,k(Tk(z)). (5.53)

This is due to the fact that for |r|< k, we have

H(Tk(r)) = Tk(r)Hm(r)−Hm,k(r),

and if r > k we have

Hm,k(r) =

Z k

0hm(σ)σexp(G(σ))dσ+k

Z r

khm(σ)exp(G(σ))dσ,

−Tk(r)Hm(r) = −k

Z k

0hm(σ)exp(G(σ))dσ−k

Z r

khm(σ)exp(G(σ))dσ,

H(k) = k

Z k

0hm(σ)exp(G(σ))dσ−k

Z k

0hm(σ)exp(G(σ))σdσ.

Combining these estimates, we conclude that

Z

Tk(un)−wiµ≥0

∂un

∂t(Tk(un)−wi

µ)hm(un)exp(G(un))dxdt ≥ ε(n, i). (5.54)

On the other hand, the second term of left hand side of (5.39) reads as

Z

Tk(un)−wiµ≥0


=Z

Tk(un)−wiµ≥0,|un|≤k

a(x, t,Tk(un),DTk(un))D(Tk(un)−wiµ)hm(un)dxdt

−

Z

Tk(un)−wiµ≥0,|un|≥k

a(x, t,un,Dun)Dwiµhm(un)dxdt.

Since m > k, hm(un) = 0 on |un| ≥ m+1, one has

Z

Tk(un)−wiµ≥0


=

Z

Tk(un)−wiµ≥0



−

Z

Tk(un)−wiµ≥0,|un|≥k

a(x, t,Tm+1(un),DTm+1(un))Dwiµhm(un)dxdt = J1 +J2. (5.55)

In the following we pass to the limit in (5.55): 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 inN

∏i=1

Lp′(Q,w∗i ), we have

that

a(x, t,Tm+1(un),DTm+1(un))hm(un)χ|un|>k → Λmχ|u|>khm(u)

strongly inN

∏i=1

Lp′(Q,w∗i ) as n tends to infinity, it follows that

J2 =

Z

Tk(u)−wiµ≥0

ΛmDwiµhm(u)χ|u|>kdxdt +ε(n)

=Z

Tk(u)−wiµ≥0

Λm(DTk(u)µ−e−µtDTk(ψi))hm(u)χ|u|>kdxdt +ε(n).

By letting µ → +∞, implies that

J2 =Z

QΛmDTk(u)dxdt +ε(n,µ).

Using now the term J1 of (5.55) one can easily show that

Z

Tk(un)−wiµ≥0


=Z

Tk(un)−wiµ≥0

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

× [DTk(un)−DTk(u)]hm(un)dxdt

+

Z

Tk(un)−wiµ≥0

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

+Z

Tk(un)−wiµ≥0

a(x, t,Tk(un),DTk(un))DTk(u)hm(un)dxdt

−

Z

Tk(un)−wiµ≥0

a(x, t,Tk(un),DTk(un))Dwiµhm(un)dxdt = K1 +K2 +K3 +K4. (5.56)

We shall go to the limit as n and µ → +∞ in the three integrals of the last side.

Starting with K2, we have by letting n → +∞

K2 = ε(n). (5.57)

About K3, we have by letting n → +∞ and using (5.33)

K3 = ε(n). (5.58)

For what concerns K4 we can write

K4 = −

Z

Tk(u)−wiµ≥0

ΛkDwiµhm(u)dxdt +ε(n),


By letting µ → +∞, implies that

K4 = −

Z

QΛkDTk(u)dxdt +ε(n,µ). (5.59)

We then conclude thatZ

Tk(un)−wiµ≥0

a(x, t,Tk(un),DTk(un))∇(Tk(un)−wiµ)hm(un)dxdt

=

Z

Tk(un)−wiµ≥0


× [DTk(un)−DTk(u)]hm(un)dxdt +ε(n,µ).

On the other hand, we have

Z

Tk(un)−wiµ≥0


× [DTk(un)−DTk(u)]dxdt

=

Z

Tk(un)−wiµ≥0



+

Z

Tk(un)−wiµ≥0

a(x, t,Tk(un),DTk(un))(DTk(un)−DTk(u))(1−hm(un))dxdt

−

Z

Tk(un)−wiµ≥0

a(x, t,Tk(un),DTk(u))(DTk(un)−DTk(u))(1−hm(un))dxdt. (5.60)

Since hm(un) = 1 in |un| ≤ m and |un| ≤ k ⊂ |un| ≤ m for m large enough, we

deduce from (5.60) that

Z

Tk(un)−wiµ≥0



=Z

Tk(un)−wiµ≥0



+

Z

Tk(un)−wiµ≥0,|un|>k

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

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

thatZ

Tk(un)−wiµ≥0




=

Z

Tk(un)−wiµ≥0


× [DTk(un)−DTk(u)]hm(un)dxdt +ε(n)

Combining (5.54), (5.56), (5.57), (5.58), (5.59) and (5.60) we obtain

Z

Tk(un)−wiµ≥0


× [DTk(un)−DTk(u)]dxdt ≤ ε(n,µ,m) (5.61)

To pass to the limit in (5.61) as n, and m tend to infinity, we obtain

limn→∞

Z

Tk(un)−wiµ≥0


× [DTk(un)−DTk(u)]dxdt = 0. (5.62)

On the other hand, take ϕ = (Tk(un)−wiµ)

−hm(un) in (5.27). Similarly, we can deduce as

in (5.62) that

limn→∞

Z

Tk(un)−wiµ≤0


× [DTk(un)−DTk(u)]dxdt = 0. (5.63)

Combining (5.62) and (5.63), we conclude

limn→∞

Z

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

× [DTk(un)−DTk(u)]dxdt = 0. (5.64)

Which implies that (by using lemma (4.5)),

Tk(un)→ Tk(u) strongly in Lp(0,T ;W1, p0 (Ω,w)) ∀k. (5.65)

Now, observe that we have, for every σ > 0

meas(x, t)∈ Ω× [0,T ] : |Dun −Du| > σ ≤ meas(x, t)∈ Ω× [0,T ] : |Dun| > k

+meas(x, t) ∈ Ω× [0,T ] : |u|> k

+meas(x, t) ∈ Ω× [0,T ] : |DTk(un)−DTk(u)|> σ

then as a consequence of (5.65) we also have, that Dun converges to Du in measure and

therefore, always reasoning for subsequence,

Dun → Du a.e in Q. (5.66)

Which implies that,

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

∏i=1

Lp′(Q,w∗i ). (5.67)


Step 3: Equi-integrability of the nonlinearity sequence.

We shall now prove that Hn(x, t,un,Dun) → H(x, t,u,Du) strongly in Ł1(Q) by using Vi-

tali’s theorem.

Since Hn(x, t,un,Dun) → H(x, t,u,Du) a.e in Q, Consider now a function ρh(s) =R s

0 g(ν)χν>hdν, take ϕ = ρh(un) =R un

0 g(s)χs>hds as test function in (5.26), we obtain

[

Z

Ωθh(un)dx

]T

0

+

Z

Qa(x, t,un,Dun)Dung(un)χun>hdxdt

≤

(

Z ∞

hg(s)χs>hds

)

exp

(

‖g‖L1(R)

α

)

(

‖γ‖L1(Q) +‖ fn‖L1(Q)

)

,

where θh(σ) =R σ

0 ρ(τ)dτ, which implies, since θh ≥ 0,

Z

Qa(x, t,un,Dun)Dung(un)χun>hdxdt

≤

(

Z ∞

hg(s)ds

)

exp

(

‖g‖L1(R)

α

)

(

‖γ‖L1(Q) +‖ fn‖L1(Q)

)

+

Z

Ωθh(u0n)dx,

using (3.13), we have

Z

un>hg(un)

N

∑i=1

wi

∣

∣

∣

∣

∂un

∂xi

∣

∣

∣

∣

p

dxdt ≤C

Z ∞

hg(s)ds.

Since g ∈ L1(R) then, we have

limh→∞

supn∈N

Z

un>hg(un)

N

∑i=1

wi

∣

∣

∣

∣

∂un

∂xi

∣

∣

∣

∣

p

dxdt = 0.

Similarly, let ϕ =R 0

ung(s)χs<−hds as a test function in (5.27), we conclude that

limh→∞

supn∈N

Z

un<−hg(un)

N

∑i=1

wi

∣

∣

∣

∣

∂un

∂xi

∣

∣

∣

∣

p

dxdt = 0.

Consequently

limh→+∞

supn∈N

Z

|un|>hg(un)

N

∑i=1

wi

∣

∣

∣

∣

∂un

∂xi

∣

∣

∣

∣

p

dxdt = 0,

which implies, for h large enough that

Z

Qg(un)

N

∑i=1

wi

∣

∣

∣

∣

∂un

∂xi

∣

∣

∣

∣

p

dxdt ≤Z

|un|<hg(un)

N

∑i=1

wi

∣

∣

∣

∣

∂un

∂xi

∣

∣

∣

∣

p

dxdt +1

≤

Z

Qg(Tk(un))

N

∑i=1

wi

∣

∣

∣

∣

∂Tk(un)

∂xi

∣

∣

∣

∣

p

dxdt +1.


Then in view of (5.65) and Vitali’s theorem, we can deduce that g(un)N

∑i=1

wi

∣

∣

∣

∂un

∂xi

∣

∣

∣

p

converges

to g(u)N

∑i=1

wi

∣

∣

∣

∂u∂xi

∣

∣

∣

p

strongly in L1(Q). Consequently, by using (3.14), we conclude that

Hn(x, t,un,Dun) → H(x, t,u,Du) strongly in Ł1(Q). (5.68)

Step 4: Convergence of (un) in C([0,T],L1(Ω))

Proposition 5.5. The sequence (un) is a Cauchy sequence in C([0,T ],L1(Ω)), moreover,

u ∈C([0,T ],L1(Ω)) and un converges to u in C([0,T ],L1(Ω)).

Proof. Let m and n be two integers, since un and um are the solutions of the prob-

lem (5.25). Taking ϕ = T1(un − um)χ[0,s[ in (5.25) ,with s ≤ T, we well have ϕ ∈

Lp(0,T ;W1, p0 (Ω,w))∩L∞(Q), hence

Z s

0

Z

Ω

∂

∂t(un −um)T1(un −um)dxdt

+

Z s

0

Z

Ω(Hn(x, t,un,Dun)−Hm(x, t,um,Dum))T1(un −um)dxdt

+

Z

Q[a(x, t,un,Dun)−a(x, t,um,Dum)]Dϕdxdt =

Z s

0

Z

Ω( fn− fm)T1(un −um)dxdt.

Which implies that,Z s

0

Z

Ω

∂

∂t(un −um)T1(un −um)dxdt

+

Z s

0

Z

Ω[a(x, t,un,Dun)−a(x, t,um,Dum)]DT1(un −um)dxdt

≤

Z

Q|Hn(x, t,un,Dun)−Hm(x, t,um,Dum)|dxdt +

Z

Q| fn− fm|dxdt. (5.69)

Recall that ϕ1 is a primitive function of T1, thus

Z

Ω

∂

∂t(un −um)T1(un −um)dx =

∂

∂t

(

Z

Ωϕ1(un−um)

)

.

Because of un ∈C([0,T ],H) we can write

Z s

0

Z

Ω

∂

∂t(un −um)T1(un −um)dxdt =

Z

Ωϕ1(un −um)(s)dx−

Z

Ωϕ1(un−um)(0)dx

Remark that, DT1(un−um) = D(un −um)χ|un−um|≤1 and also

Z s

0

Z

Ω[a(x, t,un,Dun)−a(x, t,um,Dum)]D(un−um)χ|un−um|≤1dxdt ≥ 0

due to the monotonicity of a, then we deduce from (5.69) that,

Z

Ωϕ1(un −um)(s)dx ≤

Z

Ωϕ1(u0n−u0m)dx+

Z

Q|Hn−Hm|dxdt +

Z

Q| fn − fm|dxdt.


Taking the limit as m → +∞ in (5.73) and using the estimate (5.34) show that u satisfies

(5.21) and the proof of Lemma is complete.

Let S be a function in W 2,∞(R) with S′ has a compact support in R. Let M be a positive

real number such that supp(S′) ⊂ [−M,M]. Pointwise multiplication of the approximate

equation (5.25) by S(un) leads to

∂S(un)

∂t−div[S′(un)a(x, t,un,Dun)]+S′′(un)a(x, t,un,Dun)Dun

+S′(un)Hn(x, t,un,Dun) = f S′(un) in D′(Q).

(5.74)

It was follows we pass to the limit as in (5.74) n tends to +∞.

• Limit of∂S(un)

∂t.

Since S is bounded and continuous, (un → u a.e in Q) implies that S(un) converges to

S(u) a.e in Q and L∞ weak-∗. Then∂S(un)

∂tconverges to

∂S(u)∂t

in D′(Q) as n tends to +∞.

• Limit of −div[S′(un)a(x, t,un,Dun)].

Since supp(S′)⊂ [−M,M], we have for n ≥ M

S′(un)a(x, t,un,Dun) = S′(un)a(x, t,TM(un),DTM(un)) a.e in Q.

The pointwise convergence of un to u and (5.67) as n tends to +∞ and the bounded

character of S′ permit us to conclude that

S′(un)a(x, t,un,Dun) S′(u)a(x, t,TM(u),DTM(u)) inN

∏i=1

Lp′(Q,w∗i ), (5.75)

as n tends to +∞. S′(u)a(x, t,TM(u),DTM(u)) has been denoted by S′(u)a(x, t,u,Du) in

equation (5.21).

• Limit of S′′(un)a(x, t,un,Dun)Dun.

Since supp(S′)⊂ [−M,M], we have for n ≥ M

S′′(un)a(x, t,un,Dun)Dun = S′′(un)a(x, t,TM(un),DTM(un))DTM(un) a.e in Q.

The pointwise convergence of S′′(un) to S′′(u) and (5.67)-(5.65) as n tends to +∞ and the

bounded character of S′′ permit us to conclude that

S′′(un)a(x, t,un,Dun)Dun S′′(u)a(x, t,TM(u),DTM(u))DTM(u) weakly in L1(Q). (5.76)

Recall that

S′′(u)a(x, t,TM(u),DTM(u))DTM(u) = S′′(u)a(x, t,u,Du)Du a.e in Q.

• Limit of S′(un)Hn(x, t,un,Dun).

Since supp(S′)⊂ [−M,M] and (5.68), we have

S′(un)Hn(x, t,un,Dun) → S′(u)H(x, t,u,Du) strongly in L1(Q), (5.77)

as n tends to +∞.

• Limit of S′(un) fn.


Due to (5.23) and (un → u a.e in Q), we have

S′(un) fn → S′(u) f strongly in L1(Q) as n → +∞.

As a consequence of the above convergence result, we are in a position to pass to the limit

as n tends to +∞ in equation (5.74) and to conclude that u satisfies (5.21).

It remains to show that S(u) satisfies the initial condition (5.22). To this end, firstly

remark that, S being bounded, S(un) is bounded in L∞(Q). Secondly, (5.74) and the above

considerations on the behavior of the terms of this equation show that∂S(un)

∂tis bounded

in L1(Q)+ Lp′(0,T ;W−1,p′(Ω,w∗)). As a consequence, an Aubin’s type lemma (see, e.g,

[21]) implies that S(un) lies in a compact set of C0([0,T ],L1(Ω)). It follows that on the one

hand, S(un)(t = 0) = S(un0) converges to S(u)(t = 0) strongly in L1(Ω). On the other hand,

the smoothness of S implies that

S(u)(t = 0) = S(u0) in Ω.

As a conclusion of step 1 to step 5, the proof of theorem (5.3) is complete.

6. Example

Let us consider the following special case:

H(x, s,ξ) = ρ sin(s)exp(−s2)N

∑i=1

wi(x) |ξi|p , ρ ∈ R

ai(x, t, s,d)= wi(x) |di|p−1

sgn(di), i = 1, . . .,N,

with wi(x) ( (i = 1, . . .,N).) a weight function strictly positive, x∈ Q. Then, we can consider

the Hardy inequality in the form

(

Z

Ω|u(x)|p σ(x)dx

)1p

≤ c

(

Z

Ω|Du(x)|p w(x)dx

)1p

.

It is easy to show that the ai(t,x, s,d) are Caratheodory functions satisfying the growth

condition (3.11) and the coercivity (3.13). On the order hand the monotonicity condition is

verified. In fact,N

∑i=1

(

ai(x, t,d)−a(x, t,d′))

(di −d′i)

= w(x)N−1

∑i=1

(

|di|p−1

sgn(di)−∣

∣d′i

∣

∣

p−1sgn(d′

i))

(di −d′i) > 0,

for almost all x ∈ Ω and for all d,d′ ∈ RN . This last inequality can not be strict, since for

d 6= d′ , since w > 0 a.e. in Ω.

While the Caratheodory function H(x, t, s,ξ) satisfies the condition (3.14) indeed

|H(x, t, s,ξ)| ≤ |ρ|exp(−s2)N

∑i=1

wi(x) |ξi|p = g(s)

N

∑i=1

wi(x) |ξi|p


where g(s) = |ρ|exp(−s2) is a function positive continuous which belongs to L1(R). Note

that H(x, t, s,ξ) not satisfies the sign condition (1.2) and the coercivity condition (1.4).

In particular, let us use special weight function, w, expressed in terms of the distance to

the bounded ∂Ω. Denote d(x) = dist(x,∂Ω) and set w(x) = dλ(x), σ(x) = dµ(x).

Remark 6.1. Finally, the hypotheses of Theorem 5.3 are satisfied. Therefore, for all f ∈

L1(Q), the following problem:

u ∈ C([0,T];L1(Ω));

Tk(u) ∈ Lp(0,T ;W1, p0 (Ω,w)),

limm→+∞

R

m≤|u|≤m+1a(x, t,u,Du)Dudxdt = 0;

−R

Q S(u) ∂ϕ∂t

dxdt +R

Q S′(u)N

∑i=1

wi

∣

∣

∣

∂u∂xi

∣

∣

∣

p−1

sgn(

∂u∂xi

)

∂ϕ∂xi

dxdt

+R

Q S′′(u)N

∑i=1

wi

∣

∣

∣

∂u∂xi

∣

∣

∣

p−1

sgn(

∂u∂xi

)

∂u∂xi

ϕdxdt

+R

Q ρS′(u) sin(u)exp(−u2)N

∑i=1

wi

∣

∣

∣

∂u∂xi

∣

∣

∣

p−1

ϕdxdt

=R

Q f S′(u)ϕdxdt,

S(u)(t = 0) = S(u0) in Ω,∀ ϕ ∈C∞

0 (Q) and S ∈W 2,∞(R) with S′ has a compact support in R,

(6.78)

has at least one renormalised solution.

References

[1] R. Adams, Sobolev Spaces, AC, Press, New York, 1975.

[2] L. Aharouch, E. Azroul and M. Rhoudaf, Existence result for variational degenerated

parabolic problems via pseudo-monotonicity, Procceding of the 2005 oujda Interna-

tional conference, Nonlinear Analysis, 9–20.

[3] L. Aharouch, E. Azroul and M. Rhoudaf, Strongly nonlinear variational parabolic

problems in weighted sobolev spaces, The Australian journal of Mathematical Analy-

sis and Applications, 5 (2008), No. 2, Article 13, pp. 1–25.

[4] L. Aharouch, E. Azroul and M. Rhoudaf, Existence results for Strongly nonlinear

degenerated parabolic equations via strong convergence of truncations with L1 data.

[5] Y. Akdim, J. Bennouna, M. Mekkour and M. Rhoudaf Renormalised solutions of non-

linear degenerated parabolic problems with L1 data: existence and uniqueness, to ap-

pear in “Series in Contemporary Applied Mathematics” World Scientific.

[6] L. Boccardo, D. Giachetti, J.-I. Diaz, F. Murat, Existence and Regularity of Renor-

malized Solutions of some Elliptic Problems involving derivatives of nonlinear terms,

Journal of differential equations 106 (1993), 215–237.


[7] D. Blanchard and F. Murat, Renormalized solutions of nonlinear parabolic problems

with L1 data: existence and uniqueness, Proceedings of the Royal Society of Edin-

burgh, 127A (1997), 1137–1152.

[8] J. Berkovits and V. Mustonen, Topological degree for perturbations of linear maximal

monotone mappings and applications to a class of parabolic problem, Rendiconti di

Matematica, Serie VII, 12 (1992), 597–621.

[9] A. DallAglio and L. Orsina, Nonlinear parabolic equations with natural growth con-

ditions and L1 data, Nonlinear Anal. 27 (1996), 5973.

[10] G. Dal Maso, F. Murat, L. Orsina and A. Prignet, Definition and existence of renor-

malized solutions of elliptic equations with general measure data, C. R. Acad. Sci.

Paris 325 (1997), 481–486.

[11] R.J. Diperna and P.-L. Lions, On the cauchy problem for Boltzman equations: global

existence and weak stability. Ann. of Math. (2) 130 (1989), 321–366.

[12] P. Drabek, A. Kufner and V. Mustonen, Pseudo-monotonicity and degenerated or sin-

gular elliptic operators, Bull. Austral. Math. Soc., 58 (1998), 213–221.

[13] P. Drabek, A. Kufner and F. Nicolosi, Non linear elliptic equations, singular and

degenerated cases, University of West Bohemia, 1996.

[14] A. Kufner, Weighted Sobolev Spaces, John Wiley and Sons, 1985.

[15] R. Landes, On the existence of weak solutions for quasilinear parabolic initial-

boundary value problems, Proc. Roy. Soc. Edinburgh Sect A 89 (1981), 321–366.

[16] J.-L. Lions, Quelques methodes de resolution des probleme aux limites non lineaires,

Dundo, Paris 1969.

[17] A. Porretta, Existence results for strongly nonlinear parabolic equations via strong

convergence of truncations, Ann. Mat. Pura Appl. (IV), 177 (1999), 143–172.

[18] A. Porretta, Nonlinear equations with natural growth terms and measure data, EJDE,

conference 09, 2002, 183–202.

[19] J.-M. Rakotoson, Uniqueness of renormalized solutions in a T -set for L1 data prob-

lems and the link between various formulations, Indiana University Math. Jour., 43,

(1994). No. 2.

[20] H. Redwane, Existence of a solution for a class of parabolic equations with three

unbounded nonlinearities, Adv. Dyn. Syst. Appl., 2 (2007), 241–264.

[21] J. Simon, Compact sets in the space Lp(0,T,B), Ann. Mat. Pura. Appl., 146 (1987),

65–96.

[22] E. Zeidler, Nonlinear Functional Analysis and its Applications, Springer-Verlag, New

York-Heidlberg, 1990.

Article-IJEE

Documents

Transcript of Article-IJEE