Open Access

Parabolic problems with data measure

Boundary Value Problems20112011:46

DOI: 10.1186/1687-2770-2011-46

Received: 4 April 2011

Accepted: 23 November 2011

Published: 23 November 2011

Abstract

The article deals with the existence of solutions of some unilateral problems in the Orlicz-Sobolev spaces framework when the right-hand side is a Radon measure.

Mathematics Subject Classification: 35K86.

Keywords

unilateral problem radon measure Orlicz-Sobolev spaces

1 Introduction

We deal with boundary value problems
u ψ a .e . in Q = Ω × [ 0 , T ] , u t + A ( u ) = μ in Q , u = 0 on Q = Ω × [ 0 , T ] , u ( x , 0 ) = u 0 ( x ) in Ω , ( P )
where
A ( u ) = - div ( a ( . , t , u , u ) ) ,
T > 0 and Ω is a bounded domain of R N , with the segment property. a : Ω × R × R N R N is a Carathéodory function (that is, measurable with respect to x in Ω for every (t, s, ξ) in R × R × R N , and continuous with respect to (s, ξ) in R × R N for almost every x in Ω) such that for all ξ, ξ* R N , ξξ*,
a ( x , t , s , ξ ) ξ α M ( | ξ | )
(1.1)
[ a ( x , t , s , ξ ) - a ( x , t , s , ξ * ) ] [ ξ - ξ * ] > 0 ,
(1.2)
| a ( x , t , s , ξ ) | c ( x , t ) + k 1 P ¯ - 1 M ( k 2 | s | ) + k 3 M ¯ - 1 M ( k 4 | ξ | ) ,
(1.3)
where c (x,t) belongs to E M ¯ ( Q ) , c 0 , P is an N-function such that P M and k i (i = 1,2,3,4) belongs to R+ and α to R * + .
μ M b + ( Q ) , u 0 M b + ( Ω ) ,
(1.4)
ψ L ( Ω ) W 0 1 E M ( Ω ) .
(1.5)

There have obviously been many previous studies on nonlinear differential equations with nonsmooth coefficients and measures as data. The special case was cited in the references (see [1, 2]).

It is noteworthy that the articles mentioned above differ in significant way, in the terms of the structure of the equations and data. In [1], when f L1(0,T;L1(Ω)) and u0 L1(Ω). The authors have shown the existence of solutions u of the corresponding equation of the problem ( P ) , u L q ( 0 , T ; W 0 1 , q ( Ω ) ) for every q such that q < p - N N + 1 which is more restrictive than the one given in the elliptic case q < N ( p - 1 ) N - 1 .

In this article, we are interested with an obstacle parabolic problem with measure as data. We give an improved regularity result of the study of Boccardo et al. [1].

In [1], the authors have shown the existence of a weak solutions for the corresponding equation, the function a(x, t, s, ξ) was assumed to satisfy a polynomial growth condition with respect to u and u. When trying to relax this restriction on the function a(., s, ξ), we are led to replace the space L p (0, T; W1,p(Ω)) by an inhomogeneous Sobolev space W1,xL M built from an Orlicz space L M instead of L p , where the N-function M which defines L M is related to the actual growth of the Carathéodory's function.

For simplicity, one can suppose that there exist α > 0, β > 0 such that
a ( x , t , u , u ) = a ( x , t , u ) M ( | u | ) | u | 2 u and α a ( x , t , s ) | β .

2 Preliminaries

Let M : R+R+ be an N-function, i.e. M is continuous, convex, with M(t) > 0 for t > 0 , M ( t ) t 0 as t → 0 and M ( t ) t as t → ∞. Equivalently, M admits the representation: M ( t ) = 0 t a ( τ ) d τ where a : R+R+ is non-decreasing, right continuous, with a(0) = 0, a(t) > 0 for t > 0 and a(t) → ∞ as t → ∞. The N-function M ¯ conjugate to M is defined by M ¯ ( t ) = 0 t ā ( τ ) d τ , where ā : R + R + is given by ā ( t ) = sup { s : a ( s ) t } (see [3, 4]).

The N-function M is said to satisfy the Δ2 condition if, for some k > 0:
M ( 2 t ) k M ( T ) for all t 0 ,
(2.1)

when this inequality holds only for tt0 > 0, M is said to satisfy the Δ2 condition near infinity.

Let P and Q be two N-functions. P Q means that P grows essentially less rapidly than Q; i.e., for each ε > 0,
P ( t ) Q ( ε t ) 0 as t .
Let Ω be an open subset of R N . The Orlicz class L M ( Ω ) (resp. the Orlicz space L m (Ω)) is defined as the set of (equivalence classes of) real-valued measurable functions u on Ω such that
Ω M ( u ( x ) ) d x < + resp . Ω M u ( x ) λ d x < + for some λ > 0 .

Note that L M (Ω) is a Banach space under the norm | | u | | M , Ω = inf λ > 0 : Ω M u ( x ) λ d x 1 and L M ( Ω ) is a convex subset of L M (Ω). The closure in L M (Ω) of the set of bounded measurable functions with compact support in Ω ¯ is denoted by E M (Ω). The equality E M (Ω) = L M (Ω) holds if and only if M satisfies the Δ2 condition, for all t or for t large according to whether Ω has infinite measure or not.

The dual of E M (Ω) can be identified with L M ¯ ( Ω ) by means of the pairing Ω u ( x ) v ( x ) d x , and the dual norm on L M ¯ ( Ω ) is equivalent to | | | | M ¯ , Ω . The space L M (Ω) is reflexive if and only if M and M ¯ satisfy the Δ2 condition, for all t or for t large, according to whether Ω has infinite measure or not.

We now turn to the Orlicz-Sobolev space. W1L M (Ω) (resp. W1E M (Ω)) is the space of all functions u such that u and its distributional derivatives up to order 1 lie in L M (Ω) (resp. E M (Ω)). This is a Banach space under the norm | | u | | 1 , M , Ω = | α | 1 | | D α u | | M , Ω . Thus, W1L M (Ω) and W1E M (Ω) can be identified with subspaces of the product of N + 1 copies of L M (Ω). Denoting this product by ΠL M , we will use the weak topologies σ ( L M , E M ¯ ) and σ ( L M , L M ¯ ) . The space W 0 1 E M ( Ω ) is defined as the (norm) closure of the Schwartz space D ( Ω ) in W1E M (Ω) and the space W 0 1 L M ( Ω ) as the σ ( L M , E M ¯ ) closure of D ( Ω ) in W1L M (Ω). We say that u n converges to u for the modular convergence in W1L M (Ω) if for some λ > 0 , Ω M D α u n - D α u λ d x 0 for all |α| ≤ 1. This implies convergence for σ ( L M , L M ¯ ) . If M satisfies the Δ2 condition on R+(near infinity only when Ω has finite measure), then modular convergence coincides with norm convergence.

Let W - 1 L M ¯ ( Ω ) (resp. W - 1 E M ¯ ( Ω ) ) denote the space of distributions on Ω which can be written as sums of derivatives of order ≤ 1 of functions in L M ¯ ( Ω ) (resp. E M ¯ ( Ω ) ). It is a Banach space under the usual quotient norm.

If the open set Ω has the segment property, then the space D ( Ω ) is dense in W 0 1 L M ( Ω ) for the modular convergence and for the topology σ ( L M , L M ¯ ) (cf. [5, 6]). Consequently, the action of a distribution in W - 1 L M ¯ ( Ω ) on an element of W 0 1 L M ( Ω ) is well defined.

For k > 0, s R, we define the truncation at height k,T k (s) = [k - (k - |s|)+]sign(s).

The following abstract lemmas will be applied to the truncation operators.

Lemma 2.1 [7] Let F : RR be uniformly lipschitzian, with F(0) = 0. Let M be an N-function and let u W 0 1 L M ( Ω ) (resp. W 0 1 E M ( Ω ) ).

Then F ( u ) W 0 1 L M ( Ω ) (resp. W 0 1 E M ( Ω ) ). Moreover, if the set of discontinuity points of F' is finite, then
x i F ( u ) = F ( u ) u x i a . e . i n { x Ω : u ( x ) D } 0 a . e . i n { x Ω : u ( x ) D }

Let Ω be a bounded open subset of R N , T > 0 and set Q = Ω × ]0, T[. Let m ≥ 1 be an integer and let M be an N-function. For each α IN N , denote by D x α the distributional derivative on Q of order α with respect to the variable x R N . The inhomogeneous Orlicz-Sobolev spaces are defined as follows W m , x L M ( Q ) = { u L M ( Q ) : D x α u L M ( Q ) | α | m } W m , x E M ( Q ) = { u E M ( Q ) : D x α u E M ( Q ) | α | m } .

The last space is a subspace of the first one, and both are Banach spaces under the norm | | u | | = | α | m | | D x α u | | M , Q . We can easily show that they form a complementary system when Ω satisfies the segment property. These spaces are considered as subspaces of the product space ΠL m (Q) which have as many copies as there are α-order derivatives, |α| ≤ m. We shall also consider the weak topologies σ ( L M , E M ¯ ) and σ ( L M , L M ¯ ) . If u W m, x L M (Q), then the function : t u(t) = u(t,.) is defined on [0, T] with values in W m L M (Ω). If, further, u Wm,xE M (Q) then the concerned function is a W m E M (Ω)-valued and is strongly measurable. Furthermore, the following imbedding holds: Wm,xE M (Q) L1(0,T; W m E M (Ω)). The space Wm,xL M (Q) is not in general separable, if u Wm,xL M (Q), we cannot conclude that the function u(t) is measurable on [0,T]. However, the scalar function t ||u(t)||M, is in L1(0,T). The space W 0 m , x E M ( Q ) is defined as the (norm) closure in Wm,xE M (Q) of D ( Ω ) . We can easily show as in [6] that when Ω has the segment property, then each element u of the closure of D ( Ω ) with respect of the weak * topology σ ( L M , E M ¯ ) is limit, in Wm,xL M (Q), of some subsequence ( u i ) D ( Q ) for the modular convergence; i.e., there exists λ > 0 such that for all |α| ≤ m,
Q M D x α u i - D x α u λ d x d t 0 as i ,

this implies that (u i ) converges to u in Wm,xL M (Q) for the weak topology σ ( L M , L M ¯ ) . Consequently, D ( Q ) ¯ σ ( L M , E M ¯ ) = D ( Q ) ¯ σ ( L M , L M ¯ ) , and this space will be denoted by W 0 m , x L M ( Q ) .

Furthermore, W 0 m , x E M ( Q ) = W 0 m , x L M ( Q ) E M . Poincaré's inequality also holds in W 0 m , x L M ( Q ) , i.e., there is a constant C > 0 such that for all u W 0 m , x L M ( Q ) one has | α | m | | D x α u | | M , Q C | α | = m | | D x α u | | M , Q . Thus both sides of the last inequality are equivalent norms on W 0 m , x L M ( Q ) . We have then the following complementary system:
W 0 m , x L M ( Q ) F W 0 m , x E M ( Q ) F 0

F being the dual space of W 0 m , x E M ( Q ) . It is also, except for an isomorphism, the quotient of L M ¯ by the polar set W 0 m , x E M ( Q ) , and will be denoted by F = W - m , x L M ¯ ( Q ) , and it is shown that W - m , x L M ¯ ( Q ) = f = | α | m D x α f α : f α L M ¯ ( Q ) . This space will be equipped with the usual quotient norm | | f | | = inf | α | m | | f α | | M ¯ , Q where the infimum is taken on all possible decompositions f = | α | m D x α f α , f α L M ¯ ( Q ) . The space F0 is then given by F 0 = f = | α | m D x α f α : f α E M ¯ ( Q ) and is denoted by F 0 = W - m , x E M ¯ ( Q ) .

We can easily check, using Lemma 4.4 of [6], that each uniformly lipschitzian mapping F, with F(0) = 0, acts in inhomogeneous Orlicz-Sobolev spaces of order 1: W1,xL M (Q) and W 0 1 , x L M ( Q ) .

3 Main results

First, we give the following results which will be used in our main result.

3.1 Useful results

Hereafter, we denote by X N the real number defined by X N = N C N 1 N , C N is the measure of the unit ball of R N , and for a fixed t [0, T], we denote μ(θ) = meas{(x,t) : |u(x, t)| > θ}.

Lemma 3.1 [8] Let u W 0 1 , x L M ( Q ) , and let fixed t [0, T], then we have
- μ ( θ ) - 1 X N μ ( θ ) 1 - 1 N S - 1 X N μ ( θ ) 1 - 1 N d d θ { | u | > θ } M ( | u | ) d x , θ > 0
and where S is defined by
1 S ( s ) = sup { t : B ( t ) s } , B ( s ) = M ( s ) s .
Lemma 3.2 Under the hypotheses (1.1)-(1.3), if f, u0 are regular functions and f, u0 ≥ 0, then there exists at least one positive weak solution of the problem
u t + A ( u ) = f i n Q , u = 0 o n Q , u ( x , 0 ) = u 0 ( x ) i n Ω . ( E )
such that
u t 0 , a . e . t ( 0 , T ) .

Proof

Let u be a continuous function, we say that u satisfies (*) if: there exists a continuous and increasing function β such that ||u(t) - u(s)||2β(||u0||2)|t - s|, where u0(x) = u(x, 0).

Let X : = u W 0 1 , x L M ( Q ) L 2 ( Q ) s .t . u satisfies ( * ) and d u d t L ( 0 , T , L 2 ( Ω ) ) .

Let us consider the set C = { v X : v ( t ) C , v t 0 a .e . t ( 0 , T ) } , where C is a closed convex of W 0 1 L M ( Ω ) . It is easy to see that C is a closed convex (since all its elements satisfy (*) ).

We claim that the problem
u C L 2 ( Q ) u t + A ( u ) = f in Q , u = 0 on Q , u ( x , 0 ) = u 0 in Ω . ( E )

has a weak solution which is unique in the sense defined in [9].

Indeed, let us consider the approximate problem
u n t + A ( u n ) + n T n ( Φ ( u n ) ) = f in Ω , u n ( . , 0 ) = u 0 in Ω . ( E )
where the functional Φ is defined by Φ : XR {+ ∞} such that
Φ ( v ) : = 0 if v C , + otherwise .

The existence of such u n X was ensured by Kacur et al. [10].

Following the same proof as in [9], we can prove the existence of a solution u of the problem (E') as limit of u n (for more details see [9]).

Lemma 3.3 Let v W 0 1 , x L M ( Q ) such that v t W - 1 , x L M ¯ ( Q ) + L 1 ( Q ) and v ψ , ψ L ( Ω ) W 0 1 E M ( Ω ) .

Then, there exists a smooth function (v j ) such that
v j ψ ,

v j v for the modular convergence in W 0 1 , x L M ( Q ) ,

v i t v t for the modular convergence in W - 1 , x L M ¯ ( Q ) + L 1 ( Q ) .

For the proof, we use the same technique as in [11] in the parabolic case.

3.2 Existence result

Let M be a fixed N-function, we define K as the set of N-function D satisfying the following conditions:

i) M(D-1(s)) is a convex function,

ii) 0 D o B - 1 1 r 1 - 1 N d r < + , B ( t ) = M ( t ) t ,

iii) There exists an N-function H such that H o M ¯ - 1 o M D and H ¯ D near infinity.

Theorem 3.1 Under the hypotheses (1.1)-(1.5), The problem (P) has at least one solution u in the following sense:
u ψ a . e . i n Q T k ( u ) W 0 1 , x L M ( Q ) , u W 0 1 , x L D ( Q ) D K - Q u φ t + Q a ( . , u , u ) φ d x d t - Ω φ d u 0 = Q φ d μ ,

for all φ D(RN+1) which are zero in a neighborhood of (0, T) × ∂ Ω and {T} × Ω.

Remark 3.1 (1) If ψ = - ∞ in the problem (P), then the above theorem gives the same regularity as in the elliptic case.

(2) An improved regularity is reached for all N-function satisfying the conditions (i)-(ii)-(iii).

For example, u W 0 1 , x L D ( Q ) , D ( t ) = t q L o g σ ( e + t ) , for all q < N ( p - 1 ) N - 1 , σ > 1 .

In the sequel and throughout the article, we will omit for simplicity the dependence on x and t in the function a(x, t, s, ξ) and denote ϵ(n, j, μ, s, m) all quantities (possibly different) such that
lim m lim s lim μ lim j lim n ε ( n , j , μ , s , m ) = 0 ,

and this will be in the order in which the parameters we use will tend to infinity, that is, first n, then j, μ, s, and finally m. Similarly, we will write only ϵ(n), or ϵ(n, j),... to mean that the limits are made only on the specified parameters.

3.2.1 A sequence of approximating problems

Let (f n ) be a sequence in D(Q) which is bounded in L1(Q) and converge to μ in M b (Q).

Let ( u 0 n ) be a sequence in D(Ω) which is bounded in L1(Ω) and converge to u0 in M b (Ω).

We define the following problems approximating the original (P):
u n t + A ( u n ) - n T n ( ( u n - ψ ) - ) = f n in Q , u n = 0 on Q u n ( . , 0 ) = u 0 n in Ω . ( P n )

Lemma 3.4 Under the hypotheses (1.1)-(1.3), there exists at least one solution u n of the problem (P n ) such that u n t 0 a.e. in Q.

For the proof see Lemma 3.2.

3.2.2 A priori estimates

Lemma 3.5 There exists a subsequence of (u n ) (also denoted (u n )), there exists a measurable function u such that:
u ψ , T k ( u ) W 0 1 , x L M ( Q ) f o r a l l k > 0 u n u w e a k l y i n W 0 1 , x L D ( Q ) f o r a l l D K .

Proof:

Recall that u n ≥ 0 since f n ≥ 0.

Let h > 0 and consider the following test function v = T h (u n - T k (u n )) in (P n ), we obtain
u n t , v + α { k < | u n | k + h } M ( | u n | ) d x d t - n Q T n ( ( u n - ψ ) - ) v d x d t Q f n v d x d t
We have
u n t , T h ( u n - T k ( u n ) ) = Ω 0 u n ( x , T ) T h ( s - T k ( s ) ) - Ω 0 u 0 n T h ( s - T k ( s ) ) .
So,
- Q n T n ( ( u n - ψ ) - ) T h ( u n - T k ( u n ) ) h d x d t C .

Now, let us fix k > ||ψ||, we deduce the fact that: n T n ( u n - ψ ) ( u n - k ) X { u n ψ } X { k < u n k + h } 0 .

Let h to tend to zero, one has
n Q T n ( ( u n - ψ ) - ) d x d t C .
(3.1)
Let us use as test function in (P n ),v = T k (u n ), then as above, we obtain
Q M ( | T k ( u n ) | ) C 1 k .
(3.2)

Then (T k (u n ) n ) is bounded in W 0 1 , x L M ( Q ) , and then there exist some ω k W 0 1 , x L M ( Q ) such that

T k (u n ) ω k , weakly in W 0 1 , x L M ( Q ) for σ ( L M , E M ¯ ) , strongly in E M (Q) and a.e in Q.

Let consider the C2 function defined by
μ k ( s ) s | s | k 2 k s i g n ( s ) | s | k

Multiplying the approximating equation by η k ( u n ) , we get η k ( u n ) t - d i v ( a ( . , u n , u n ) η k ( u n ) ) + a ( . , u n u n ) n k ( u n ) = f n η k ( u n ) + n ( T n ( ( u n - ψ ) - ) ) η k ( u n ) in the distributions sense. We deduce then that η k (u n ) being bounded in W 0 1 , x L M ( Q ) and η k ( u n ) t in W - 1 , x L M ¯ ( Q ) + L 1 ( Q ) . By Corollary 1 of [12], η k (u n ) is compact in L1 (Q).

Following the same way as in [2], we obtain for every k > 0,
T k ( u n ) T k ( u ) , weakly in W 0 1 , x L M ( Q ) for σ ( L M , E M ¯ ) , strongly in L 1 ( Q ) and a .e in Q .
(3.3)
Using now the estimation (3.1) and Fatou's lemma to obtain
( u - ψ ) - = 0 and so , u ψ .
Let fixed a t [0, T]. We argue now as for the elliptic case, the problem becomes:
u n t - div ( a ( . , u n , u n ) ) = f n + n T n ( ( u n - ψ ) - ) in Ω . ( P n )

We denote g n := nT n ((u n - ψ)-).

Let φ be a truncation defined by
φ ( ξ ) = 0 0 ξ θ 1 h ( ξ - t ) θ < ξ < θ + h 1 ξ θ + h - φ ( - ξ ) ξ < 0
(3.4)

for all θ, h > 0.

Using v = φ (u n ) as a test function in the approximate elliptic problem ( P n ) , we obtain by using the same technique as in [8]
- d d θ { | u n | > θ } M ( | u n | ) d x C { | u n | θ } ( f n + g n - u n t ) d x .
(3.5)

here and below C denote positive constants not depending on n.

By using Lemma 3.1, we obtain (supposing -μ'(θ) > 0 which does not affect the proof) and following the same way as in [8], we have for D K,
- d d θ { | u n | > θ } D ( | u n | ) d x ( - μ ( θ ) ) D o B - 1 - 1 X N μ ( θ ) 1 - 1 N d d θ { | u n | > θ } M ( | u n | d x ) .
(3.6)
Let denote k ( t , s ) : = 0 s u n * ( t , ρ ) d ρ , then
k t ( t , s ) = 0 s u n * ( t , ρ ) t d ρ , u n > θ u n t d x = k t ( t , μ ( θ ) ) .
Using Lemma 3.1, denoting F ( t , μ ( θ ) ) : = 0 μ ( θ ) ( f n * + g n * ) ( ρ ) d ρ one has
1 - μ ( θ ) X N μ ( θ ) 1 - 1 N B - 1 1 X N μ ( θ ) 1 - 1 N F ( t , μ ( θ ) ) - k t ( t , μ ( θ ) ) .

Remark also that F ( t , s ) k t ( t , s ) and using Lemma 3.2, we have k t ( t , s ) F ( t , s ) .

Combining the inequalities (3.5) and (3.6) we obtain,
- d d θ { | u n | > θ } D ( | u n | ) d x ( - μ ( θ ) ) D o B - 1 - 1 X N μ ( θ ) 1 - 1 N F ( t , μ ( θ ) ) - k t ( t , μ ( θ ) ) .
(3.7)
and since the function θ { | u n | > θ } D ( | u n | ) d x is absolutely continuous, we get
Ω D ( | u n | ) d x = 0 + - d d θ { | u n | > θ } D ( | u n | ) d x d t 1 C 0 C | Ω | D o B - 1 C s 1 - 1 N d s ( using 3 . 1 and 3 . 7 ) .

Then, the sequence (u n ) is bounded in W 0 1 , x L D ( Q ) and we deduce that u W 0 1 , x L D ( Q ) for all N-function D K.

3.3 Almost everywhere convergence of the gradients

Lemma 3.6 The subsequence (u n ) obtained in Lemma 3.5 satisfies:
u n u a . e . i n Q .

Proof:

Let m > 0, k > 0 such that m > k. Let ρ m be a truncation defined by
ρ m ( s ) = 1 | s | m , m + 1 - | s | m < | s | < m + 1 , 0 | s | m + 1 . R m ( s ) = 0 s ρ m ( t ) d t and ω μ , j = T k ( v j ) μ .

where v j D(Q) such that v j ψ and v j T k (u) with the modular convergence in W 0 1 , x L M ( Q ) (for the existence of such function see [11] since ψ L ( Ω ) W 0 1 E M ( Ω ) ).

ω μ is the mollifier function defined in Landes [13], the function ωμ,jhave the following properties:
ω μ , j t = μ ( T k ( v j ) - ω μ , j ) , ω μ , j ( 0 ) = 0 , | ω μ , j | k , ω μ , j T k ( u ) μ in W 0 1 , x L M ( Q ) for the modular convergence with respect to j , T k ( u ) μ T k ( u ) in W 0 1 , x L M ( Q ) for the modular convergence with respect to μ .
Set v = (T k (u n ) - ωμ,j) ρ m (u n ) as test function, we have
u n t , v + Q a ( . , u n , u n ) ( T k ( u n ) - ω μ , j i ) ρ m ( u n )
(1)
+ Q a ( . , u n , u n ) u n ( T k ( u n ) - ω μ , j ) ρ m ( u n ) = Q f n v d x d t + n Q T n ( ( u n - ψ ) - ) v d x d t = : ( 3 ) + ( 4 ) .
(2)
Let us recall that for u n W 0 1 , x L M ( Q ) , there exists a smooth function u (see [14]) such that
u n σ u n for the modular convergence in W 0 1, x L M ( Q ) , u n σ t u n t for the modular convergence in W - 1 , x L M ¯ ( Q ) + L 1 ( Q ) .
u n t , v = lim σ 0 + Q ( u n σ ) ( T k ( u n σ ) ω μ , j ) ρ m ( u n σ ) = lim σ 0 + ( Q ( R m ( u n σ ) T k ( u n σ ) ) ( T k ( u n σ ) ω μ , j ) d x d t + Q ( T k ( u n σ ) ( T k ( u n σ ) ω μ , j ) d x d t ) = lim σ 0 + [ Ω ( R m ( u n σ ) T k ( u n σ ) ) ( T k ( u n σ ) ω μ , j ) d x ] 0 T Q ( R m ( u n σ ) T k ( u n σ ) ) ( T k ( u n σ ) ω μ , j ) d x d t + Q ( T k ( u n σ ) ( T k ( u n σ ) ω μ , j ) d x d t = : I 1 + I 2 + I 3 .
Remark also that,
R m ( u n σ ) T k ( u n σ ) if u n σ < k and R m ( u n σ ) > k = T k ( u n σ ) | ω μ , j | if u n σ k . I 1 = Ω ( R m ( u n σ ) ( T ) - T k ( u n σ ) ( T ) ) ( T k ( u n σ ) ( T ) - ω μ , j ( T ) ) d x I 1 u n σ ( T ) k ( R m ( u n σ ) ( T ) - T k ( u n σ ) ( T ) ) ( T k ( u n σ ) ( T ) - ω μ , j ( T ) ) d x

and it is easy to see that limsup σ 0 + I 1 ε ( n , j , μ ) .

Concerning I2,
I 2 = - u n σ k ( R m ( u n σ ) - T k ( u n σ ) ) ( T k ( u n σ ) - ω μ , j ) d x d t + u n σ > k ( R m ( u n σ ) - T k ( u n σ ) ) ( ω μ , j ) d x d t = : I 2 1 + I 2 2 .

As in I1, we obtain I 2 1 ε ( n , j , μ ) ,

and
I 2 2 = u n σ > k ( R m ( u n σ ) - T k ( u n σ ) ) ( ω μ , j ) d x d t μ u n σ > k ( R m ( u n σ ) - T k ( u n σ ) ) ( T k ( v j ) - T k ( u n σ ) ) d x d t ,

thus by using the fact that ( R m ( u n σ ) - T k ( u n σ ) ) ( T k ( u n σ ) - ω μ , j ) X u n σ > k 0 .

So, limsup σ 0 + I 2 2 μ u n > k ( R m ( u n ) - T k ( u n ) ) ( T k ( v j ) - T k ( u n ) ) d x d t = ε ( n , j ) .

About I3,
I 3 = Q ( T k ( u n σ ) ) ( T k ( u n σ ) - ω μ , j ) d x d t = Q ( T k ( u n σ ) - ω μ , j ) ( T k ( u n σ ) - ω μ , j ) d x d t + Q ( ω μ , j ) ( T k ( u n σ ) - ω μ , j ) d x d t .
Set Φ(s) = s2/2, Φ ≥ 0,then
I 3 = Ω Φ ( T k ( u n σ ) - ω μ , j ) d x 0 T + μ Q ( T k ( v j ) - ω μ , j ) ( T k ( u n σ ) - ω μ , j ) d x d t ε ( n , j , μ ) + μ Q ( T k ( v j ) - T k ( u n σ ) ) ( T k ( u n σ ) - ω μ , j ) d x d t ( as i n I 2 ) .
So,
limsup σ 0 + I 3 ε ( n , j , μ ) + μ Q ( T k ( v j ) - T k ( u n ) ) ( T k ( u n ) - ω μ , j ) d x d t = ε ( n , j , μ ) + μ Q ( T k ( v j ) - T k ( u ) ) ( T k ( u ) - ω μ , j ) d x d t + ε ( n ) ,

and easily we deduce, limsup σ 0 + I 3 ε ( n , j , μ ) .

Finally we conclude that: u n t , ( T k ( u n ) - ω μ , j ) ρ m ( u n ) ε ( n , j , μ ) .

We are interested now with the terms of (1)-(4).

About (1):
Q a ( . , u n , u n ) ( T k ( u n ) - ω μ , j ) ρ m ( u n ) d x d t = u n k a ( . , u n , u n ) ( T k ( u n ) - ω μ , j ) ρ m ( u n ) d x d t + u n > k a ( . , u n , u n ) ( T k ( u n ) - ω μ , j ) ρ m ( u n ) d x d t = Q a ( . , T k ( u n ) , T k ( u n ) ) ( T k ( u n ) - ω μ , j ) d x d t + u n > k a ( . , u n , u n ) ( T k ( u n ) - ω μ , j ) ρ m ( u n ) d x d t

recall that ρ m (u n ) = 1 on {|u n | ≤ k}.

Let s > 0 , Q s = { ( x , t ) Q : | T k ( u ) | s } , Q j s = { ( x , t ) Q : | T k ( v j ) | s } .
Q a ( . , u n , u n ) ( T k ( u n ) - ω μ , j ) ρ m ( u n ) d x d t = Q a ( . , T k ( u n ) , T k ( u n ) ) - a ( . , T k ( u n ) , T k ( v j ) X j s ) ( T k ( u n ) - T k ( v j ) X j s ) d x d t + Q a ( . , T k ( u n ) , T k ( v j ) X j s ) ( T k ( u n ) - T k ( v j ) X j s ) d x d t + Q a ( . , T k ( u n ) , T k ( u n ) ) T k ( v j ) X j s d x d t - Q a ( . , u n , u n ) ω μ , j ρ m ( u n ) d x d t = : J 1 + J 2 + J 3 + J 4 .
By using the inequality (1.3), we can deduce the existence of some measurable function h k such that
a ( . , T k ( u n ) , T k ( u n ) ) h k in ( L M ¯ ( Q ) ) N for σ ( L M , E M ¯ ) , J 2 = Q a ( . , T k ( u ) , T k ( v j ) X j s ) ( T k ( u ) - T k ( v j ) X j s ) d x d t + ε ( n ) ,
since
a ( . , T k ( u n ) , T k ( v j ) X j s ) a ( . , T k ( u ) , T k ( v j ) X j s ) strongly in ( E M ¯ ( Q ) ) N , a ( . , T k ( u ) , T k ( v j ) X j s ) a ( . , T k ( u ) , T k ( u ) X j s ) strongly in ( E M ¯ ( Q ) ) N ,

and T k ( v j ) X j s T k ( u ) X s strongly in ( L M ¯ ( Q )) N .

Then,
J 2 = ε ( n , j ) .
Following the same way as in J2, one has
J 3 = Q h k T k ( u ) d x d t + ε ( n , j , μ , s ) .
Concerning the terms J4 :
J 4 = - Q a ( . , T m + 1 ( u n ) , T m + 1 ( u n ) ) ω μ , j i ρ m ( u n ) d x d t = - | u n | k a ( . , T m + 1 ( u n ) , T m + 1 ( u n ) ) ω μ , j ρ m ( u n ) d x d t - k < | u n | m + 1 a ( . , T m + 1 ( u n ) , T m + 1 ( u n ) ) ω μ , j ρ m ( u n ) d x d t .
Letting n → ∞, then
J 4 = - k < | u | m + 1 h m + 1 ω μ , j ρ m ( u ) d x d t - | u | k h k ω μ , j ρ m ( u ) d x d t + ε ( n ) .
Taking now the limits j → ∞ and after μ → ∞ in the last equality, we obtain
J 4 = - Q h k T k ( u ) d x d t + ε ( n , j , μ ) .
Then,
( 1 ) = Q a ( . , T k ( u n ) , T k ( u n ) ) - a ( . , T k ( u n ) , T k ( v j ) X j s ) ( T k ( u n ) - T k ( v j ) X j s ) + ε ( n , j , μ , s ) .
About (2):
| Q a ( . , u n , u n ) u n ( T k ( u n ) - ω μ , j ) ρ m ( u n ) | d x d t C ( k ) m < | u n | m + 1 a ( . , u n , u n ) u n d x d t .
Since (u n ) is bounded in W 0 1 , x L D ( Q ) and using (iii), we have (a(., u n ,u n )) is bounded in L H (Q), then
| m < | u n | m + 1 a ( . , u n , u n ) u n d x d t | | | a ( . , u n , u n ) | | H , Q | | u n | | D , m < | u n | m + 1 ε ( n , m ) ,
so,
( 2 ) ε ( n , m ) .

About (4):

Since uψ, then T k (u) ≥ T k (ψ) and there exist a smooth function v j T k (ψ) such that v j T k (u) for the modular convergence in W 0 1 , x L M ( Q ) .
( 4 ) = n Q T n ( ( u n - ψ ) - ) ( T k ( u n ) - T k ( v j ) μ ) ρ m ( u n ) d x d t ε ( n , j , μ ) .
Taking into account now the estimation of (1), (2), (4)and (5), we obtain
Q a ( . , T k ( u n ) , T k ( u n ) ) - a ( . , T k ( u n ) , T k ( v j ) X j s ) ( T k ( u n ) - T k ( v j ) X j s ) d x d t ε ( n , j , μ , s , m ) .
(3.8)
On the other hand,
Q ( a ( . , T k ( u n ) , T k ( u n ) ) - a ( . , T k ( u n ) , T k ( u ) X s ) ) ( T k ( u n ) - T k ( u ) X s ) d x d t - Q a ( . , T k ( u n ) , T k ( u n ) ) - a ( . , T k ( u n ) , T k ( v j ) X j s ) ( T k ( u n ) - T k ( v j ) X j s ) d x d t = Q a ( . , T k ( u n ) , T k ( u n ) ) ( T k ( v j ) X j s - T k ( u ) X s ) d x d t - Q a ( . , T k ( u n ) , T k ( u ) X s ) ( T k ( v j ) X j s - T k ( u ) X s ) d x d t + Q a ( . , T k ( u n ) , T k ( v j ) X j s ) ( T k ( u n ) - T k ( v j ) X j s ) d x d t ,
each term of the last right hand side is of the form ϵ(n, j, s), which gives
Q ( a ( . , T k ( u n ) , T k ( u n ) ) - a ( . , T k ( u n ) , T k ( u ) X s ) ) ( T k ( u n ) - T k ( u ) X s ) d x d t = Q a ( . , T k ( u n ) , T k ( u n ) ) - a ( . , T k ( u n ) , T k ( v j ) X j s ) ( T k ( u n ) - T k ( v j ) X j s ) d x d t + ( n , j , s ) .
Following the same technique used by Porretta [2], we have for all r < s :
Q r ( a ( . , T k ( u n ) , T k ( u n ) ) - a ( . , T k ( u n ) , T k ( u ) ) ) ( T k ( u n ) - T k ( u ) ) d x d t 0 .
Thus, as in the elliptic case (see [7]), there exists a subsequence also denoted by u n such that
u n u a .e . in Q .
We deduce then that,
a ( . , T k ( u n ) , T k ( u n ) ) a ( . , T k ( u ) , T k ( u ) ) in ( L M ¯ ( Q )) N for σ ( L M , E M ¯ ) .
Lemma 3.7 For all k > 0,
T k ( u n ) T k ( u ) f o r t h e m o d u l a r c o n v e r g e n c e i n ( L M ( Q ) ) N .

Proof:

We have proved that
Q a ( . , T k ( u n ) , T k ( u n ) ) - a ( . , T k ( u n ) , T k ( v j ) X j s ) ( T k ( u n ) - T k ( v j ) X j s ) d x d t

≤ ϵ (n, j, μ, s, m) (see (3.8)).

We can also deduce that
Q ( a ( . , T k ( u n ) , T k ( u n ) ) - a ( . , T k ( u n ) , T k ( u ) X s ) ) ( T k ( u n ) - T k ( u ) X s ) d x d t = Q a ( . , T k ( u n ) , T k ( u n ) ) - a ( . , T k ( u n ) , T k ( v j ) X j s ) ( T k ( u n ) - T k ( v j ) X j s ) d x d t + ε ( n , j , s ) .
Then
Q a ( . , T k ( u n ) , T k ( u n ) ) T k ( u n ) d x d t Q a ( . , T k ( u n ) , T k ( u n ) ) T k ( u ) X s d x d t + Q a ( . , T k ( u n ) , T k ( u ) X s ) ( T k ( u n ) T k ( u ) X s ) d x d t + ε ( n , j , μ , s , m ) . lim ¯ n Q a ( . , T k ( u n ) , T k ( u n ) ) T k ( u n ) d x d t Q a ( . , T k ( u ) , T k ( u ) ) T k ( u ) X s d x d t + lim n ε ( n , j , μ , s , m )
then,
lim ¯ n Q a ( . , T k ( u n ) , T k ( u n ) ) T k ( u n ) Q a ( . , T k ( u ) , T k ( u ) ) T k ( u ) lim ¯ n Q a ( . , T k ( u n ) , T k ( u n ) ) T k ( u n ) .
Letting n → ∞, we deduce
a ( . , T k ( u n ) , T k ( u n ) ) T k ( u n ) a ( . , T k ( u ) , T k ( u ) ) T k ( u ) X s in L 1 ( Q ) .
Using the same argument as above, we obtain
a ( . , T k ( u n ) , T k ( u n ) ) T k ( u n ) a ( . , T k ( u ) , T k ( u ) ) T k ( u ) in L 1 ( Q ) ,
and Vitali's theorem and (1.1) gives
T k ( u n ) T k ( u ) for the modular convergence in ( L M ( Q ) ) N .

3.3.1 The convergence of the problems (P n ) and the completion of the proof of Theorem 3.1

The passage to the limit is an easy task by using the last steps, then
a ( . , u n , u n ) a ( . , u , u ) weakly in L H ( Q ) and a .e . in Q ,
then,
- Q u φ t + Q a ( . , u , u ) φ d x d t - Ω φ d u 0 = Q φ d μ ,

for all φ D(RN+1) which are zero in a neighborhood of (0,T) × ∂Ω and {T} × Ω.

4 Conclusion

In this article, we have proved the existence of solutions of some class of unilateral problems in the Orlicz-Sobolev spaces when the right-hand side is a Radon measure.

Declarations

Authors’ Affiliations

(1)
Department of Mathematics, King Khalid University

References

  1. Boccardo L, Dall'Aglio A, Gallouet T, Orsina L: Nonlinear parabolic equations with measures data. J Funct Anal 1997, 147(1):237-258. 10.1006/jfan.1996.3040View ArticleMathSciNet