2005-Oujda International Conference on Nonlinear Analysis. Electronic Journal of Differential Equations, Conference 14, 2006, pp. 9–20. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp)

EXISTENCE RESULT FOR VARIATIONAL DEGENERATED PARABOLIC PROBLEMS VIA PSEUDO-MONOTONICITY

LAHSEN AHAROUCH, ELHOUSSINE AZROUL, MOHAMED RHOUDAF

Abstract. In this paper, we study the existence of weak solutions for the initial-boundary value problems of the nonlinear degenerated parabolic equation

\frac{∂u}{∂t} − div a(x, t, u, ∇u) + a_{0}(x, t, u, ∇u) = f,

where Au = − div a(x, t, u, ∇u) is a classical divergence operator of Leray-lions acting from L^{p}(0, T, W_{0}^{1,p}(Ω, w)) to its dual. The source term f is assumed to belong to L^{p′}(0, T, W^{−1,p′}(Ω, w^{∗})).

1. Introduction

Let Ω be a bounded open subset of ℝ^{N} and let Q be the cylinder Ω × (0, T) with some given T > 0. Consider the parabolic initial-boundary value problem

\frac{∂u}{∂t} + A(u) = f in Q

u(x, t) = 0 on ∂Ω × (0, T)

u(x, 0) = u_{0}(x) in Ω,

(1.1)

where Au = − div a(x, t, u, ∇u) is a classical divergence operator of Leray-lions form with respect to the Sobolev space L^{p}(0, T, W_{0}^{1,p}(Ω)) for some 1 < p < ∞. The right-hand side f is supposed lying in L^{p′}(0, T, W_{0}^{−1,p′}(Ω)).

We consider, first, the case where A satisfies the classical Leray-lions conditions, in particular the classical coercivity

a(x, t, s, ξ)ξ ≥ α|ξ|^{p}.  (1.2)

Then A is a bounded pseudo-monotone and coercive operator from the space L^{p}(0, T, W_{0}^{1,p}(Ω)) into its dual L^{p′}(0, T, W_{0}^{−1,p′}(Ω)). In this setting, problems of the form (1.1) were solved by Lions [16] and Breszis-Browder [7] in the case p ≥ 2 and by Landes [12] and Landes-Mustonen [13] when 1 < p < 2 (see also [5],[6],[8] for related topics ). When the classical coercivity (1.2) is replaced by the more general condition

2000 Mathematics Subject Classification. 35J60.

Key words and phrases. Weighted Sobolev spaces; boundary value problems; truncations;

parabolic problems.

\begin{matrix}c \\ ◯ \end{matrix}2006 Texas State University - San Marcos.

Published September 20, 2006.

a(x, t, s, ξ)ξ ≥ c ∑_{i=1}^{N} w_{i}(x)|ξ_{i}|^{p},  (1.3)

where now w(x) = {w_{i}(x), 1 ≤ i ≤ N} is a family of weight functions on Ω, the problem (1.1) can not be solved in the classical Sobolev settings L^{p}(0, T, W_{0}^{1,p}(Ω)). However, to do this, we must to change this classical setting by the general one L^{p}(0, T, W_{0}^{1,p}(Ω, w)) related to the so-called weighted Sobolev space W_{0}^{1,p}(Ω, w). In this direction, we list in particular the work [17] where the authors have studied the existence of weak solution of the variational parabolic boundary-value problems

\frac{∂u}{∂t} + A(u) + A_{0}(x, t, u, ∇u) = f in Q

u(x, t) = u_{0}(x) on ∂Ω × (0, T)

u(x, t) = 0 in Ω,

(1.4)

but under more restrictions on the weight family w (compare with Remark 2.1).

Note that, little information is known for the degenerate parabolic. Similar problems for degenerate nonlinear elliptic equations have been studied in [9] and [2]. Our aim of this paper is to study the same variational degenerate parabolic problems (1.1) in some general case of weight. For that some important lemmas is firstly proved and the approach of pseudo-monotonicity is used. A simple model of our problem is as follows

\frac{∂u}{∂t} − div(|x|^{s}|Du|^{p−2}Du) + σ(x)|u|^{p−2}u = f in Q

u(x, t) = u_{0}(x) on ∂Ω × (0, T)
u(x, t) = 0 in Ω.

The present paper is organized as follows: We start with the introduction of a basic assumptions and main result in section 2, which is proved in section 3. Finally, we give an example in section 4.

2. Assumptions and Main results

Hypotheses. Let Ω be a bounded open set of ℝ^{N}, p be a real number such that 2 < p < ∞ and w = {w_{i}(x) : 1 ≤ i ≤ N} be a vector of weight functions, i.e., every component w_{i}(x) is a measurable function which is strictly positive a.e. in Ω. Further, we suppose in all our considerations that,

w_{i} ∈ L_{loc}^{1}(Ω),  (2.1)
wi^{\frac{−1}{p−1}} ∈ L_{loc}^{1}(Ω),  (2.2)

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

\frac{∂u}{∂x_{i}} ∈ L^{p}(Ω, w_{i}) for i = 1, …, N.

This is a Banach space under the norm

∥u∥_{1,p,w} = [ ∫_{Ω} |u(x)|^{p}w_{0} dx + ∑_{i=1}^{N} ∫_{Ω} |\frac{∂u(x)}{∂x_{i}}|^{p}w_{i}(x) dx]^{1/p}.  (2.3)

The condition (2.1) implies that C_{0}^{∞}(Ω) is a subset of W^{1,p}(Ω, w) and consequently, we can introduce the subspace W_{0}^{1,p}(Ω, w) of W^{1,p}(Ω, w) as the closure of C_{0}^{∞}(Ω) with respect to the norm (2.3). Moreover, the condition (2.2) implies that W^{1,p}(Ω, w) as well as W_{0}^{1,p}(Ω, w) are reflexive Banach spaces. We recall that the dual space of weighted Sobolev spaces W_{0}^{1,p}(Ω, w) is equivalent to W^{−1,p′}(Ω, w^{∗}), where w^{∗} = {w_{i}^{∗} = w_{i}^{1−p′}, i = 0, …, N} and where p′ is the conjugate of p i.e. p′ = \frac{p}{p−1}. For more details, we refer the reader to [10].

Now we state the some assumptions.

(H1) For 2 ≤ p < ∞, the expression

∥|u∥| = (∑_{i=1}^{N}∫_{Ω} |\frac{∂u}{∂x_{i}} |^{p}w_{i}(x) dx)^{1/p}  (2.4)

is a norm on W_{0}^{1,p}(Ω, w) and it’s equivalent to (2.3). There exists a weight function σ on Ω such that

σ ∈ L^{1}(Ω) and σ^{\frac{−1}{p−1}} ∈ L_{loc}^{1}(Ω).  (2.5)

The Hardy inequality

(∫_{Ω} |u(x)|^{p}σ dx)^{1/p} ≤ c(∑_{i=1}^{N}∫_{Ω} |\frac{∂u}{∂x_{i}} |^{p}w_{i}(x) dx)^{1/p},  (2.6)

holds for every u ∈ W_{0}^{1,p}(Ω, w) with a constant c > 0 independent of u. Moreover, the imbedding

W_{0}^{1,p}(Ω, w) ↪ L^{p}(Ω, σ)  (2.7)

expressed by the inequality (2.6) is compact.

Note that (W_{0}^{1,p}(Ω, w), ∥|.∥|) is a uniformly convex (and thus reflexive) Banach space.

Remark 2.1. Assume that w_{0}(x) ≡ 1 and there exists ν ∈]\frac{N}{P} , +∞[∩[\frac{1}{P −1}, +∞[ such that

w_{i}^{−ν} ∈ L^{1}(Ω) for all i = 1, …, N.  (2.8)

Note that the assumptions (2.1) and (2.8) imply that,

∥|u∥| = (∑_{i=1}^{N}∫_{Ω} |\frac{∂u}{∂x_{i}} |^{p}w_{i}(x) dx)^{1/p}  (2.9)

is a norm defined on W_{0}^{1,p}(Ω, w) and it’s equivalent to (2.3) and that, the imbedding

W_{0}^{1,p}(Ω, w) ↪ L^{p}(Ω)  (2.10)

is compact [10, pp 46]. Thus the hypothesis (H_{1}) is satisfied for σ ≡ 1.

(H2) For i = 1, …, N,

|a_{0}(x, t, s, ξ)| ≤ βσ^{1/p}(x) [c_{0}(x, t) + σ^{\frac{1}{p′}} |s|^{p−1} + ∑_{j=1}^{N}wj^{\frac{1}{p′}} (x)|ξ_{j}|^{p−1}],  (2.11)

|a_{i}(x, t, s, ξ)| ≤ βw_{i}^{1/p}(x) [c_{1}(x, t) + σ^{\frac{1}{p′}} |s|^{p−1} + ∑_{j=1}^{N}wj^{\frac{1}{p′}} (x)|ξ_{j}|^{p−1}],  (2.12)

∑_{i=1}^{N}[a_{i}(x, t, s, ξ) − a_{i}(x, t, s, η)](ξ_{i} − η_{i}) > 0 ∀ξ ≠ η ∈ ℝ^{N},  (2.13)

a_{0}(x, t, s, ξ).s + ∑_{i=1}^{N}a_{i}(x, t, s, ξ).ξ_{i} ≥ α∑_{i=1}^{N}w_{i}|ξ_{i}|^{p},  (2.14)

where c_{0}(x, t) and c_{1}(x, t) are some positive functions in L^{p′}(Q), and α and β are some strictly positive constants.

Some lemmas. In this subsection we establish some imbedding and compactness results in weighted Sobolev Spaces which allow in particular to extend in the settings of weighted Sobolev spaces. Let V = W_{0}^{1,p}(Ω, w), H = L^{2}(Ω, σ) and let V^{∗} = W^{−1,p′}(Ω, w^{∗}), with (2 ≤ p < ∞). Let X = L^{p}(0, T, V). The dual space of X is X^{∗} = L^{p′}(0, T, V^{∗}) where \frac{1}{p′} + \frac{1}{p} = 1 and denoting the space W_{p}^{1}(0, T, V, H) = {v ∈ X : v′ ∈ X^{∗}} endowed with the norm

∥u∥_{w_{p}^{1}} = ∥u∥_{X} + ∥u′∥_{X^{∗}},  (2.15)

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

∫_{0}^{T} u′(t)φ(t) dt = − ∫_{0}^{T} u(t)φ′(t) dt for all φ ∈ C_{0}^{∞}(0, T).

Lemma 2.2. The Banach space H is an Hilbert space and its dual H′ can be identified with him self; i.e., H′ ≃ H

Lemma 2.3. The evolution triple V ⊆ H ⊆ V^{∗} is verified.

Lemma 2.4. Let g ∈ L^{r}(Q, γ) and let g_{n} ∈ L^{r}(Q, γ), with ∥g_{n}∥_{L^{r}(Q,γ)} ≤ c, 1 < r < ∞. If g_{n}(x) → g(x) a.e in Q, then g_{n} ⇀ g in L^{r}(Q, γ), where ⇀ denotes weak convergence and γ is a weight function on Q.

Lemma 2.5. 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 u_{n} ⇀ u weakly in L^{p}(0, T, W_{0}^{1,p}(Ω, w)) and

∫_{Q}[a(x, t, u_{n}, ∇u_{n}) − a(x, t, u_{n}, ∇u)][∇u_{n} − ∇u] dt dx → 0.  (2.16)

Then u_{n} → u in L^{p}(0, T, W_{0}^{1,p}(Ω, w)).

Now we recall the well-known general Sobolev imbedding theorems for evolution equations.

Lemma 2.6 ([18]). Let V ⊆ H ⊆ V^{∗} be an evolution triple. Then the imbedding

W_{p}^{1}(0, T, V, H) ⊆ C([0, T]), H)

is continuous.

Lemma 2.7 ([18]). Let Z_{1}, Y, Z_{2} be real reflexive Banach spaces. Assume that the imbeddings Z_{1} ⊆ Y ⊆ Z_{2} are continuous, and the imbedding Z_{1} ⊆ Y is compact, 0 < T < ∞, 1 < p, q < ∞. Then W = {u ∈ L^{p}(0, T, Z_{1}) : u′ ∈ L^{q}(0, T, Z_{2})} equipped with the norm

∥u∥_{w} = ∥u∥_{L^{p}(0,T,Z_{1})} + ∥u′∥_{L^{q}(0,T,Z_{2})}

is a Banach space and the imbedding W ⊆ L^{p}(0, T, Y) is compact.

Existence results.

Definition 2.8. 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 \frac{∂}{∂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} = ∫_{0}^{T} ⟨u′(t), v(t)⟩_{V} dt u ∈ D(L), v ∈ X.  (2.17)

Lemma 2.9 ([18]). 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 demicontinuous map of monotone type from X to X^{∗}.

Definition 2.10. A mapping S is pseudomonotone with respect to D(L), if for any sequence {u_{n}} in D(L) with u_{n} ⇀ u, Lu_{n} ⇀ Lu and lim_{n→∞} sup⟨S(u_{n}), u_{n} − u⟩ ≤ 0, we have lim_{n→∞}⟨S(u_{n}), u_{n} − u⟩ = 0 and S(u_{n}) ⇀ S(u) as n → ∞.

Consider the non linear parabolic problem

\frac{∂u}{∂t} + A(u) + A_{0}(x, t, u, ∇u) = f in Q

u(x, t) = 0 on ∂Ω × (0, T)
u(x, 0) = u_{0} in Ω.

Definition 2.11. A function u is said to be a weak solution of the initial-boundary value problem (1.4) if u ∈ C([0, T], H) ∩ L^{p}(0, T, V), \frac{∂u}{∂t} ∈ L^{p′}(0, T, V^{∗}) and u satisfies the equation

\frac{∂u}{∂t} + Au + A_{0}u = f 0 < t < T, u(0) = u_{0},

where the operator A + A_{0} : X → X^{∗} is defined by

⟨(A + A_{0})(u), v⟩ = ∫_{Q} a(x, t, u, ∇u)∇v dx dt + ∫_{Q} a_{0}(x, t, u, ∇u)v dx dt

Proposition 2.12. The operator A + A_{0} : X → X^{∗} is :

(a) bounded and demicontinuous;

(b) pseudomonotone with respect to D(L)

(c) strongly coercive, i.e.,

\frac{⟨(A + A_{0})(u), u⟩_{X}}{∥u∥_{X}} → +∞, as ∥u∥_{X} → +∞.

We first consider the Zero-initial value problem,

\frac{∂u}{∂t} + A(u) + A_{0}(x, t, u, ∇u) = f in Q

u(x, t) = 0 on ∂Ω × (0, T)
u(x, 0) = 0 in Ω,

(2.18)

Theorem 2.13. Assume that the conditions (H1)-(H2) hold, then problem (2.18) admits a weak solution for any f ∈ X^{∗}.

Theorem 2.14. Assume that the conditions (H1)-(H2) hold and u_{0} ∈ W_{0}^{1,p}(Ω, w), then the initial-boundary value problem (1.4) admits a weak solution for any f ∈ X^{∗}.

3. Proofs of Main results

Proof of lemma 2.2. Let the map F : H × H → ℝ be defined by

F (f, g) = ∫_{Ω} fgσ dx.

Note that F is a symmetric bilinear form, which is also continuous and defined positively, since

∫_{Ω} fgσ dx = ∫_{Ω} fσ^{1/2}gσ^{1/2} dx ≤ ( ∫_{Ω} |f|^{2}σ dx)^{1/2}( ∫_{Ω} |g|^{2}σ dx)^{1/2}.

Then the Banach space H is an Hilbert space. Finally by a standard argument, we can identify H with its dual H′; i.e., H′ ≃ H. □

Proof of lemma 2.3. By the imbedding (2.7) and the fact that 2 ≤ p < ∞, and σ ∈ L^{1}(Ω), we can write

W_{0}^{1,p}(Ω, w) ↪↪ L^{p}(Ω, σ) ↪ H ≃ H′ ↪ W^{−1,p′}(Ω, w^{∗}).
□

Proof of lemma 2.4. Since g_{n}γ^{1/r} is bounded in L^{r}(Q) and g_{n}(x)γ^{1/r}(x) → gγ^{1/r}, a.e. in Q, then by [15, lemma 3.2], we have

g_{n}γ^{1/r} ⇀ gγ^{1/r} in L^{r}(Q).

Moreover, for all φ ∈ L^{r′}(Q, γ^{1−r′}), we have φγ^{\frac{−1}{r}} ∈ L^{r′}(Q). Then

∫_{Q} g_{n}φ dx → ∫_{Q} gφ dx, i.e. g_{n} ⇀ g in L^{r}(Q, γ).
□

Proof of lemma 2.5. Let D_{n} = [a(x, t, u_{n}, ∇u_{n}) − a(x, t, u_{n}, ∇u)][∇u_{n} − ∇u]. Then by (2.13), D_{n} is a positive function and by (2.16), D_{n} → 0 in L^{1}(Q). Extracting a subsequence, still denoted by u_{n}, and using (2.7) we can write

u_{n} → u a.e. in Q, D_{n} → 0 a.e. in Q.

Then, there exists a subset B of Q, of zero measure such that for (t, x) ∈ Q \ B, |u_{n}(x, t)| < ∞, |∇u(x, t)| < ∞, |c_{1}(x, t)| < ∞, w_{i}(x) > 0 and u_{n}(x, t) → u(x, t), D_{n}(x, t) → 0. We set ϵ_{n} = ∇u_{n}(x, t) and ϵ = ∇u(x, t). Then

D_{n}(x, t) = [a(x, t, u_{n}, ϵ_{n}) − a(x, t, u_{n}, ϵ)](ϵ_{n} − ϵ)

≥ α ∑_{i=1}^{N} w_{i}|ϵ_{n}^{i}|^{p} + α ∑_{i=1}^{N} w_{i}|ϵ^{i}|^{p}

− ∑_{i=1}^{N} βw_{i}^{1/p}[c_{1}(x, t) + σ^{\frac{1}{p′}} |u_{n}|^{p−1} + ∑_{j=1}^{N} wj^{\frac{1}{p′}} |ϵ_{n}^{j}|^{p−1}]|ϵ^{i}|

− ∑_{i=1}^{N} βw_{i}^{1/p}[c_{1}(x, t) + σ^{\frac{1}{p′}} |u_{n}|^{p−1} + ∑_{j=1}^{N} wj^{\frac{1}{p′}} |ϵ^{j}|^{p−1}]|ϵ_{n}^{i}|;

i.e,

D_{n}(x, t) ≥ α ∑_{i=1}^{N} w_{i}|ϵ_{n}^{i}|^{p} − c_{x,t}[1 + ∑_{j=1}^{N} wj^{\frac{1}{p′}} |ϵ_{n}^{j}|^{p−1} + ∑_{i=1}^{N} w_{i}^{1/p}|ϵ_{n}^{i}|],  (3.1)

where c_{x,t} is a constant which depends on x, but does not depend on n. Since u_{n}(x, t) → u(x, t), we have |u_{n}(x, t)| ≤ M_{x,t}, where M_{x,t} is some positive constant. Then by a standard argument |ϵ_{n}| is bounded uniformly with respect to n. Indeed, (3.1) becomes

D_{n}(x, t) ≥ ∑_{i=1}^{N} |ϵ_{n}^{i}|^{p}(αw_{i} − \frac{c_{x,t}}{N|ϵ_{n}^{i}|^{p}} − \frac{c_{x,t}wi^{\frac{1}{p′}}}{|ϵ_{n}^{i}|} − \frac{c_{x,t}w_{i}^{1/p}}{|ϵ_{n}^{i}|^{p−1}} ).

If |ϵ_{n}| → ∞ (for a subsequence) there exists at least one i_{0} such that |ϵ_{n}^{i_{0}}| → ∞, which implies that D_{n}(x, t) → ∞ which gives a contradiction.

Let now ϵ^{∗} be a cluster point of ϵ_{n}. We have |ϵ^{∗}| < ∞ and by the continuity of a with respect to the two last variables we obtain

(a(x, t, u(x, t), ϵ^{∗}) − a(x, t, u(x, t), ϵ))(ϵ^{∗} − ϵ) = 0.

In view of (2.13) we have ϵ^{∗} = ϵ. The uniqueness of the cluster point implies

∇u_{n}(x, t) → ∇u(x, t) a.e. in Q.

Since the sequence a(x, t, u_{n}, ∇u_{n}) is bounded in the space ∏_{i=1}^{N} L^{p′}(Q, w_{i}^{∗}) and a(x, t, u_{n}, ∇u_{n}) → a(x, t, u, ∇u) a.e. in Q, Lemma 2.4 implies

a(x, t, u_{n}, ∇u_{n}) ⇀ a(x, t, u, ∇u) in ∏_{i=1}^{N} L^{p′}(Q, w_{i}^{∗}) and a.e. in Q.

We set ȳ_{n} = a(x, t, u_{n}, ∇u_{n})∇u_{n} and ȳ = a(x, t, u, ∇u)∇u. As in [4, lemma lemma 5] we can write ȳ_{n} → ȳ in L^{1}(Q). By (2.14), we have

α ∑_{i=1}^{N} w_{i}|\frac{∂u_{n}}{∂x_{i}} | ≤ a(x, t, u_{n}, ∇u_{n})∇u_{n}.

Let

z_{n} = ∑_{i=1}^{N} w_{i}|\frac{∂u_{n}}{∂x_{i}} |^{p}, z = ∑_{i=1}^{N} w_{i}|\frac{∂u}{∂x_{i}} |^{p},

y_{n} = \frac{ȳ_{n}}{α} , y = \frac{ȳ}{α}.

Then, by Fatou’s lemma we obtain

∫_{Q} 2y dxdt ≤ lim_{n→∞} inf ∫_{Q} y + y_{n} − |z_{n} − z| dx dt;

i.e., 0 ≤ lim_{n→∞} sup ∫_{Q} |z_{n} − z| dxdt, hence

0 ≤ lim_{n→∞} inf ∫_{Q} |z_{n} − z| dxdt ≤ lim_{n→∞} sup ∫_{Q} |z_{n} − z| dxdt ≤ 0.

This implies

∇u_{n} → ∇u in ∏_{i=1}^{N} L^{p}(Q, w_{i}),

which with (2.4) completes the present proof. □

Proof of proposition 2.12. (a) We set B = A+A_{0}. Using (2.11), (2.12) and Hölder’s inequality we can show that B is bounded. For showing that B is demicontinuous, let v_{ϵ} → v in X as ϵ → 0, and prove that,

⟨B(v_{ϵ}), φ⟩ → ⟨B(v), φ⟩ for all φ ∈ X.

Since, a_{i}(x, t, v_{ϵ}, ∇v_{ϵ}) → a_{i}(x, t, v, ∇v) as ϵ → 0, for a.e. x ∈ Ω, by the growth conditions (2.12), (2.11) and lemma 2.4 we get

a_{i}(x, t, v_{ϵ}, ∇v_{ϵ}) ⇀ a_{i}(x, t, v, ∇v) in L^{p′}(Q, w_{i}^{1−p′}) as ϵ → 0

for (i = 1, …, N) and

a_{0}(x, t, v_{ϵ}, ∇v_{ϵ}) ⇀ a_{0}(x, t, v, ∇v) in L^{p′}(Q, σ^{1−p′}) as ϵ → 0.

Finally for all φ ∈ X,

⟨B(v_{ϵ}), φ⟩ → ⟨B(v), φ⟩ as ϵ → 0

(since φ ∈ L^{p}(Q, σ) for all φ ∈ X).

(b) Suppose that {u_{j}} is a sequence in D(L) with (i) u_{j} ⇀ u weakly in X (ii) Lu_{j} → Lu weakly in X^{∗}, (iii) lim sup⟨A + A_{0}(u_{j}), u_{j} − u⟩_{X} ≤ 0.

By the definition of the operator L in (2.17), we obtain that {u_{j}} is a bounded sequence in W_{p}^{1}(0, T, V, H). By virtue of lemma 2.7, we get,

u_{j} → u strongly in L^{p}(Q, σ).

On the other hand,

⟨A_{0}u_{j}, u_{j} − u⟩ = ∫_{Q} a_{0}(x, t, u_{j}, ∇u_{j})(u_{j} − u) dx dt

Thus the Hölder’s inequality and (i) imply

⟨A_{0}u_{j}, u_{j} − u⟩ ≤ ( ∫_{Q} |a_{0}|^{p′}σ^{1−p′} dx dt)^{1/p′}∥u_{j} − u∥_{L^{p}(Q,σ)}
≤ ∥a_{0}∥_{L^{p′} (Q,σ∗)}∥u_{j} − u∥_{L^{p}(Q,σ)},

i.e, ⟨A_{0}u_{j}, u_{j} − u⟩ → 0 as j → ∞. Combining the last convergence with (iii), we obtain

lim_{j→∞} sup⟨Au_{j}, u_{j} − u⟩ ≤ 0.

And by the pseudo-monotonicity of A (see [9, Proposition 1]), we have

Au_{j} ⇀ Au in X^{∗} and lim_{j→∞}⟨Au_{j}, u_{j} − u⟩ = 0.

Then

lim_{j→∞}⟨Au_{j} + A_{0}(u_{j}), u_{j} − u⟩ = 0.

On the other hand, lim_{j→∞}⟨Au_{j}, u_{j} − u⟩ = 0, which implies

0 = lim_{j→∞} ∫_{Q} a(x, t, u_{j}, ∇u_{j})∇(u_{j} − u) dx dt

= lim_{j→∞} ∫_{Q}[a(x, t, u_{j}, ∇u_{j}) − a(x, t, u_{j}, ∇u)][∇u_{j} − ∇u] dx dt

+ lim_{j→∞} ∫_{Q} a(x, t, u_{j}, ∇u)(∇u_{j} − ∇u) dx dt.

The last integral in the right hand tends to zero, since by the continuity of the Nemytskii operator, a(x, t, u_{j}, ∇u) → a(x, t, u, ∇u) in ∏_{i=1}^{N} L^{p′}(Q, w_{i}^{1−p′}) as j → +∞. So that

lim_{j→∞} ∫_{Q}[a(x, t, u_{j}, ∇u_{j}) − a(x, t, u_{j}, ∇u)][∇u_{j} − ∇u] dx dt = 0.

By lemma 2.5 we have

∇u_{j} → ∇u a.e. in Q.

Hence a_{0}(x, t, u_{j}, ∇u_{j}) → a_{0}(x, t, u, ∇u) a.e. in Q as j → ∞ and since

a_{0}(x, t, u_{j}, ∇u_{j}) ∈ L^{p′}(Q, σ^{1−p′})

by Lemma 2.4, we obtain

a_{0}(x, t, u_{j}, ∇u_{j}) ⇀ a_{0}(x, t, u, ∇u) in L^{p′}(Q, σ^{1−p′}).

Finally,

B(u_{j}) ⇀ B(u) in X^{∗}.

(c) The strongly coercivity follows from (2.14) □

Proof of Theorem 2.13. By proposition 2.12 the operator A + A_{0} : X → X^{∗} is pseudomonotone with respect to D(L), and the operator A+A_{0} satisfies the strong coercivity condition which implies that both of the conditions (i) and (ii) in [3, theorem 4] hold. So all the conditions in [3, theorem 4] are met. Therefore, there exists a solution u ∈ D(L) of the evolution equation

\frac{∂u}{∂t} + Au + A_{0}u = f

for any f ∈ X^{∗}. In order to prove that u is also a weak solution of the problem (2.18), we have to show that u ∈ C([0, T], H). By the definition of D(L) and lemma 2.6, we obtain

D(L) ⊆ W_{p}^{1}(0, T, V, H) ⊆ C([0, T], H).

Which implies that u ∈ C([0, T], H). □

Proof of Theorem 2.14. Now we turn to problem (1.4). Assume that (H1)-(H2) hold and u_{0} ∈ W_{0}^{1,p}(Ω, w). Let

\overline{a}_{i}(x, t, u, ∇u) = a_{i}(x, t, u + u_{0}, ∇u + ∇u_{0})

for all i = 0,…,N. Then it is easy to see that \overline{a}_{i} also satisfies the conditions (H1)- (H2). But β , α and the function c_{0}(x, t) and c_{1}(x, t) in (H1)-(H2) may depend on the function u_{0}. Analogously, \overline{A} + \overline{A}_{0} : X → X^{∗} is defined by

⟨(\overline{A} + \overline{A}_{0})(u), v⟩ = ∫_{Q} \overline{a}(x, t, u, ∇u)∇v dx dt + ∫_{Q} \overline{a}_{0}(x, t, u, ∇u)v dx dt

for u, v ∈ L^{p}(0, T, V), where \overline{A} = −div\overline{a}(x, t, u, ∇u). Then, by Theorem 2.13, we have Theorem 2.14 □

4. An example

Let Ω be a bounded domain of ℝ^{N}(N ≥ 1), satisfying the cone condition. Let us consider the Carathéodory functions

a_{i}(x, t, s, ϵ) = w_{i}|ϵ_{i}|^{p−1} sgn(ϵ_{i}) for i = 1, …, N,
a_{0}(x, t, s, ϵ) = ρσ(x)s|s|^{p−2}, ρ > 0,

where σ and w_{i}(x) (i =, 1, …, N) are given weight functions, strictly positive almost everywhere in Ω. We shall assume that the weight functions satisfy, w_{i}(x) = w(x), x ∈ Ω, for all i = 1, …, N. Then, we can consider the Hardy inequality (2.6) in the form

( ∫_{Q} |u(x, t)|^{p}σ(x) dx)^{1/p} ≤ c( ∫_{Q} |∇u(x, t)|^{p}w dx)^{1/p}.

It is easy to show that the functions a_{i}(x, t, s, ϵ) are Carathéodory functions satisfying the growth condition (2.12) and the coercivity (2.13). On the other hand, the monotonicity condition is satisfied, in fact,

∑_{i=1}^{N}(a_{i}(x, t, s, ϵ) − a_{i}(x, t, s, ϵ̂))(ϵ_{i} − ϵ̂_{i})

= w(x) ∑_{i=1}^{N}(|ϵ_{i}|^{p−1} sgn ϵ_{i} − |ϵ̂_{i}|^{p−1} sgn ϵ̂_{i})(ϵ_{i} − ϵ̂_{i}) > 0

for almost all (x, t) ∈ Q and for all ϵ, ϵ̂ ∈ ℝ^{N} with ϵ ≠ ϵ̂, since w > 0 a.e. in Ω. In particular, let us use the special weight functions w and σ expressed in terms of the distance to the boundary ∂Ω. Denote d(x) = dist(x, ∂Ω) and set

w(x) = d^{λ}(x), σ(x) = d^{μ}(x).

In this case, the Hardy inequality reads

( ∫_{Q} |u(x, t)|^{p}d^{μ}(x) dx)^{1/p} ≤ (c ∫_{Q} |∇u(x, t)|^{p}d^{λ}(x) dx)^{1/p}.

For λ < p − 1, \frac{μ−λ}{p} + 1 > 0; See for example [9]

Corollary 4.1. The parabolic initial-boundary value problem

∫_{Q} \frac{∂u(x, t)}{∂t}φ dx dt + ∫_{Q} d^{λ}(x) ∑_{i=1}^{N} |\frac{∂u(x, t)}{∂x_{i}}|^{p−1} sgn(\frac{∂u}{∂x_{i}})\frac{∂φ(x, t)}{∂x_{i}} dx dt

+ ∫_{Q} ρd^{μ}(x)u(x, t)|u(x, t)|^{p−2}φ(x, t)

= ∫_{Q} fφ dx dt ∀φ ∈ D(Q)

admits at least one solution u in L^{p}(0, T, W_{0}^{1,p}(Ω, d^{λ})), for any function f in L^{p′}(0, T, W^{−1,p′}(Ω, d^{λ′})) where λ′ = λ(1 − p′) and u_{0} ∈ W_{0}^{1,p}(Ω, d^{λ}).

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Lahsen Aharouch Département de Mathématiques et Informatique, Faculté des Sciences Dhar-Mahraz, B.P 1796 Atlas Fès, Maroc

E-mail address: l aharouch@yahoo.fr

Elhoussine Azroul Département de Mathématiques et Informatique, Faculté des Sciences Dhar-Mahraz, B.P 1796 Atlas Fès, Maroc

E-mail address: azroul elhoussine@yahoo.fr

Mohamed Rhoudaf Département de Mathématiques et Informatique, Faculté des Sciences Dhar-Mahraz, B.P 1796 Atlas Fès, Maroc

E-mail address: rhoudaf mohamed@yahoo.fr