2002-Fez conference on Partial Differential Equations, Electronic Journal of Differential Equations, Conference 09, 2002, pp 117–126. http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp)

Existence of stable periodic solutions for quasilinear parabolic problems in the presence of

well-ordered lower and upper-solutions ^{∗}

Abderrahmane El Hachimi & Abdelilah Lamrani Alaoui

Abstract

We present existence and stability results for periodic solutions of quasilinear parabolic equation related to Leray-Lions’s type operators. To prove existence and localization, we use the penalty method; while for stability we use an approximation scheme.

1 Introduction

In the last few years, many works have been devoted to the existence and stability of periodic solutions of problem

\frac{∂u}{∂t} + A(u) + F (u, ∇u) = 0 in Ω × ℝ^{+},

u = 0 on ∂Ω × ℝ^{+},
u(0) = u(T) in Ω,

(1.1)

where Ω is a bounded and open subset of ℝ^{N}, N ≥ 1. For the usual Leray-Lions’s operator A, Deuel and Hess [4] obtained existence of periodic solutions under the presence of well-ordered lower and upper-solutions. Unfortunately, uniqueness and therefore stability, can not be derived from the definition they used for solutions of (1.1).

For A(u) = −Δg(u) and F depending only on (x, t), Harraux and Kenmochi [7] proved both existence and stability results by using subdifferential theory on Hilbert spaces.

Recently, Boldrini and Crema [2] considered the case where A(u) is the plaplacian operator, with p ≥ 2, and F is independent of ∇u. They obtained an existence result via Shauder’s fixed point theorem.

∗Mathematics Subject Classifications: 35K55, 35B20, 35B40.

Key words: Leray-lions operator, penalization, lower and upper-solutions,

monotone process, periodic solutions, stabilization.

\begin{matrix}c \\ ◯ \end{matrix}2002 Southwest Texas State University.

Published December 28, 2002.

More recently, De coster and Omari [3] considered problem (1.1) with a linear uniformly elliptic operator A

Au := − ∑_{i,j=1}^{N} ∂_{x_{i}}(a_{i,j}∂_{x_{j}}u) + ∑_{i=1}^{N} a_{i}∂_{x_{i}}u + a_{0}u

and F independent of ∇u. These last authors obtained stability in a suitable sense of the maximal and minimal solutions in the presence of well-ordered lower and upper-solutions.

The aim of this paper is to show that the result of De Coster and Omari still holds for general problem (1.1). Our existence result is obtained by using a classical method of penalization as it was done by Grenon in [4]; while the stability one follows essentially the principal arguments of De Coster and Omari with some changes imposed by the nonlinear character of the equation in (1.1).

This paper is organized as follows: In section 2 we recall some known results related to the initial boundary value problem associated with (1.1), and give hypotheses and definitions of solutions. In section 3, we give existence and uniqueness results concerning periodic solutions of problem (1.1), while section 4 is devoted to the stability result of periodic solutions. Finally, in section 5 we give an application to a periodic-parabolic problem associated to the p-laplacian operator.

2 Hypotheses, definitions, and known results

Let Ω be an open bounded subset of ℝ^{N} with boundary ∂Ω and T > 0 a fixed real. We shall denote

Q_{T} := Ω×]0, T [, Σ_{T} := ∂Ω×]0, T [,

and for a real p with 1 < p < +∞, we denote by V the space V := L^{p}(0, T; W_{0}^{1,p}) and by V′ := L^{p′}(0, T; W^{−1,p′}) its dual, with p′ the real conjugate of p : \frac{1}{p} + \frac{1}{p′} = 1.

Let us consider the Leray-Lions’s operator

A(v) := − ∑_{i=1}^{N} \frac{∂}{∂x_{i}} A_{i}(x, t, v, ∇v), for each v ∈ V.  (2.1)

We shall use the following assumptions:

(A1) A_{i} are caratheodory functions such that there exists β_{i} > 0, and k_{i} ∈ L^{p′}(Q_{T}), so that for all s ∈ ℝ and all ξ ∈ ℝ^{N}:

|A_{i}(x, t, s, ξ)| ≤ β_{i}(|s|^{p−1} + |ξ|^{p−1} + k_{i}(x, t)), ∀i = 1, …, N,

(A2) For all s ∈ ℝ and all ξ, ξ^{∗} ∈ ℝ^{N}, with ξ ≠ ξ^{∗}, we have

∑_{i=1}^{N}[A_{i}(x, t, s, ξ) − A_{i}(x, t, s, ξ^{∗})](ξ − ξ^{∗}) > 0 a.e. in Q_{T} .

(A3) There exists α > 0, so that for all s ∈ ℝ and all ξ ∈ ℝ^{N}, we have

∑_{i=1}^{N} A_{i}(x, t, s, ξ)ξ_{i} ≥ α|ξ|^{p} a.e. in Q_{T} .

(A4) The function f is of caratheodory type on Q_{T} × ℝ × ℝ^{N}, and there exist functions b : ℝ^{+} → ℝ^{+} increasing, and h ∈ L^{1}(Q_{T}), h ≥ 0, such that

|f(x, t, s, ξ)| ≤ b(|s|)(h(x, t) + |ξ|^{p}), for (x, t, s, ξ) ∈ Q_{T} × ℝ × ℝ^{N}.

We denote by F the Nemyskii operator related to f and defined by

F (u, ∇u)(x, t) := f(x, t, u, ∇u).

To obtain (among other results) global existence for initial boundary-value problems associated with (1.1), we shall assume the following.

(A5) There exists c_{i} > 0 and l_{i} ∈ L^{p′}(Q_{T}) with l_{i} ≥ 0, such that for all s, s^{∗} ∈ ℝ and all ξ ∈ ℝ^{N},

|A_{i}(x, t, s, ξ) − A_{i}(x, t, s^{∗}, ξ)| ≤ c_{i}|s − s^{∗}|[l_{i}(x, t) + |ξ|^{p−1}] a.e. in Q_{T} .

(A6) All data (coefficients and second member) are periodic in time with period T.

We are interested in the existence and stability of the solutions of problem

\frac{∂u}{∂t} + A(u) + F (u, ∇u) = 0 in Q_{T} ,

u = 0 on Σ_{T} ,
u(0) = u(T) in Ω.

(2.2)

To this end, we consider the problem (P_{t_{1},t_{2};u_{0}}):

\frac{∂u}{∂t} + A(u) + F (u, ∇u) = 0 in Ω×]t_{1}, t_{2}[,

u = 0 on ∂Ω×]t_{1}, t_{2}[,
u(t_{1}) = u_{0} in Ω,

(2.3)

where 0 ≤ t_{1} < t_{2} ≤ +∞ and u_{0} is a given function in L^{∞}(Ω).

Definition We say that α is a lower-solution of (2.2) if

α ∈ V ∩ L^{∞}(Q_{T}), \frac{∂α}{∂t} ∈ V′ + L^{1}(Q_{T})

and (in the distributional sense)

\frac{∂α}{∂t} + A(α) + F (α, ∇α) ≤ 0 in Q_{T} ,

α(0) ≤ α(T) in Ω.

An upper-solution is defined by reversing the sense of inequalities. And a solution is a function which is simultaneously lower and upper-solution.

Definition We say that α is a lower-solution of problem (P_{t_{1},t_{2};u_{0}}) if: α ∈ L^{p}(t_{1}, t_{2}; W^{1,p}) ∩ L^{∞}(Ω×]t_{1}, t_{2}[), \frac{∂α}{∂t} ∈ L^{p′}(t_{1}, t_{2}; W^{−1,p′}) + L^{1}(Ω×]t_{1}, t_{2}[) and

\frac{∂α}{∂t} + A(α) + F (α, ∇α) ≤ 0 in Ω×]t_{1}, t_{2}[,

α ≤ 0 on ∂Ω×]t_{1}, t_{2}[,
α(0) ≤ u_{0} in Ω.

Upper-solutions and solutions of (P_{t_{1},t_{2};u_{0}}) are defined exactly as in the periodic case.

Remarks 1.) As the function f does not satisfy any Lipschitz condition, the use of systematic results concerning stability questions, as the Poincaré operator in connection with the theory of monotone operators and discrete dynamical systems (see [8] or [1]), seems not to be possible. 2.) As in [5], our definitions allow us to use solutions as lower or upper-solutions. This enables us to prove an uniqueness result among a class of periodic solutions. This can not be done when using the definitions of [4], where it is supposed that lower and upper-solutions are more regular than solutions.

Now, we recall some known results concerning solutions of (P_{t_{1},t_{2};u_{0}}) with T′ > 0 and u_{0} ∈ L^{∞}(Ω). We refer the reader to [5] for proofs.

Lemma 2.1 Assume (A1)–(A5) and let (α_{1}, α_{2}) and (β_{1}, β_{2}) be respectively pairs of lower and upper-solutions of (P_{0,T′;u_{0}}) such that

sup(α_{1}, α_{2}) ≤ inf(β_{1}, β_{2}) a.e. in Q_{T′}.

Then, there exists a solution u ∈ C([0, T′]; L^{q}(Ω)) for any q ≥ 1, of (P_{t_{1},t_{2};u_{0}}) such that sup(α_{1}, α_{2}) ≤ u ≤ inf(β_{1}, β_{2}) a.e. in Q_{T′}. Moreover, when α_{1} = α_{2} and β_{1} = β_{2}, the Hypothesis (A5) can be removed.

Lemma 2.2 Assume (A1)–(A5) and let α and β be respectively lower and upper-solutions of (P_{0,T′;u_{0}}), for any T′ > 0. Then, there exists u (resp. v) ∈ C([0, +∞[; L^{q}(Ω)), ∀q ≥ 1 such that for any T′ > 0, the restriction of u (resp. v) on [0, T′] is the minimal (resp. maximal) solution of (P_{0,T′;u_{0}}) located between α and β. Moreover, if u_{0} and v_{0} are in L^{∞}(Ω) and satisfy

α(0) ≤ u_{0} ≤ v_{0} ≤ β(0) a.e. in Ω

and u_{min}(u_{0}) (resp. u_{min}(v_{0})) is the minimal solution of (P_{0,T′;u_{0}}) with u_{0} (resp. (P_{0,T′;v_{0}}) with v_{0}) laying between α and β, then

u_{min}(u_{0}) ≤ u_{min}(v_{0}), a.e. in Q_{T′}.

Furthermore, the same holds for maximal solutions.

Lemma 2.3 Assume (A1)–(A5). Let 0 < T_{1} < T_{2} and α and β be respectively lower and upper-solution of (P_{0,T′;u_{0}}) with T′ > 0. Let u_{1} (resp. u_{2}) be the minimal solution of (P_{0,T_{1};u_{0}}) (resp. (P_{0,T_{2};u_{0}})) located between α and β. Then u_{1} is the restriction of u_{2} on [0, T_{2}] and the same holds for maximal solutions.

3 Existence and uniqueness of periodic solutions

The first result of this section is the following.

Theorem 3.1 Assume (A1)–(A4) and let α and β be respectively lower and upper-solutions of (2.2) with α ≤ β a.e. in Q_{T} . Then, problem (2.2) has a weak solution u satisfying u ∈ C([0, T]; L^{q}(Ω)), for any q ≥ 1, and α ≤ u ≤ β a.e. in Q_{T}.

Proof The proof is similar to that of the corresponding initial boundary-value problem treated in [5]. We shall give here only a sketch.

(i) We regularize (2.2) by taking

A_{i}^{∗}(u, ∇u) := A_{i}(Su, ∇u), i ∈ {1, …, N}

F_{ϵ}^{∗}(u, ∇u) := \frac{F (Su, ∇Su)}{1 + ϵ|F (Su, ∇Su)|}

where Su := u+(α−u)^{+} −(u−β)^{−}, ϵ > 0, and using the penalization operator θ_{η} related to the convex

K := {v ∈ V such that − k ≤ v ≤ k a.e. in Q_{T} },

where k is such that −k ≤ α − 1 ≤ β + 1 ≤ k. For η and ϵ > 0 fixed, consider the problem

u_{η,ϵ} ∈ V, \frac{∂u_{η,ϵ}}{∂t} ∈ V′,

\frac{∂u_{η,ϵ}}{∂t} + ∑_{i=1}^{N} \frac{∂}{∂x_{i}} A_{i}^{∗}(u_{η,ϵ}, ∇u_{η,ϵ}) + F_{ϵ}^{∗}(u_{η,ϵ}, ∇u_{η,ϵ}) + θ_{η}(u_{η,ϵ}) = 0 in Q_{T} ,

u_{η,ϵ}(0) = u_{η,ϵ}(T) in Ω.

(3.1)

By [9, Theorem 1.1] (see also section 2.2 of chapter 3, p. 328), this problem has a solution u_{η,ϵ}. moreover the estimates of [5, lemmas 3.6, 39] still apply and eventually after extracting a subsequence, we get

lim_{η→0^{+}} u_{η,ϵ} = u_{ϵ} in V,

with u_{ϵ} a solution of the variational inequality

\begin{matrix}⟨\frac{∂u_{ϵ}}{∂t} , v − u_{ϵ}⟩ + ∫_{Q_{T}} A^{∗}(u_{ϵ}, ∇u_{ϵ})∇(v − u_{ϵ}) + ∫_{Q_{T}} F_{ϵ}^{∗}(u_{ϵ}, ∇u_{ϵ})(v − u_{ϵ}) ≥ 0 \\ u_{ϵ} ∈ K, for v ∈ K. \end{matrix}

(3.2)

and of the system of equations

\frac{∂u_{ϵ}}{∂t} − div(A^{∗}(u_{ϵ}, ∇u_{ϵ})) + F_{ϵ}^{∗}(u_{ϵ}, ∇u_{ϵ}) + g_{ϵ} = 0 in Q_{T}

u_{ϵ} = 0 on Σ_{T} ,
u_{ϵ}(0) = u_{ϵ}(T) in Ω,

(3.3) where

lim_{η→0^{+}} θ_{η}(u_{η,ϵ}) = g_{ϵ} in L^{p′}(Q_{T}) weak .

As in [5, p. 93], there exists u ∈ V such that lim_{ϵ→0^{+}} u_{ϵ} = u in V and lim_{ϵ→0^{+}} \frac{∂u_{ϵ}}{∂t} = \frac{∂u}{∂t} in V′ + L^{1}(Q_{T}), with u satisfying

\frac{∂u}{∂t} = div(A(Su, ∇u)) + F (Su, ∇Su).

To conclude that u ∈ C([0, T]; L^{q}(Ω)) for any q ≥ 1, it suffices to show that u(0) ∈ L^{∞}(Ω) and then use [5, Lemma 3.2]. In fact, u_{ϵ} ∈ L^{p}(0, T; W_{0}^{1,p}(Ω) ∩ L^{∞}(Ω)), and \frac{∂u_{ϵ}}{∂t} ∈ V′ so that u_{ϵ} ∈ C([0, T]; L^{2}(Ω)) by Lions’s lemma [9, p. 156]. But u_{ϵ} ∈ K, so the following claim gives −k ≤ u_{ϵ}(0) ≤ k a.e. in Ω.

Claim. Let u, v ∈ C([0, T]; L^{1}(Ω)) with u ≥ v a.e. in Q_{T}. Then u(t) ≥ v(t) a.e. in Ω for all t ∈ [0, T]. To prove this claim take w := (v−u)^{+}, so that w = 0 a.e. in Q_{T}. The continuity and the non negativity of t → ∫_{Ω} w(x, t)dx on [0, T] gives the result. (ii) A careful application of [5, Lemma 3.1] shows that

⟨⟨\frac{∂α}{∂t} − \frac{∂u_{ϵ}}{∂t} , (α − u_{ϵ})^{+}⟩⟩ ≥ 0.

Where ⟨⟨., .⟩⟩ is the duality between V ∩ L^{∞}(Q_{T}) and V′ + L^{1}(Q_{T}). So we get: α ≤ u a.e. in Q_{T} and by similar arguments, we also obtain u ≤ β a.e. in Q_{T}.

Now we state a uniqueness result concerning maximal and minimal solutions.

Theorem 3.2 Assume (A1)–(A5) and let α and β be respectively lower and upper-solutions of (2.2) such that α ≤ β. Then, there exist a minimal solution v and a maximal solution w of (2.2) such that α ≤ v ≤ w ≤ β a.e. in Q_{T}.

The proof is based on the following lemma.

Lemma 3.3 Assume (A1)–(A5) and let α_{1}, α_{2} be two lower-solutions and β be an upper-solutions of (2.2) such that sup(α_{1}, α_{2}) ≤ β_{1} a.e. in Q_{T}. Then, there exists at least one weak solution of (2.2) such that sup(α_{1}, α_{2}) ≤ u ≤ β a.e. in Q_{T}

The proof of this lemma is the same as that in [5, Theorem 3.2], except for what concerns the inequality of [5, Lemma 3.18], which must be replaced by

⟨⟨\frac{∂α_{1}}{∂t} , [1 − β_{δ}(α_{2} − α_{1})]ω_{δ}⟩⟩ + ⟨⟨\frac{∂α_{2}}{∂t} , β_{δ}(α_{2} − α_{1})ω_{δ}⟩⟩ + ⟨−\frac{∂u_{ϵ}}{∂t} , ω_{δ}⟩ ≥ φ(δ)

where γ_{δ}, β_{δ} and ω_{δ} are defined as in [5, p. 31], φ is given by the uniform continuity of the function s → s^{+} on some compact set associated to K and is such that φ(δ) → 0 as δ → 0^{+}, and where ⟨., .⟩ designates the duality between V an V′.

4 Stability result

The aim of this section is to prove the following theorem.

Theorem 4.1 Assume (A1)–(A6) and let α and β be respectively lower and upper-solution of (2.2) with α ≤ β a.e. in Q_{T} and α(0), β(0) ∈ L^{∞}(Ω). Denote by v (resp. ω) the minimal (resp. maximal) solution of (2.2) located between α and β. Then, for all u_{0} ∈ L^{∞}(Ω) satisfying α(0) ≤ u_{0} ≤ v(0) (resp. ω(0) ≤ u_{0} ≤ β(0)), the set U(u_{0}, α, v) (resp. U(u_{0}, β, ω)) of all solutions u of (P_{0,+∞;u_{0}}) satisfying α ≤ u ≤ v (resp. ω ≤ u ≤ β)in Ω × (0, +∞), is nonempty and is such that for any q ≥ 1, we have

lim_{t→+∞}∥u(., t) − v(., t)∥_{L^{q}(Ω)} = 0

(resp. lim_{t→+∞} ∥u(., t) − ω(., t)∥_{L^{q}(Ω)} = 0),

(4.1)

This theorem is a consequence of the following lemma.

Lemma 4.2 Assume (A1)–(A6) and let Z be a solution of (2.2) such that Z(0) ∈ L^{∞}(Ω). Then, we have:

(a) If α is a lower-solution of (2.2) with α(0) ∈ L^{∞}(Ω) such that α ≤ Z a.e. in Q_{T} , with strict inequality in a subset of positive measure, and such that every solution v of (2.2) satisfying α ≤ v ≤ Z is equal to Z. Then the minimal solution α̃ of (P_{0,+∞;α(0)}) is such that α ≤ α̃ ≤ Z, and

lim_{t→+∞} ∥α̃(., t) − Z(., t)∥_{L^{q}(Ω)} = 0, ∀q ≥ 1.  (4.2)

(b) If β is an upper-solution of (2.2) with β(0) ∈ L^{∞}(Ω) such that Z ≤ β a.e. in Q_{T} , with strict inequality in a subset of positive measure, and such that every solution v of (2.2) satisfying Z ≤ v ≤ β is equal to Z. Then the maximal solution β̃ of (P_{0,+∞;β(0)}) is such that Z ≤ β̃ ≤ β, and

lim_{t→+∞} ∥β̃(., t) − Z(., t)∥_{L^{q}(Ω)} = 0, ∀q ≥ 1.

Proof. With the help of the lemmas in section 2, we apply the method of De coster and Omari [3]. First we show (a), and then (b) can be obtained by similar way. The proof is divided into three steps.

(i) We construct a sequence of lower-solutions of (2.2) converging to Z: Let α be a lower-solution of (P_{0,T;α(0)}), and Z verify Z(0) ≥ α(0). Then Z is an uppersolution of (P_{0,T;α(0)}). By lemma 2.2, there exists a minimal solution α̃_{0} of (P_{0,T;α(0)}) such that α ≤ α̃_{0} ≤ Z a.e. in Q_{T}. So α̃_{0}(T) ≥ α(T) ≥ α(0) = α̃_{0}(0). Now, we define by induction, the sequence (α̃_{n})_{n} such that α̃_{n} is the minimal solution u of

\frac{∂u}{∂t} + A(u) + F (u, ∇u) = 0 in Q_{T} ,

u = 0 on Σ_{T} ,
u(0) = α̃_{n−1}(T) in Ω,

satisfying

α̃_{n−1} ≤ u ≤ Z a.e. in Q_{T} .

Hence α̃_{n} is a lower-solution of (2.2). Consequently,

α ≤ α̃_{n−1} ≤ α̃_{n} ≤ Z, for all n.  (4.3)

and

α̃_{n−1}(T) = α̃_{n}(0), for all n.  (4.4)

By Lebesgue dominated convergence theorem, there exists u ∈ L^{∞}(Q_{T}) such that α ≤ u ≤ Z a.e. in Q_{T} and lim_{n→+∞} α̃_{n} = u in L^{q}(Q_{T}), for any q ≥ 1. Moreover, α̃_{n}, u ∈ C([0, T]; L^{q}(Ω)). By (4.3) and this claim, we get

lim_{n→+∞}α̃_{n}(t) = u(t) in L^{q}(Ω), ∀q ≥ 1.  (4.5)

Let f_{n}(t) := ∫_{Ω}(u − α̃_{n})^{q}(x, t)dx, for any n ≥ 1. We have, (f_{n})_{n} ⊂ C([0, T]; ℝ) and converges simply to zero. By Dini’s theorem one has

lim_{n→+∞} sup_{[0,T]} ∥α̃_{n}(t) − u(t)∥_{q} = 0.

(ii) Using [5, Theorem 3.6], we deduce that u satisfies the first two equations in (2.2). The third equation, the periodicity condition, is a consequence of (4.4). Then u is a solution of (2.2) with α ≤ u ≤ β. Therefore, we have u = Z a.e. in Q_{T} and

lim_{n→+∞} sup_{[0,T]} ∥α̃_{n}(t) − Z(t)∥_{q} = 0.  (4.6)

Let α̃(x, t) := α̃_{n}(x, t − nT) for (x, t) ∈ Ω × [nT, (n + 1)T [. Then α̃ is a solution of (P_{0,α(0)}) satisfying (4.2). Indeed, we have

∥α̃(., t) − Z(., t)∥_{L^{q}(Ω)} ≤ sup_{θ∈[0,T]} ∥α̃_{n_{t}}(., θ) − Z(., θ)∥_{L^{q}(Ω)},

where n_{t} = [t/T] is the integer part of t/T. Now, (4.2) is a consequence of (4.6). (iii)The minimality of α̃ as a solution of (P_{0,α(0)}) satisfying α ≤ α̃ ≤ Z is obtained exactly as in [3]. □

Remark In the sequel we shall identify a lower or an upper-solution ϕ defined on Ω×[0, T) to its prolongment on Ω×[0, +∞) defined by ϕ̃(x, t) := ϕ(x, t−nT) ∀(x, t) ∈ Ω × [nT, (n + 1)T [.

Proof of Theorem 4.1 We prove the result concerning the minimal solution, the one corresponding to the maximal solution is obtained in a similar way. Let u_{0} be such that α(0) ≤ u_{0} ≤ v(0). We first show that: U(u_{0}, α, v) ≠ ∅. v (resp. α) is an upper (resp. lower) solution of (P_{0,T′;u_{0}}), for any T′ > 0. By lemma 2.3 the maximal and minimal solutions of P_{0,+∞;u_{0}} are defined globally. Let u ∈ U(u_{0}, α, v) and u_{min} the minimal solution of (P_{0,T′;u_{0}}), T′ > 0. We have α ≤ u_{min} ≤ u ≤ v on Ω × (0, +∞). And from lemma 2.2, we get

α ≤ α̃ ≤ u_{min} ≤ u ≤ v,  (4.7)

where α̃ is the minimal solution of (P_{0,α(0)}) and u_{0} satisfying α(0) ≤ u_{0}. Hence the proof is completed □

5 Applications

In this section we give some sufficient conditions on the data in order to obtain existence of lower and upper-solutions for a periodic-parabolic problem associated with the p-laplacian operator. Consider the problem

\frac{∂u}{∂t} − Δ_{p}u + g(u) = h(x, t) in Ω × ℝ^{+},

u = 0 on ∂Ω × ℝ^{+},
u(0) = u(T) in Ω,

(5.1)

where Δ_{p}u = div(|∇u|^{p−2}∇u), with p such that 1 < p < +∞ and T a fixed positive real number. h ∈ L^{∞}(Ω × ℝ) is a caratheodory function which is T - periodic in time, and g is continuous function from [0, +∞[ to [0, +∞[ such that there is a non decreasing function b from ℝ^{+} to ℝ^{+} with g(s) ≤ b(|s|) for any s ∈ ℝ. We denote by G the primitive of g vanishing at zero: G(t) := ∫_{0}^{t} g(s)ds. By applying [6, Theorem 2.1] successively to the elliptic problems

\begin{matrix}−Δ_{p}u + g(u) = −∥h∥_{∞} in Ω, \\ u = 0 on ∂Ω, \end{matrix}  (5.2)

and

\begin{matrix}−Δ_{p}u + g(u) = ∥h∥_{∞} in Ω, \\ u = 0 on ∂Ω, \end{matrix}  (5.3)

we obtain the following

Theorem 5.1 Suppose that

lim inf_{|s|→+∞} \frac{pG(s)}{|s|^{p}} < μ′ := \frac{1}{R(Ω)^{p}} \frac{p − 1}{p}β(\frac{1}{p}, 1 − \frac{1}{p}),

where R(Ω) and β(r, s) are as in [6]. Then the conclusions of Theorem 4.1 are verified.

References

[1] N. D. Alikakos and P. Hess. On stabilization of discrete monotone dynamical systems. preprint.

[2] J. L. Boldrini and J. Crema. on forced periodic solutions of superlinear quasi-parabolic problems. Electron. J. Diff. Eqns., vol. 1998, No. 14: 1-18, (1998).

[3] C. De Coster and P. Omari. unstable periodic solutions of a parabolic problem in the presence of non-well-ordered lower and upper solutions. Journal of functional analysis, 175:52-88, (2000).

[4] J. Deuel and P. Hess. Nonlinear parabolic boundary value problems with upper and lower solutions, Israel J. Math. 29:92-104,(1978).

[5] N. Grenon. Résultat d’existence et de comportement asymptotique pour les équations paraboliques quasilinéaire, thèse de Doctorat, Université d’Orléans. 1989-1990.

[6] A. El Hachimi and J.-P. Gossez. A note on nonresonance condition for a quasilinear elliptic problem. Nonlinear Analysis, Theory Methods & Applications, vol 22, n2 p:229-236, (1994)

[7] A. Harraux and N. Kenmochi. asymptotic behaviour of solutions to some degenerate parabolic equations. Funkcialj Ekvacioj, 34:19-38,(1991).

[8] P. Hess. Periodic-parabolic boundary value problems and positivity, Pitman res. Notes Mathematics, vol. 247. longman. sci. tech, Harlow, 1991.

[9] J. L. Lions. Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Paris, 1969.

Abderrahmane El Hachimi UFR Mathématiques Appliquées et Industrielles Faculté des Sciences B.P 20, El Jadida - Maroc e-mail adress: elhachimi@ucd.ac.ma Abdelilah Lamrani Alaoui UFR Mathématiques Appliquées et Industrielles Faculté des Sciences B.P 20, El Jadida - Maroc e-mail adress: a lamrani@hotmail.com