Electronic Journal of Qualitative Theory of Differential Equations 2008, No. 13, 1-20; http://www.math.u-szeged.hu/ejqtde/

On a Parabolic Strongly Nonlinear

Problem on Manifolds

Alexandro O. Marinho,^{∗}

Universidade Federal do Rio de Janeiro-IM CEP 21920-430 - Rio de Janeiro - RJ - Brazil e-mail: alexmaiver@hotmail.com

Aldo. T. Lourêdo,^{†} Osmundo. A. Lima Universidade Estadual da Paraíba-DM CEP 58109095 - Campina Grande - PB- Brazil

e-mail: aldotl@bol.com.br, e-mail:osmundo@hs24.com.br.

Dedicated to Professor Marcondes Clark on the occasion of his 54th birthday

Abstract: In this work we will prove the existence uniqueness and asymptotic behavior of weak solutions for the system (*) involving the pseudo Laplacian operator and the condition \frac{∂u}{∂t} + ∑_{i=1}^{n} |\frac{∂u}{∂x_{i}} |^{p−2} \frac{∂u}{∂x_{i}} ν_{i} + |u|^{ρ}u = f on Σ_{1}, where

Σ_{1} is part of the lateral boundary of the cylinder Q = Ω × (0, T) and f is a given function defined on Σ_{1}. Key Words: Weak solutions, manifolds, monotonic operator, compactness. Mathematics Subject Classications: 35L85, 35L05, 35L20, 35L70, 49A29

∗supported by CNPq-Brazil, Corresponding author

†supported by Capes-Brazil

1 Introduction

Let Ω be a bounded open set of ℝ^{n}(n ≥ 1) with smooth boundary Γ. We consider {Γ_{0}, Γ_{1}} a partition of Γ, that is, Γ = Γ_{0} ∪ Γ_{1}, with Γ_{0} and Γ_{1} having positive Lebesgue measure and with \overline{Γ_{0}} ∩ \overline{Γ_{1}} = ∅.

Let ν be the outward normal to Γ and T > 0 is a real number. We denote by Q = Ω × (0, T) the cylinder of the ℝ^{n+1}.

The goal of this work is to solve following strongly nonlinear boundary problem:

(∗) | \begin{matrix}Au = 0 & in Q = Ω × (0, T) \\ u = 0 & on Σ_{0} = Γ_{0} × (0, T) \\ \frac{∂u}{∂t} + ∑_{i=1}^{n} |\frac{∂u}{∂x_{i}} |^{p−2} \frac{∂u}{∂x_{i}} ν_{i} + |u|^{ρ}u = f & on Σ_{1} = Γ_{1} × (0, T) \\ u(x, 0) = u_{0}(x) & on Γ, \end{matrix}

where A is the pseudo Laplacian operator defined by

\begin{aligned}A & : & W_{0}^{1,p}(Ω) & → & W^{−1,p′}(Ω), & \frac{1}{p} + \frac{1}{p′} = 1 \\ & & w & ↦ & Aw \end{aligned}

with Aw = − ∑_{i=1}^{n} \frac{∂}{∂x_{i}} (|\frac{∂w}{∂x_{i}} |^{p−2} \frac{∂w}{∂x_{i}} ), 2 < p < ∞; ρ is a positive real constant

satisfying the conditions (1) and f is a known real value function.

As the solution of system depend of x and t and the equation (∗)_{1} does not have temporal derivative of the function u, this system is not Cauchy-Kovalevsky type.

This problem associated with evolution equation on lateral boundary, with p = 2, was study in Araruna-Antunes-Medeiros [1] and Domingos-Cavalcante [4], both motivated by the idea applied in Lions ([6], pp. 134), which consists to reduce the problem in a model of mathematical physics on the manifolds Σ_{1}. Also, Araruna-Araujo in [2] studied the system (∗) in your form more simple, that is, p = 2. Recently, O.A.Lima, at al has been researching in

EJQTDE, 2008 No. 13, p. 2 Partial Differential Equations involving the pseudo Laplacian operator [10]. In this work we use a technique due to Lions [6], which transforms the system (∗) in a Cauchy-Kovalevsky type one by means of a suitable perturbations in the equation (∗)_{1}. The solution of (∗) is obtained as limit of solutions of the perturbed problem.

For p > 2, the operator A brings great difficulties, because it is non-linear, mainly to establish concepts of solutions, in passage to the limit, to work with the trace application and immersion in spaces W^{s,p}(Ω), s ∈ ℝ (for this we consult Nĕcas [5]) and to obtain a estimative for derived of the approximate function(here we use strongly the proprieties of the trace application). Finally all the difficulties will be overcome through careful handling of the proprieties of the operator A.

This paper is organized as follows: In Section 1, we will give some notations, hypothesis and results. In Section 2, we will introduce the perturbed problem. In Section 3, we will prove the existence of the solutions for the perturbed problem. In Section 4, we will treat of the uniqueness for the solution of the perturbed problem. Finally in Section 5, we will prove the main result of this work.

2 Hypotheses and Notations

We denote by W^{\frac{1}{p′} ,p}(Γ) the vectorial space of functions v|_{Γ} when v ∈ W^{1,p}(Ω), for \frac{1}{p} + \frac{1}{p′} = 1. By W^{− \frac{1}{p′} ,p′}(Γ) denotes the dual of the space W^{\frac{1}{p′} ,p}(Γ). Let p > 2 be and V_{0} the Banach space given by V_{0} = {v ∈ W^{1,p}(Ω); v|_{Γ_{0}} = 0} equipped with the norm ∥v∥_{V_{0}} = ( ∑_{i=1}^{n} ∫_{Ω} |\frac{∂v}{∂x_{i}} |^{p} dx)^{\frac{1}{p}} . Note that the appli-

cation γ : V_{0} ⟶ W^{\frac{1}{p′} ,p}(Γ_{1}) is linear, continuous and surjective.

In (∗) assume that ρ is a real number such that

ρ > 0, if n = 1 or 0 < ρ ≤ \frac{(n + 2)p − 2(n + 1)}{n − p + 1} se n ≥ 2. (1)

With the choice we have W^{\frac{1}{p′} ,p}(Γ_{1}) ⊂ L^{ρ+2}(Γ_{1}) with continuous and dense immersion, for \frac{1}{p′} + \frac{1}{p} = 1. Therefore L^{\frac{ρ+2}{ρ+1}} (Γ_{1}) = (L^{ρ+2}(Γ_{1}))′ ⊂ W^{− \frac{1}{p′} ,p′}(Γ_{1}) with immersions are dense and continuous.

Let a : V_{0} × V_{0} → ℝ defined by

a(u, v) = ∑_{i=1}^{n} ∫_{Ω} |\frac{∂u}{∂x_{i}} |^{p−2} \frac{∂u}{∂x_{i}} \frac{∂v}{∂x_{i}} dx.  (2)

which is linear with respect the second variable. Note that, the application v → a(u, v) is continuous on V_{0} for u ∈ V_{0} fix.

Let V = L^{p}(0, T; V_{0}) and the operator B from V in V′ defined by

(B(u), v)_{V′×V} = ∫_{0}^{T} a(u(t), v(t)) dt,  ∀ u, v ∈ V.  (3)

Thus B is a hemicontinuous, monotonic operator and ∥Bu∥_{V′} ≤ C∥u∥_{V}^{p−1}, ∀ u ∈ V.

To facilitate the understand of this work, introduce the followings notations:

(f, g)_{Ω} = ∫_{Ω} fg dx, (φ, ψ)_{Γ} = ∫_{Γ} φψ dΓ.

3 Perturbed Problem

The Problem (∗) is not of the Cauchy-Kowaleska’s type. Thus, consider the following perturbed problem:

For all ε > 0, the family of functions u_{ε}(x, t) is defined by:

(∗∗) | \begin{matrix}ε\frac{∂u_{ε}}{∂t} + Au_{ε} = 0 & in Q = Ω × (0, T), \\ u_{ε} = 0 & on Σ_{0} = Γ_{0} × (0, T), \\ \frac{∂u_{ε}}{∂t} + ∑_{i=1}^{n} |\frac{∂u_{ε}}{∂x_{i}} |^{p−2} \frac{∂u_{ε}}{∂x_{i}} ν_{i} + |u_{ε}|^{ρ}u_{ε} = f & on Σ_{1} = Γ_{1} × (0, T), \\ u_{ε}(x, 0) = w_{0}(x) & x ∈ Ω, \end{matrix}

where w_{0} = γ^{−1}u_{0} ∈ V_{0}. The solution concept for (∗∗) is established by Gauss’s Theorem as follows:

EJQTDE, 2008 No. 13, p. 4 For v ∈ V_{0} ∩ C^{2}(\overline{Ω}) we have

∫_{Ω} \frac{∂}{∂x_{i}} \{(|\frac{∂u_{ε}}{∂x_{i}} |^{p−2} \frac{∂u_{ε}}{∂x_{i}} ).v\}dx = ∫_{Γ_{1}} |\frac{∂u_{ε}}{∂x_{i}} |^{p−2} \frac{∂u_{ε}}{∂x_{i}} .ν_{i}.v dΓ,

hence

∫_{Ω} \frac{∂}{∂x_{i}} (|\frac{∂u_{ε}}{∂x_{i}} |^{p−2} \frac{∂u_{ε}}{∂x_{i}} ).v dx+∫_{Ω} |\frac{∂u_{ε}}{∂x_{i}} |^{p−2} \frac{∂u_{ε}}{∂x_{i}} .\frac{∂v}{∂x_{i}} dx = ∫_{Γ_{1}} |\frac{∂u_{ε}}{∂x_{i}} |^{p−2} \frac{∂u_{ε}}{∂x_{i}} .ν_{i}.v dΓ.

Summing up from i = 1 to n on both sides of the above equation yields:

(A(u_{ε}), v)_{Ω} = a(u_{ε}, v) − ∑_{i=1}^{n} ∫_{Γ_{1}} |\frac{∂u_{ε}}{∂x_{i}} |^{p−2} \frac{∂u_{ε}}{∂x_{i}} .ν_{i}.v dΓ.

From this and observing that (see (∗∗)_{3})

− ∑_{i=1}^{n} ∫_{Γ_{1}} |\frac{∂u_{ε}}{∂x_{i}} |^{p−2} \frac{∂u_{ε}}{∂x_{i}} .ν_{i}.v dΓ = ∫_{Γ_{1}} \frac{∂u_{ε}}{∂t} vdΓ_{1} + ∫_{Γ_{1}} |u_{ε}|^{ρ}u_{ε}vdΓ_{1} − ∫_{Γ_{1}} fvdΓ_{1},

we obtain (A(u_{ε}), v)_{Ω} = a(u_{ε}, v)+(γu′_{ε}, γv)_{Γ_{1}} +(|γu_{ε}|^{ρ}γu_{ε}, γv)_{Γ_{1}} −(f, γv)_{Γ_{1}}. Substituting this identity in (∗∗)_{1} we get:

ε(u′_{ε}, v)_{Ω} + (γu′_{ε}, γv)_{Γ_{1}} + a(u_{ε}, v) + (|γu_{ε}|^{ρ}γu_{ε}, γv)_{Γ_{1}} = (f, γv)_{Γ_{1}}, where u′_{ε} means \frac{∂u_{ε}}{∂t}. Therefore, a solution of the problem (∗∗) is understood in the following sense. Definition 3.1. A real value function u_{ε}(x, t) is a solution of the problem (∗∗) if, only if,

u_{ε} ∈ L^{p}(0, T; V_{0}) ∩ L^{∞}(0, T; L^{2}(Ω)),  (4)
γu_{ε} ∈ L^{ρ+2}(0, T; L^{ρ+2}(Γ_{1})) ∩ L^{∞}(0, T; L^{2}(Γ_{1})),  (5)
u′_{ε} ∈ L^{p′}(0, T; W^{−1,p′}(Ω)),  (6)
γu′_{ε} ∈ L^{p′}(0, T; W^{\frac{−1}{p′} ,p′}(Γ_{1})),  (7)
ε(u′_{ε}, v)_{Ω}+(γu′_{ε}, γv)_{Γ_{1}} +a(u_{ε}, v)+(|γu_{ε}|^{ρ}γu_{ε}, γv)_{Γ_{1}} = (f, γv)_{Γ_{1}}, ∀ v ∈ V_{0}. (8)

and satisfying the initial conditions

u_{ε}(0) = w_{0} in Ω,  (9)
γu_{ε}(0) = u_{0} in Γ_{1}  (10)

with u_{0} belongs to W^{\frac{1}{p′} ,p}(Γ_{1}).

4 Existence Theorem

In this section we will establish a theorem of existence of solutions. Theorem 1. Suppose f ∈ L^{p′}(0, T, W^{− \frac{1}{p′} ,p′}(Γ_{1})) and w_{0} ∈ V_{0}. Then, for each ε > 0 the problem (∗∗) has a unique solution u_{ε} in the sense of Definition (3.1). Remark 1. Note that, the date w_{0} is taken such that γw_{0} = u_{0}, since, given u_{0} ∈ W^{\frac{1}{p′} ,p}(Γ_{1}) there exists w_{0} ∈ V_{0} such that γw_{0} = u_{0} because the application γ : V_{0} → W^{\frac{1}{p′} ,p}(Γ_{1}) is surjective.

Proof: We will employ the Faedo-Galerkin’s method. In fact, for V_{0} we construct a special Hilbertian basis (w_{μ})_{μ∈ℕ} of V_{0}. By V_{0m} = [w_{1}, ....., w_{m}] we will denote the subspace spanned by the m first vectors of V_{0}. The approximated problem consist to find a function u_{εm}(t) ∈ V_{0m} of the type u_{εm}(x, t) = ∑_{j=1}^{m} g_{jεm}(t)w_{j}(x) solution of the initial value problem for the system of ordinary differential equations:

| \begin{matrix}ε(u′_{εm}(t), v)_{Ω} + (γu′_{εm}(t), γv)_{Γ_{1}} + a(u_{εm}(t), v) + \\ + (|γu_{εm}(t)|^{ρ}γu_{εm}(t), γv)_{Γ_{1}} = (f(t), γv)_{Γ_{1}} ∀ v ∈ V_{0} \\ u_{εm}(0) = u_{ε0m} ⟶ w_{0} in V_{0}. \end{matrix}  (11)

The system (11) has a local solution on the interval [0, t_{m}[, with t_{m} < T. This solution can be extended to the whole interval [0, T] as consequence of the a priori estimates that shall be proved in the next step. Estimates Taking v = u_{εm}(t) in (11) and integrating from 0 to t < t_{m} we obtain

\frac{ε}{2}|u_{εm}(t)|_{L^{2}(Ω)}^{2} + \frac{1}{2}∥γu_{εm}(t)∥_{L^{2}(Γ_{1})}^{2} + ∫_{0}^{t} ∥u_{εm}(s)∥_{V_{0}}^{p}ds + ∫_{0}^{t} ∥γu_{εm}(s)∥_{L^{ρ+2}(Γ_{1})}^{ρ+2}ds ≤

∫_{0}^{t} ∥f(s)∥_{W^{− \frac{1}{p′} ,p′}(Γ_{1})}∥γu_{εm}(s)∥_{W ^{\frac{1}{p′} ,p}(Γ_{1})}ds ≤ C ∫_{0}^{t} ∥f(s)∥_{W^{− \frac{1}{p′} ,p′}(Γ_{1})}∥u_{εm}(s)∥_{V_{0}}ds +

\frac{ε}{2}|u_{ε0m}|_{L^{2}(Ω)}^{2} + \frac{1}{2}∥γu_{ε0m}∥_{L^{2}(Γ_{1})}^{2} ≤ \frac{C}{p′} ∫_{0}^{T} ∥f(s)∥_{W^{− \frac{1}{p′} ,p′}(Γ_{1})}^{p′} + \frac{1}{p} ∫_{0}^{t} ∥u_{εm}(s)∥_{V_{0}}^{p}ds +

\frac{ε}{2}|u_{ε0m}|_{L^{2}(Ω)}^{2} + \frac{1}{2}∥γu_{ε0m}∥_{L^{2}(Γ_{1})}^{2}.

EJQTDE, 2008 No. 13, p. 6 From hypotheses about the initial conditions and the continuity of the application γ, we obtain:

\begin{matrix}\frac{ε}{2}|u_{εm}(t)|_{L^{2}(Ω)}^{2} + \frac{1}{2}∥γu_{εm}(t)∥_{L^{2}(Γ_{1})}^{2} + \frac{1}{p′} ∫_{0}^{t} ∥u_{εm}(s)∥_{V_{0}}^{p}ds+ \\ ∫_{0}^{t} ∥γu_{εm}(s)∥_{L^{ρ+2}(Γ_{1})}^{ρ+2}ds ≤ C \end{matrix}  (12)

where C is constant which is independent of t and m. This estimate implies that we can prolong the approximate solution u_{εm}(t) to interval [0, T] and too we obtain:

| \begin{matrix}(u_{εm}) is bounded in L^{∞}(0, T; L^{2}(Ω)); \\ (γu_{εm}) is bounded in L^{∞}(0, T; L^{2}(Γ_{1})); \\ (u_{εm}) is bounded in L^{p}(0, T; V_{0}); \\ (γu_{εm}) is bounded in L^{p}(0, T; W^{\frac{1}{p′} ,p}(Γ_{1})); \\ (γu_{εm}) is bounded in L^{ρ+2}(0, T; L^{ρ+2}(Γ_{1})); \\ (u_{εm}(T)) is bounded in L^{2}(Ω); \\ (γu_{εm}(T)) is bounded in L^{2}(Γ_{1}). \end{matrix}  (13)

Note that ∫_{0}^{T} ∫_{Γ_{1}} ||γu_{εm}|^{ρ}γu_{εm}|^{\frac{ρ+2}{ρ+1}} dt = ∫_{0}^{T} ∫_{Γ_{1}} |γu_{εm}|^{ρ+2}dt ≤ C. Thus

(|γu_{εm}|^{ρ}γu_{εm}) is bounded in L^{\frac{ρ+2}{ρ+1}} (0, T; L^{\frac{ρ+2}{ρ+1}} (Γ_{1})).  (14)

From (11)_{1} we get

⟨{εu′_{εm}, γu′_{εm}}, {v, γv}⟩ = (f(t) − |γu_{εm}(t)|^{ρ}γu_{εm}(t), γv)_{Γ_{1}} − a(u_{εm}(t), v),

where < ., . > represent the duality paring between V′_{0} × W^{− \frac{1}{p′} ,p′}(Γ_{1}) and \{V_{0} × W^{\frac{1}{p′} ,p}(Γ_{1})\}.

Hence

|⟨{εu′_{εm}, γu′_{εm}}, {v, γv}⟩| ≤ |a(u_{εm}(t), v)| +

(∥f(t)∥_{W^{− \frac{1}{p′} ,p′}(Γ_{1})} + ∥γu_{εm}|^{ρ}γu_{εm}∥_{W^{− \frac{1}{p′} ,p′}(Γ_{1})})∥γv∥_{W ^{\frac{1}{p′} ,p}(Γ_{1})} ≤

(∥f(t)∥_{W^{− \frac{1}{p′} ,p′}(Γ_{1})} + C∥γu_{εm}|^{ρ}γu_{εm}∥_{L^{\frac{ρ+2}{ρ+1}} (Γ_{1})})∥γv∥_{W ^{\frac{1}{p′} ,p}(Γ_{1})}+

∥u_{εm}(t)∥_{V_{0}}^{p−1}∥v∥_{V_{0}} ≤ (∥γv∥_{W ^{\frac{1}{p′} ,p}(Γ_{1})} + ∥v∥_{V_{0}}) ×

[∥f(t)∥_{W^{− \frac{1}{p′} ,p′}(Γ_{1})} + C∥γu_{εm}|^{ρ}γu_{εm}∥_{L^{\frac{ρ+2}{ρ+1}} (Γ_{1})} + ∥u_{εm}(t)∥_{V_{0}}^{p−1}].

From estimates above we have |⟨{εu′_{εm}, γu′_{εm}}, {v, γv}⟩| ≤

[∥f(t)∥_{W^{− \frac{1}{p′} ,p′}(Γ_{1})} + C∥γu_{εm}|^{ρ}γu_{εm}∥_{L^{\frac{ρ+2}{ρ+1}} (Γ_{1})} + ∥u_{εm}(t)∥_{V_{0}}^{p−1}]× ∥\{v, γv\}∥_{V_{0}×W ^{\frac{1}{p′} ,p}(Γ_{1})}.

Therefore, from bounded (14) and (13)_{3}, we get

(\{εu′_{εm}, γu′_{εm}\}) is bounded in L^{p′}(0, T; V′_{0} × W^{− \frac{1}{p′} ,p′}(Γ_{1})).  (15)

Passage to the Limit From estimates (13), (14) and (15) we obtain

| \begin{matrix}u_{εm} ⇀^{∗} u_{ε} weak-star in L^{∞}(0, T; L^{2}(Ω)); \\ γu_{εm} ⇀^{∗} γu_{ε} weak-star in L^{∞}(0, T; L^{2}(Γ_{1})); \\ u_{εm} ⇀ u_{ε} weak in L^{p}(0, T; V_{0}); \\ γu_{εm} ⇀ γu_{ε} weak in L^{p}(0, T; W^{\frac{1}{p′} ,p}(Γ_{1})); \\ γu_{εm} ⇀ γu_{ε} weak in L^{ρ+2}(0, T; L^{ρ+2}(Γ_{1})) ≡ L^{ρ+2}(Σ_{1}); \\ u_{εm}(T) ⇀ χ weak in L^{2}(Ω); \\ γu_{εm}(T) ⇀ ς weak in L^{2}(Γ_{1}) \\ |γu_{εm}|^{ρ}γu_{εm} ⇀ η weak in L^{\frac{ρ+2}{ρ+1}} (0, T; L^{\frac{ρ+2}{ρ+1}} (Γ_{1})) ≡ L^{\frac{ρ+2}{ρ+1}} (Σ_{1}) \\ u′_{εm} ⇀ u′_{ε} weak in L^{p′}(0, T; V′_{0}); \\ γu′_{εm} ⇀ γu′_{ε} weak in L^{p′}(0, T; W^{− \frac{1}{p′} ,p′}(Γ_{1})) \end{matrix}  (16)

Remark 2. Note that by the convergence (16)_{1}, (16)_{9} and (16)_{2}, (16)_{10} it makes sense to calculate u_{ε}(0) , u_{ε}(T) and γu_{ε}(0) , γu_{ε}(T) respectively.

Let V = L^{p}(0, T; V_{0}) and B the operator from V given by

(B(u), v)_{V′×V} = ∫_{0}^{T} a(u(t), v(t)) dt,  ∀ u, v ∈ V,  (17)

hence, B is hemicontinuous, monotonic operator and ∥Bu∥_{V′} ≤ C∥u∥_{V}^{p−1}, ∀ u ∈ V. Thus, from estimative (16)_{3} we have (Bu_{εm})_{m} is bounded in V′, hence

Bu_{εm} ⇀ ζ is V′.  (18)

From convergence (16)_{9} we obtain < u′_{εm}, φ >→< u′_{ε}, φ >, ∀ φ ∈ L^{p}(0, T; V_{0}), that is,

∫_{0}^{T} (u′_{εm}, v)_{Ω}θdt → ∫_{0}^{T} (u′_{ε}, v)_{Ω}θdt, v ∈ V_{0m} ⊂ V_{0}, ∀ θ ∈ D(0, T) ⊂ L^{p}(0, T),

or

(u′_{εm}, v)_{Ω} → (u′_{ε}, v)_{Ω} v ∈ V_{0m}, in D′(0, T).  (19)

From convergence (16)_{10} we have < γu′_{εm}, γφ >→< γu′_{ε}, γφ >, ∀ φ ∈ L^{p}(0, T; V_{0}), that is,

(γu′_{εm}, γv)_{Γ_{1}} → (γu′_{ε}, γv)_{Γ_{1}} v ∈ V_{0m}, in D′(0, T).  (20)

Analogously, we have

(Bu_{εm}, v) → (ζ, v) v ∈ V_{0m}, in D′(0, T),  (21)

and

(|γu_{εm}|^{ρ}γu_{εm}, γv)_{Γ_{1}} → (η, γv)_{Γ_{1}} v ∈ V_{0m}, in D′(0, T).  (22)

Thus, taking the limit as m → ∞ in the approximated equation (11)_{1}, using the convergence (19) − (22) and the density of V_{0m} in V_{0}, we obtain:

\begin{matrix}ε(u′_{ε}, v)_{Ω} + (γu′_{ε}, γv)_{Γ_{1}} + (ζ, v)_{Ω} + (η, γv)_{Γ_{1}} = (f(t), γv)_{Γ_{1}} \\ ∀ v ∈ V_{0} in D′(0, T). \end{matrix}  (23)

To follow we will proof that: |γu_{ε}|^{ρ}γu_{ε} = η and Bu_{ε} = ζ.

EJQTDE, 2008 No. 13, p. 9 Proof. In fact, from the estimate above we have

(γu_{εm}) is bounded in L^{p}(0, T; W^{\frac{1}{p′} ,p}(Γ_{1}));
(γu′_{εm}) is bounded in L^{p′}(0, T; W^{− \frac{1}{p′} ,p′}(Γ_{1})).

As W^{\frac{1}{p′} ,p}(Γ_{1}) ↪^{c} L^{p}(Γ_{1}) ↪ L^{2}(Γ_{1}) ↪ L^{p′}(Γ_{1}) ↪ W^{− \frac{1}{p′} ,p′}(Γ_{1}), we have

W^{\frac{1}{p′} ,p}(Γ_{1}) ↪^{c} L^{p}(Γ_{1}) ↪ W^{− \frac{1}{p′} ,p′}(Γ_{1}). Thus, of the Aubin-Lions’s Theorem we obtain a subsequence, still denoted by (γu_{εm}), such that γu_{εm} → γu_{ε} in L^{p}(Σ_{1}), where still we can extract other subsequence which we insist in denote by (γu_{εm}), such that: γu_{εm} → γu_{ε} a.e Σ_{1}, thus,

|γu_{εm}|^{ρ}γu_{εm} → |γu_{ε}|^{ρ}γu_{ε} a.e Σ_{1}.  (24)

From estimative (14) we obtain:

∥|γu_{εm}|^{ρ}γu_{εm}∥_{L^{\frac{ρ+2}{ρ+1}} (Σ_{1})} ≤ C  (25)

Hence, from (24), (25) and the Lions’s Lema, we obtain

|γu_{εm}|^{ρ}γu_{εm} ⇀ |γu_{ε}|^{ρ}γu_{ε} weak in L^{\frac{ρ+2}{ρ+1}} (Σ_{1}).

Thus |γu_{ε}|^{ρ}γu_{ε} = η.

Now we will prove that Bu_{ε} = ζ.

Proof. In fact for this purpose we needed prove that: (i) u_{ε}(0) = w_{0}, (ii) χ = u_{ε}(T), (iii) γu_{ε}(0) = u_{0} and (iv) γu_{ε}(T) = ς.

In fact, to prove (i) we use the convergence (16)_{1} and (16)_{9} which yield

∫_{0}^{T} (u_{εm}, v)φ′ dt → ∫_{0}^{T} (u_{ε}, v)φ′ dt ∀ v ∈ L^{2}(Ω),
φ ∈ C^{1}([0, T]), φ(0) = 1, φ(T) = 0

and

∫_{0}^{T} \frac{d}{dt}(u_{εm}, v)φ dt → ∫_{0}^{T} \frac{d}{dt}(u_{ε}, v)φ dt ∀ v ∈ L^{2}(Ω) ⊂ V′_{0},

φ ∈ C^{1}([0, T]), φ(0) = 1, φ(T) = 0,

where

∫_{0}^{T} \frac{d}{dt}\{(u_{εm}, v)φ\} dt → ∫_{0}^{T} \frac{d}{dt}\{(u_{ε}, v)φ\} dt ∀ v ∈ L^{2}(Ω),

φ ∈ C^{1}([0, T]), φ(0) = 1, φ(T) = 0,

hence,

(u_{εm}(0), v) → (u_{ε}(0), v), ∀ v ∈ L^{2}(Ω),

that is,

u_{εm}(0) ⇀ u_{ε}(0) in L^{2}(Ω).

As u_{εm}(0) → w_{0} in V_{0} ↪ L^{2}(Ω), we have u_{εm}(0) → w_{0} in L^{2}(Ω), where u_{εm}(0) ⇀ w_{0} in L^{2}(Ω). Hence

u_{ε}(0) = w_{0}.

Analogously, working as φ(0) = 0 and φ(T) = 1 we obtain

u_{ε}(T) = χ.

In fact, to prove (iii) we use the convergence (16)_{2} and (16)_{10} which yield

(γu_{εm}(0), v) → (γu_{ε}(0), v), ∀ v ∈ L^{2}(Γ_{1}),

where

γu_{εm}(0) ⇀ γu_{ε}(0) em L^{2}(Γ_{1}).

As u_{εm}(0) → w_{0} in V_{0} and γ continuous from V_{0} in W^{\frac{1}{p′} ,p}(Γ_{1}), we have that

γu_{εm}(0) → γw_{0} in W^{\frac{1}{p′} ,p}(Γ_{1}). Being W^{\frac{1}{p′} ,p}(Γ_{1}) ↪ L^{2}(Γ_{1}), for p > 2 > \frac{2n}{2n + 1},

by Fractionary Sobolev’s Theorem, we have γu_{εm}(0) → γw_{0} in L^{2}(Γ_{1}). Therefore γu_{ε}(0) = γw_{0} = u_{0}, by remark 1. Analogously, we have γu_{ε}(T) = ς

We will show that: Bu_{ε} = ζ. In fact, being the operator B : V → V′ mononotonic, we obtain:

(Bu_{εm}, u_{εm}) − (Bu_{εm}, v) − (Bv, u_{εm} − v) ≥ 0.  (26)

Taking v = u_{εm} and integrating of 0 the T in the approximated equation (11)_{1} we obtain:

\begin{matrix}(Bu_{εm}, u_{εm}) = ∫_{0}^{T} a(u_{εm}, u_{εm})dt = ∫_{0}^{T} (f, u_{εm})_{Γ_{1}}dt − ε ∫_{0}^{T} (u′_{εm}, u_{εm})_{Ω}dt − \\ ∫_{0}^{T} (γu′_{εm}, γu_{εm})_{Γ_{1}}dt − ∫_{0}^{T} (|γu_{εm}|^{ρ}γu_{εm}, γu_{εm})_{Γ_{1}}dt. \end{matrix}

Thus, substituting in (26), we have

\begin{matrix}0 ≤ ∫_{0}^{T} (f, u_{εm})_{Γ_{1}}dt − \frac{ε}{2}|u_{εm}(T)|_{L^{2}(Ω)}^{2} + \frac{ε}{2}|u_{εm}(0)|_{L^{2}(Ω)}^{2} − \frac{1}{2}|γu_{εm}(T)|_{L^{2}(Γ_{1})}^{2} + \\ \frac{1}{2}|γu_{εm}(0)|_{L^{2}(Γ_{1})}^{2} − ∫_{0}^{T} ∥γu_{εm}∥_{L^{ρ+2}(Γ_{1})}^{ρ+2}dt − (Bu_{εm}, v) − (Bv, u_{εm} − v). \end{matrix}

Using the convergence obtained and applying the lim inf_{m→∞} in both sides of the inequality above we have:

\begin{matrix}0 ≤ ∫_{0}^{T} (f, u_{ε})_{Γ_{1}}dt − \frac{ε}{2}|u_{ε}(T)|_{L^{2}(Ω)}^{2} + \frac{ε}{2}|w_{0}|_{L^{2}(Ω)}^{2} − \frac{1}{2}|γu_{ε}(T)|_{L^{2}(Γ_{1})}^{2} + \\ \frac{1}{2}|u_{0}|_{L^{2}(Γ_{1})}^{2} − ∫_{0}^{T} ∥γu_{ε}∥_{L^{ρ+2}(Γ_{1})}^{ρ+2}dt − (ζ, v) − (Bv, u_{ε} − v). \end{matrix} (27)

Taking v = u_{ε} and integrating of 0 the T in the equation (23) we obtain:

\begin{matrix}∫_{0}^{T} (f, u_{ε})_{Γ_{1}}dt = (ζ, u_{ε}) + \frac{ε}{2}|u_{ε}(T)|_{L^{2}(Ω)}^{2} − \frac{ε}{2}|w_{0}|_{L^{2}(Ω)}^{2} + \\ \frac{1}{2}|γu_{ε}(T)|_{L^{2}(Γ_{1})}^{2} − \frac{1}{2}|u_{0}|_{L^{2}(Γ_{1})}^{2} + ∫_{0}^{T} ∥γu_{ε}∥_{L^{ρ+2}(Γ_{1})}^{ρ+2}dt. \end{matrix}

If we substitute this expression in (27), we obtain

0 ≤ (ζ − Bv, u_{ε} − v), ∀ v ∈ V.

Consider u_{ε} − v = λw, λ > 0. Thus, using the hemicontinuity of the operator B, we obtain 0 ≤ (ζ − Bu_{ε}, w), ∀ w ∈ V. Working with λ < 0 we have:

EJQTDE, 2008 No. 13, p. 12 (ζ − Bu_{ε}, w) ≤ 0, ∀ w ∈ V. Therefore (ζ − Bu_{ε}, w) = 0, ∀ w ∈ V, thus Bu_{ε} = ζ. Note that (Bu_{εm}, w) → (Bu_{ε}, w), ∀ w ∈ V = L^{p}(0, T; V_{0}), hence,

∫_{0}^{T} a(u_{εm}, v)θdt → ∫_{0}^{T} a(u_{ε}, v)θdt, ∀ v ∈ V_{0}, ∀ θ ∈ D(0, T) ⊂ L^{p}(0, T).

Thus a(u_{εm}, v) → a(u_{ε}, v) ∀ v ∈ V_{0} in D′(0, T). Therefore,

| \begin{matrix}ε(u′_{ε}, v)_{Ω} + (γu′_{ε}, γv)_{Γ_{1}} + a(u_{ε}, v) + \\ (|γu_{ε}|^{ρ}γu_{ε}, γv)_{Γ_{1}} = (f(t), γv)_{Γ_{1}} ∀ v ∈ V_{0} in D′(0, T), \end{matrix}  (28)

Uniqueness of the Solution To obtain the uniqueness of the solution, we suppose that there exists two solutions such that u_{ε}, û_{ε} in the conditions of the Theorem 1. It following that w_{ε} = u_{ε} − û_{ε} satisfy:

w_{ε} ∈ L^{p}(0, T; V_{0}) ∩ L^{∞}(0, T; L^{2}(Ω)),  (29)
γw_{ε} ∈ L^{ρ+2}(0, T; L^{ρ+2}(Γ_{1})) ∩ L^{∞}(0, T; L^{2}(Γ_{1})),  (30)
w′_{ε} ∈ L^{p′}(0, T; W^{−1,p′}(Ω)),  (31)
γw′_{ε} ∈ L^{p′}(0, T; W^{− \frac{1}{p′} ,p′}(Γ_{1})),  (32)
\begin{matrix}ε(w′_{ε}, v)_{Ω} + (γw′_{ε}, γv)_{Γ_{1}} + a(u_{ε}, v) − a(û_{ε}, v) + \\ (|γu_{ε}|^{ρ}γu_{ε} − |γû_{ε}|^{ρ}γû_{ε}, γv)_{Γ_{1}} = 0, ∀ v ∈ V_{0}. \end{matrix}  (33)

Taking v = w_{ε} in (33) and integrating from 0 the t ≤ T we obtain:

\begin{matrix}\frac{ε}{2}|w_{ε}(t)|_{L^{2}(Ω)}^{2} + \frac{1}{2}|γw_{ε}(t)|_{L^{2}(Γ_{1})}^{2} + ∫_{0}^{t}(a(u_{ε}, w) − a(û_{ε}, w))dt+ \\ ∫_{0}^{t}(|γu_{ε}|^{ρ}γu_{ε} − |γû_{ε}|^{ρ}γû_{ε}, γu_{ε} − γû_{ε})dt = 0. \end{matrix}

Using the monotoneity of the function h(s) = |s|^{ρ}s and a(u_{ε}, w)−a(û_{ε}, w) ≥ 0, we have

\frac{ε}{2}|w_{ε}(t)|_{L^{2}(Ω)}^{2} + \frac{1}{2}|γw_{ε}(t)|_{L^{2}(Γ_{1})}^{2} ≤ 0.

Therefore, we have that w_{ε}(t) = 0 ∀ t ∈ [0, T]. Thus the Theorem is proved.

5 Main Result

In this Section we will prove the following result Theorem 2. When ε → 0 we have

u_{ε} ⇀ u in L^{p}(0, T; V_{0}),  (34)

being u the solution of the problem (∗).

Proof: Making v = u_{ε}(t) in (8) and proceeding as in the previous Theorem we obtain

| \begin{matrix}(\sqrt{ε}u_{ε}) is bounded in L^{∞}(0, T; L^{2}(Ω)); \\ (γu_{ε}) is bounded in L^{∞}(0, T; L^{2}(Γ_{1})); \\ (u_{ε}) is bounded in L^{p}(0, T; V_{0}); \\ (γu_{ε}) is bounded in L^{p}(0, T; W^{\frac{1}{p′} ,p}(Γ_{1})); \\ (γu_{ε}) is bounded in L^{ρ+2}(0, T; L^{ρ+2}(Γ_{1})); \\ (u_{ε}(T)) is bounded in L^{2}(Ω); \\ (γu_{ε}(T)) is bounded in L^{2}(Γ_{1}). \end{matrix}  (35)

and

| \begin{matrix}(εu′_{ε}) is bounded in L^{p′}(0, T; V′_{0}); \\ (|γu_{ε}|^{ρ}γu_{ε}) is bounded in L^{\frac{ρ+2}{ρ+1}} (0, T; L^{\frac{ρ+2}{ρ+1}} (Γ_{1})) \\ (γu_{ε}) is bounded in L^{p}(0, T; W^{\frac{1}{p′} ,p}(Γ_{1})); \\ (γu′_{ε}) is bounded in L^{p′}(0, T; W^{− \frac{1}{p′} ,p′}(Γ_{1})) \\ (Bu_{ε}) is bounded in V′ = L^{p′}(0, T; V′_{0}). \end{matrix}  (36)

Hence there exists an subsequence, still represented by (u_{ε}), such that, when ε → 0

| \begin{matrix}\sqrt{ε}u_{ε} ⇀^{∗} 0 weak-star in L^{∞}(0, T; L^{2}(Ω)); \\ γu_{ε} ⇀^{∗} γu weak-star in L^{∞}(0, T; L^{2}(Γ_{1})); \\ u_{ε} ⇀ u weak in V = L^{p}(0, T; V_{0}); \\ γu_{ε} ⇀ γu weak in L^{p}(0, T; W^{\frac{1}{p′} ,p}(Γ_{1})); \\ γu_{ε} ⇀ γu weak in L^{ρ+2}(0, T; L^{ρ+2}(Γ_{1})) ≡ L^{ρ+2}(Σ_{1}); \\ γu_{ε}(T) ⇀ χ weak in L^{2}(Γ_{1}) \\ εu′_{ε} ⇀ 0 weak in V′ = L^{p′}(0, T; V′_{0}); \\ |γu_{ε}|^{ρ}γu_{ε} ⇀ η weak in L^{\frac{ρ+2}{ρ+1}} (0, T; L^{\frac{ρ+2}{ρ+1}} (Γ_{1})) \\ γu′_{ε} ⇀ γu′ weak in L^{p′}(0, T; W^{− \frac{1}{p′} ,p′}(Γ_{1})); \\ Bu_{ε} ⇀ ζ weak in V′ = L^{p′}(0, T; V′_{0}). \end{matrix}  (37)

Analogously to the Theorem 1 we can to show that |γu|^{ρ}γu = η and Bu = ζ. Using the convergence (37) in (28) we obtain the variational formulation of the problem (∗), when ε → 0

(γu′, γv)_{Γ_{1}} +a(u, v) + (|γu|^{ρ}γu, γv)_{Γ_{1}} = (f(t), γv)_{Γ_{1}} ∀ v ∈ V_{0} in D′(0, T).

On the other hand, with a analogously analysis as in the Remark 2, we have as in the Theorem 1: γu_{ε}(0) ⇀ γu(0) in L^{2}(Γ_{1}). Therefore

γu(0) = u_{0} on Γ_{1}.

In this sense, we have the solution of the problem (∗) as limit of the perturbed problem (∗∗).

6 Boundary Stabilization

The aim of section is study the algebric decay for the energy E(t) associated to weak solution of the problem (∗). To asymptotic behavior, we use the Nakao’s method [9]. This energy is given by

E(t) = \frac{1}{2}|γu(t)|_{L^{2}(Γ_{1})}^{2}  (38)

Remark 3. Note that, the solution of the problem (∗) we can be extend to [0, ∞), when f = 0.

Theorem 3. Let E(t) a energy associated the weak solution of problem (∗). Then, there exists a constant δ > 0 such that the energy satisfies

E(t) ≤ C \frac{1}{(1 + t)^{\frac{1}{δ}}} , ∀t ≥ 0.

Considering (28) with f = 0 and v = u_{ϵ} we get:

\frac{1}{2} \frac{d}{dt}|\sqrt{ϵ}u_{ϵ}(t)|^{2} + \frac{1}{2} \frac{d}{dt}|γu_{ϵ}(t)|^{2} = −∥u_{ϵ}(t)∥_{V_{0}}^{p} − ∥γu_{ϵ}(t)∥_{L^{ρ+2}(Ω)}^{ρ+2}

Let E_{ε}(t) = \frac{1}{2}|\sqrt{ϵ}u_{ϵ}(t)|^{2} + \frac{1}{2}|γu_{ϵ}(t)|^{2}, then

\frac{d}{dt}E_{ε}(t) ≤ 0, ∀t ≥ 0  (39)

and

E_{ε}(t) ≤ E_{ε}(0) = \frac{1}{2}|\sqrt{ϵ}u_{ϵ}(0)|^{2} + \frac{1}{2}|γu_{ϵ}(0)|^{2} ≤ C(γu_{0})

because, \frac{1}{2} \sqrt{ε}u_{ε}(0) → 0 in L^{2}(Ω) and γu_{ε}(0) → γu(0) in L^{2}(Γ_{1}) when ε → 0. Therefore, E_{ε}(t) is increasing and bounded.

We have that

\frac{d}{dt}E_{ε}(t) = −∥u_{ε}(t)∥_{V_{0}}^{p} − ∥γu_{ε}(t)∥_{L^{ρ+2}(Ω)}^{ρ+2}  (40)

then

E_{ε}(t + 1) − E_{ε}(t) = − ∫_{t}^{t+1} ∥u_{ε}(t)∥_{V_{0}}^{p} − ∫_{t}^{t+1} ∥γu_{ε}(t)∥_{L^{ρ+2}(Ω)}^{ρ+2}.

Therefore

∫_{t}^{t+1} ∥u_{ε}(t)∥_{V_{0}}^{p} + ∫_{t}^{t+1} ∥γu_{ε}(t)∥_{L^{ρ+2}(Ω)}^{ρ+2} = E_{ε}(t) − E_{ε}(t + 1)  (41)

Since V_{0} ↪ L^{2}(Ω) and L^{ρ+2}(Γ_{1}) ↪ L^{2}(Γ_{1}), we have

∫_{t}^{t+1} |u_{ε}(t)|_{L^{2}(Ω)}^{p} + ∫_{t}^{t+1} |γu_{ε}(t)|_{L^{2}(Γ_{1})}^{p} ≤ C_{1}(E_{ε}(t) − E_{ε}(t + 1))  (42)

Therefore exist t^{∗} ∈ [t, t + 1], such that

|u_{ε}(t^{∗})|_{L^{2}(Ω)}^{p} + |γu_{ε}(t^{∗})|_{L^{2}(Γ_{1})}^{p} ≤ C_{1}[E_{ε}(t) − E_{ε}(t + 1)],

where,

\begin{matrix}\frac{1}{2}|u_{ε}(t^{∗})|^{2} ≤ C_{2}[E_{ε}(t) − E_{ε}(t + 1)]^{\frac{2}{p}} \\ \frac{1}{2}|γu_{ε}(t^{∗})|^{2} ≤ C_{2}[E_{ε}(t) − E_{ε}(t + 1)]^{\frac{2}{p}} . \end{matrix}

Thus

\begin{matrix}\frac{ε}{2}|u_{ε}(t^{∗})|^{2} ≤ εC_{2}[E_{ε}(t) − E_{ε}(t + 1)]^{\frac{2}{p}} \\ \frac{1}{2}|γu_{ε}(t^{∗})|^{2} ≤ C_{2}[E_{ε}(t) − E_{ε}(t + 1)]^{\frac{2}{p}} \end{matrix}

we obtain

\frac{1}{2}|\sqrt{ε}u_{ε}(t^{∗})|^{2} + \frac{1}{2}|γu_{ε}(t^{∗})|^{2} ≤ (1 + ε)C_{2}[E_{ε}(t) − E_{ε}(t + 1)]^{\frac{2}{p}}

Therefore,

E_{ε}(t^{∗}) ≤ (1 + ε)C_{2}[E_{ε}(t) − E_{ε}(t + 1)]^{\frac{2}{p}} .  (43)

Integrating from (40) of t to t^{∗}, we obtain

\begin{matrix}E_{ε}(t) = E_{ε}(t^{∗}) + ∫_{t}^{t^{∗}} ∥u_{ε}(t)∥_{V_{0}}^{p} + ∫_{t}^{t^{∗}} ∥γu_{ε}(t)∥_{L^{p}(Γ_{1})}^{p} ≤ \\ ((1 + ε)C_{2})(E_{ε}(t) − E_{ε}(t + 1))^{\frac{2}{p}} + (E_{ε}(t) − E_{ε}(t + 1)) \end{matrix}

Therefore,

\begin{matrix}E_{ε}(t)^{\frac{p}{2}} ≤ max\{1, (1 + ε)C_{2}\}^{\frac{p}{2}} (E_{ε}(t) − E_{ε}(t + 1)) + (E_{ε}(t) − E_{ε}(t + 1))^{\frac{p}{2}} ≤ \\ max\{1, (1 + ε)C_{2}\}^{\frac{p}{2}} [E_{ε}(t) − E_{ε}(t + 1)][1 + (E_{ε}(t) − E_{ε}(t + 1))^{\frac{p−2}{2}} ] . \end{matrix}

Being, E_{ε}(t) limited for all ε > 0, follows that

E_{ε}(t)^{\frac{p}{2}} ≤ C_{3} max\{1, (1 + ε)C_{2}\}^{\frac{p}{2}} (E_{ε}(t) − E_{ε}(t + 1)) EJQTDE, 2008 No. 13, p. 17 or

E_{ε}(t)^{\frac{p}{2}} ≤ C_{1ε}(E_{ε}(t) − E_{ε}(t + 1)),

where C_{1ε} = C_{3} max\{1, (1 + ε)C_{2}\}^{\frac{p}{2}} .

Thus, by Nakao’s Lemma, there exists δ > 0 such that

E_{1ε}(t) ≤ C_{1ε} \frac{1}{(1 + t)^{\frac{1}{δ}}}

Note that C_{1ε} → max\{1, C_{2}\}^{\frac{p}{2}} , γu_{ε}(t) ⇀ γu(t) weak in L^{2}(Γ_{1}) and \sqrt{ε}u_{ε}(t) ⇀ 0 weak in L^{2}(Ω). From

lim inf_{ε→0} E_{1ε}(t) ≤ lim inf_{ε→0} C_{1ε} \frac{1}{(1 + t)^{\frac{1}{δ}}} , ∀t ≥ 0

implies that

lim inf_{ε→0} \{\frac{1}{2}|\sqrt{ε}u_{ε}(t)|_{L^{2}(Ω)}^{2} + \frac{1}{2}|γu_{ε}(t)|_{L^{2}(Γ_{1})}^{2}\} ≤ max\{1, C_{2}\}^{\frac{p}{2}} \frac{1}{(1 + t)^{\frac{1}{δ}}} , ∀t ≥ 0

Thus,

\frac{1}{2}|γu(t)|_{L^{2}(Γ_{1})}^{2} ≤ C_{1} \frac{1}{(1 + t)^{\frac{1}{δ}}} , ∀t ≥ 0,

where C_{1} = max\{1, C_{2}\}^{\frac{p}{2}} .

Acknowledgements.

The authors wish to acknowledge Professor L.A. Medeiros who gave their attention to this question and for assistance during the developments of the work and also to the referee of EJQTDE for his constructive remarks and corrections in the manuscript.

References

[1] Araruna, F.D., Antunes, G. O. & Medeiros, L.A.: Semilinear Wave Equation on Manifolds, Annales de la Faculté des Science de Toulouse, XI(1), 2002, pp. 7-18.

EJQTDE, 2008 No. 13, p. 18 [2] Araruna, F.D, Araujo, M.: On an Evolution Problem on Manifolds. Pro-

ceedings of 62ō Seminário Brasileiro de Análise, UNIRIO, Rio de Janeiro-Brasil 2006.

[3] Brezis, H: Analyse Fonctionelle-theorie et Applications. Masson, Paris (1983).

[4] Cavalcanti, M.M. and Domingos Cavalcanti, V.N., On Solvability of Solutions of Degenerate Nonlinear Equations on Manifolds, Differential and Integral Equation, 13(10-12), 2000, pp. 1445-1458.

[5] J. Necas, Les Méthodes Directes en Théorie des Équations Elliptiques, Masson, Pars, 1967.

[6] Lions, J.L : Quelques Méthodes des Resolution des Probléms aux Limites non Lineaires. Dunod, Paris (1969).

[7] Lions, J.L. & Magenes, E., Problèmes aux Limites Non Homogenes et Applications, Dunod, Gauthier-Villars, Paris, vol. 1, 1968.

[8] Medeiros, L. A. and Miranda M.M., Introdução aos Espaços de Sobolev e às Equações Diferenciais Parciais, IM-UFRJ, Rio de Janeiro, 1993.

[9] Nakao, M. Decay of solutions of some nonlinear evolution equation, J. Math. Anal. Appl., 60 (1977), 542-549.

[10] O.A.Lima, A.T.Louredo & A.M. Oliveira : Weak solutions for a stronglycoupled nonlinear system, Electronic Journal of Differential Equations. Vol. 2006, No. 130, pp. 1-18.

[11] Segal, I. : Nonlinear Partial Differential Equations in Quantum Field Theory. Proc. Symb. Appl. Math. A.M.S, 17 (1965), 210 − 226.

[12] Sobolev, S.L.: Applications of Functional Analysis in Mathematical Physics, AMS, 1963.

[13] Temam, R.: Navier-Stokes Equations, Theory and Numerical Analysis, North Holland, 1979.

[14] Yosida, K.: Functional Analysis, Springer Verlag, 1965.

(Received August 25, 2007)