On a Parabolic Strongly Nonlinear Problem on Manifolds

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 ∂u ∂t + n ∑ i=1 ∣∣ ∂u ∂xi ∣∣p−2 ∂u ∂xi νi + |u|u = f on Σ1, where Σ1 is part of the lateral boundary of the cylinder Q = Ω × (0, T ) and f is a given function defined on Σ1.

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: where A is the pseudo Laplacian operator defined by 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 Nȇcas [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.

Hypotheses and Notations
We denote by Let p > 2 be and V 0 the Banach space given by . Note that the application γ : with immersions are dense and continuous. Let which is linear with respect the second variable.Note that, the application Thus B is a hemicontinuous, monotonic operator and To facilitate the understand of this work, introduce the followings notations:

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: ( * * ) where The solution concept for ( * * ) is established by Gauss's Theorem as follows: EJQTDE, 2008 No. 13, p. 4 Summing up from i = 1 to n on both sides of the above equation yields: From this and observing that (see ( * * ) 3 ) Substituting this identity in ( * * ) 1 we get: where u ε means ∂u ε ∂t .Therefore, a solution of the problem ( * * ) is understood in the following sense.
and satisfying the initial conditions with u 0 belongs to W In this section we will establish a theorem of existence of solutions.
) 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 ) there exists w 0 ∈ V 0 such that γw 0 = u 0 because the application 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 g jεm (t)w j (x) solution of the initial value problem for the system of ordinary differential equations: 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 EJQTDE, 2008 No. 13, p. 6 From hypotheses about the initial conditions and the continuity of the application γ, we obtain: 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: Note that From (11) 1 we get where < ., .> represent the duality paring between V 0 × W − 1 p ,p (Γ 1 ) and From estimates above we have .
Therefore γu ε (0) = γw 0 = u 0 , by remark 1. Analogously, we have γu ε (T ) = ς EJQTDE, 2008 No. 13, p. 11 We will show that: Bu ε = ζ.In fact, being the operator B : V → V mononotonic, we obtain: Taking v = u εm and integrating of 0 the T in the approximated equation ( 11) we obtain: Thus, substituting in (26), we have Using the convergence obtained and applying the lim inf m→∞ in both sides of the inequality above we have: Taking v = u ε and integrating of 0 the T in the equation ( 23) we obtain: If we substitute this expression in (27), we obtain

Uniqueness of the Solution
To obtain the uniqueness of the solution, we suppose that there exists two solutions such that u ε , u ε in the conditions of the Theorem 1.It following that w ε = u ε − u ε satisfy: Taking v = w ε in (33) and integrating from 0 the t ≤ T we obtain: Using the monotoneity of the function h(s) = |s| ρ s and a(u ε , w)−a( u ε , w) ≥ 0, Therefore, we have that w

Definition 3 . 1 .
A real value function u ε (x, t) is a solution of the problem ( * * ) if, only if,