Existence of solutions for a nonlinear hyperbolic-parabolic equation in a non-cylinder domain

Marcondes Rodrigues Clark

doi:10.1155/s0161171296000221

Abstract

In this paper, we study the existence of global weak solutions for the equationk2(x)u″+k1(x)u′+A(t)u+|u|ρu=f (I)in the non-cylinder domainQinRn+1;k1andk2are bounded real functions,A(t)is the symmetric operatorA(t)=−∑i,j=1n∂∂xj(aij(x,t)∂∂xi) whereaijandfare real functions given inQ. For the proof of existence of global weak solutions we use the Faedo-Galerkin method, compactness arguments and penalization.

Highlights

In the cylinder x(0, T), where ft C N" is a bounded open set, Bensoussan et al [1] and Lions [7] have studied the homogenization for the following Cauchy problem: k(x)u" + k,(x)u’ + Au f in gt (II)
Many authors have been investigating the solvability of solution for the nonlinear equations associated with problem (I)see: Larkin [4], Lima [5], Medeiros [9], Medeiros [10], Medeiros [11], Melo [12], Maciel [13], Neves [14] and Vagrov [16]
In the non-cylindrical domain Q, Lions, J.L. [8] studied the existence and uniqueness of global weak solutions for nonlinear equations associated with problem (II) with nonlinearity of type ulu

Summary

Introduction

INTRODUCTION AND TERMINOLOGYLet T _> 0 be a positive real number, O a bounded open set of R" and Q c O x [0, T) a noncylindrical domain in R" +In the cylinder x(0, T), where ft C N" is a bounded open set, Bensoussan et al [1] and Lions [7] have studied the homogenization for the following Cauchy problem: k(x)u" + k,(x)u’ + Au f in gt (II)u(x,O) Uo(X and k(x)u’(x,O) k/(x)u,(x),x fMany authors have been investigating the solvability of solution for the nonlinear equations associated with problem (I)see: Larkin [4], Lima [5], Medeiros [9], Medeiros [10], Medeiros [11], Melo [12], Maciel [13], Neves [14] and Vagrov [16].In the non-cylindrical domain Q, Lions, J.L. [8] studied the existence and uniqueness of global weak solutions for nonlinear equations associated with problem (II) with nonlinearity of type ulu." Let F/t=Qf’l{t=s} be a plane in +. Let a(t,u,v) denote the bilinear form associated to the operator A(t) Froxn (H.4) and, umng Cauchy-Schwartz, we obtain: by Poincar-Friedrichs inequality and of (H.4), there exists a > 0, real, such that" 1, p p+2 and u is a solution (1) in the weak sense in Q, i.e., d dt For each e > 0, there exists one function U,:O (O,T)--N, solution of the problem (P,), such that: U" L(O,T;H(O))

Results

Conclusion