Remarks on a Nonlinear Wave Equation in a Noncylindrical Domain

• Autor(a) principal: LOPEZ,I.F.
• Data de Publicação: 2015
• Outros Autores: ANTUNES,G.O., SILVA,M.D.G. DA, MEDEIROS,L.A.J.
• Tipo de documento: Artigo
• Idioma: eng
• Título da fonte: TEMA (Sociedade Brasileira de Matemática Aplicada e Computacional. Online)
• Texto Completo: http://old.scielo.br/scielo.php?script=sci_arttext&pid=S2179-84512015000300195
• topic: nonlinear problem; non-cylindrical domain; hyperbolic equation

ABSTRACT In this paper we investigate the existence of solution for an initial boundary value problem of the following nonlinear wave equation: u'' - Δu + |u|ρ =f in . where represents a non-cylindrical domain ofRn+ 1. The methodology, cf. Lions3, consists of transforming this problem, by means of a perturbation depending on a parameter ε > 0, into another one defined in a cylindrical domain Qcontaining . By solving the cylindrical problem, we obtain estimates that depend on ε. These ones will enable a passage to the limit, when ε goes to zero, that will guarantee, later, a solution for the non-cylindrical problem. The nonlinearity |uε|ρ introduces some obstacles in the process of obtaining a priori estimates and we overcome this difficulty by employing an argument due to Tartar(8) plus a contradiction process.