Remarks on a Nonlinear Wave Equation in a Noncylindrical Domain

Abstract

In this paper we investigate the existence and uniqueness of solution for a initial boundary value problem for the following nonlinear wave equation:

u′′ − ∆ u + | u | ˆρ = f in Q

where Q represents a non-cylindrical domain of R^{n+1}. The methodology, cf. Lions [3], consists of transforming this problem, by means of a perturbation depending on a parameter ε > 0, into another one defined in a cylindrical domain Q containing Q. 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 |u_ε|^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.