Remarks on a Nonlinear Wave Equation in a Noncylindrical Domain^†

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 of^{_Rn+ 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.

Keywords:
nonlinear problem; non-cylindrical domain; hyperbolic equation