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. Lions33 J.-L. Lions. Une remarque sur les problémes de évolution nonlineares dans domains non cilìndriques. Revue Roumaine de Mathématique Pure et Appliquées, 9 (1964), 11-18., 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(88 L. Tartar. "Topics in Nonlinear Analysis". Un. Paris Sud. Dep. Math., Orsay, France (1978).) plus a contradiction process.
Keywords:
nonlinear problem; non-cylindrical domain; hyperbolic equation