Remarks on a Nonlinear Wave Equation in a Noncylindrical Domain
<div class="page" title="Page 1"><div class="layoutArea"><div class="column"><p><span>In this paper we investigate the existence and uniqueness of solution for a initial boundary value problem for the following nonlinear wave equation: </span></p><p><span>u′′</span> − ∆ u + | u | ˆρ = f in Q</p><div class="page" title="Page 1"><div class="layoutArea"><div class="column"><p><span>where </span><span>Q </span><span>represents a non-cylindrical domain of </span><span>R^{</span><span>n</span><span>+</span><span>1}</span><span>. The methodology, cf. Lions [3], consists of transforming this problem, by means of a perturbation depending on a parameter </span><span>ε > </span><span>0, into another one defined in a cylindrical domain </span><span>Q </span><span>containing </span><span>Q</span><span>. By solving the cylindrical problem, we obtain estimates that depend on </span><span>ε</span><span>. These ones will enable a passage to the limit, when </span><span>ε </span><span>goes to zero, that will guarantee, later, a solution for the non-cylindrical problem. The nonlinearity </span><span>|</span><span>u_</span><span>ε</span><span>|^</span><span>ρ </span><span>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. </span></p></div></div></div></div></div></div>