On the existence of periodic solutions of an hyperbolic equation in a thin domain (Q1390738)

The aim of this note is to present an existence result for \(T\)-periodic solutions with respect to \(t\) of a damped wave equation in a thin domain both in the non-autonomous and autonomous cases. The considered equation in the non-autonomous case is of the form: \[ {\partial^2 u\over \partial t^2} =\Delta_X u+{\partial^2u \over \partial Y^2} -\beta {\partial u\over \partial t} -\alpha u+ g(t,X,Y,u), \tag{1} \] where \(\alpha\) and \(\beta\) are positive constants and \(g\) is a suitable smooth function, which we assume \(T\)-periodic in time. \((X,Y)\) is a generic point of the thin domain \(Q_\varepsilon =\Omega \times(0, \varepsilon) \subset \mathbb{R}^{N+1}\), where \(\Omega\) is a \(C^2\)-smooth bounded domain in \(\mathbb{R}^N\) and \(\varepsilon \in(0, \varepsilon_0)\) is a small parameter. Associated to equation (1) we consider the Neumann boundary condition \[ {\partial u\over \partial v_\varepsilon} =0\quad \text{on } \partial Q_\varepsilon. \tag{2} \] By using topological degree arguments for compact operators we show the existence of a \(T\)-periodic solution to (1)--(2). The main assumption is that the reduced problem at \(\varepsilon =0\) has an isolated \(T\)-periodic solution whose topological index is different from zero. Then suitable admissible homotopies allow us to derive the existence of a \(T\)-periodic solution of (1)-(2) for sufficiently small \(\varepsilon>0\).