Local solutions for a hyperbolic equation

Abstract

Let $\Omega$ be an open bounded set of $\mathbb{R}^n$ with its boundary $\Gamma$ constituted of two disjoint parts $\Gamma_0$ and $\Gamma_1$ with $\overline{\Gamma}_0 \cap \overline{\Gamma}_1=\emptyset.$ This paper deals with the existence of local solutions to the nonlinear hyperbolic problem \begin{equation} \left| \begin{aligned} &u'' - \triangle u + |u|^\rho=f &\quad& \mbox{in} \ \Omega \times (0, T_0), \\ &u=0 &\quad&\mbox{on} \ \Gamma_0 \times (0, T_0), \\ & \displaystyle\frac{\partial u}{\partial \nu} + h(\cdot,u')=0 &\quad&\mbox{on} \ \Gamma_1 \times (0, T_0), \end{aligned} \right. \tag{$\ast$} \end{equation} where $\rho >1$ is a real number, $\nu(x)$ is the exterior unit normal at $x\in \Gamma_1$ and $h(x,s)$ (for $x \in \Gamma_1$ and $s \in \mathbb{R}$) is a continuous function and strongly monotone in $s$. We obtain existence results to problem ($\ast$) by applying the Galerkin method with a special basis, Strauss' approximations of continuous functions and trace theorems for non-smooth functions. As usual, restrictions on $\rho$ are considered in order to have the continuous embedding of Sobolev spaces.

Keywords

• galerkin method
• local solutions
• hyperbolic equation