Local solutions for a hyperbolic equation

摘要

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 egin{equation} left| egin{aligned} &u'' - riangle u + |u|^ho=f &quad& mbox{in} Omega imes (0, T_0), &u=0 &quad&mbox{on} Gamma_0 imes (0, T_0), & displaystylerac{partial u}{partial u} + h(cdot,u')=0 &quad&mbox{on} Gamma_1 imes (0, T_0), end{aligned} ight. ag{$st$} end{equation} where $ho >1$ is a real number, $u(x)$ is the exterior unit normal at $xin Gamma_1$ 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 ($st$) 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 $ho$ are considered in order to have the continuous embedding of Sobolev spaces.