Local solutions for a hyperbolic equation

Let W be an open bounded set of R n with its boundary G constituted of two disjoint parts G0 and G1 with G0\ G1 = ˘. This paper deals with the existence of local solutions to the nonlinear hyperbolic problem u 00 4 u +juj r = f in W (0, T0), u = 0 on G0 (0, T0), ¶u ¶n + h( , u 0 ) = 0 on G1 (0, T0), ( ) where r > 1 is a real number, n(x) is the exterior unit normal at x 2 G1 and h(x, s) (for x2 G1 and s2 R) is a continuous function and strongly monotone in s. We obtain existence results to problem ( ) 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 r are considered in order to have the continuous embedding of Sobolev spaces.