Abstract
We treat an initial boundary value problem for a nonlinear wave equation
in the domain
,
. The boundary condition at the boundary point
of the domain for a solution
involves a time convolution term of the boundary value of
at
, whereas the boundary condition at the other boundary point is of the form
with
and
given nonnegative constants. We prove existence of a unique solution of such a problem in classical Sobolev spaces. The proof is based on a Galerkin-type approximation, various energy estimates, and compactness arguments. In the case of
, the regularity of solutions is studied also. Finally, we obtain an asymptotic expansion of the solution
of this problem up to order
in two small parameters
,
.