Initial-boundary value problems for degenerate hyperbolic equations

Abstract

The aim of the paper is to study solvability in Sobolev spaces initial-boundary value problems for differential equations $$u_{tt}-\varphi(t)Au+c(x,t)u=f(x,t)$$ in which $A$ is an elliptic operator acting in the spatial variables $x_1$,\ldots,$x_n$ and $\varphi(t)$ is a non-negative function on the segment $[0,T]$. Existence theorems of regular solutions are proven. Some generalizations of the results are also described.