Regular solutions of initial-boundary value problems for linear and nonlinear wave-equations I

Abstract

This paper deals with the question of the existence of classical solutions for the equations

$$\frac{{\partial ^{2} u}{\partial t^{2} }} + \sum_{\begin{subarray}{l} |\alpha| \leqslant m \\ | \beta | \leqslant m \end{subarray}} D^{\alpha} (A_{\alpha \beta } (x,t) D^{\beta} u) = f (t,x,u)$$

on [0,T] × G. G is a bounded or unbounded domain; the differential operator in the space variables is elliptic; the initial values of u are prescribed and D^αu (t,x) vanishes for (t,x) ∈ [0,T] × ∂G, |α|≤ m−1. First we develop a method for solving regularly linear wave equations. In contrast to the usual compatibility conditions, our method requires less differentiability in t but imposes some boundary conditions on f(t). It allows some applications to nonlinear problems which will be treated in the second part of this paper and which e.g. enable us to solve ∂² u/∂t²−A(t)u+u³=f.