Wave Equations with Time–Dependent

Spatial Operators of Higher Order

Peter Lesky jun.

Abstract. We study the initial–boundary value problem for ∂_{t}^{2}u(t, x) + A(t)u(t, x) + B(t)∂_{t}u(t, x) = f(t, x) on [0, T] × Ω (Ω ⊂ IR^{n}) with homogeneous Dirichlet boundary condition; here A(t) denotes a family of uniformly strongly elliptic operators of order 2m, B(t) denotes a family of spatial differential operators of order ≤ m, and u is a scalar function. We prove the existence of a unique strong solution u. Furthermore an energy estimate for u is given.

Keywords: Linear wave equation, Energy estimate

AMS classification 35L05, 35L30, 35L35, 35G15

Published in: Math. Meth. in the Appl. Sci. 15 (1992), 453-468