Local Existence for Solutions of Fully Nonlinear Wave Equations

P. Lesky jun., Stuttgart

Abstract

Let Ω be a domain in IR^{n} and let m ∈ IN be given. We study the initial–boundary value problem for the equation F (t, x, \overline{D}_{x}^{2m}u(t, x), \overline{D}_{x}^{m}∂_{t}u(t, x), ∂_{t}^{2}u(t, x)) = f(t, x) with homogeneous Dirichlet’s boundary condition; here u is a scalar function, \overline{D}_{x}^{m}u := (∂_{x}^{α}u)_{|α|≤m} and certain restrictions are made on F guaranteeing that energy estimates are possible. We prove the existence of a T > 0 such that a unique classical solution u exists on [0, T] × Ω. Furthermore we show that T → ∞ if the data tend to zero.

AMS classification code: 35G30, 35A07

Published in: Math. Meth. in the Appl. Sci. 14 (1991), 483-508