Investigations of Solutions of Nonlinear Hyperbolic Equations with a Small Nonlinearity and Applications

Abstract

The Cauchy problem for the nonlinear hyperbolic equation $L_\varepsilon u + \varepsilon ^p N_\varepsilon (t,x,D^\alpha u,| \alpha | \leq q) = f_\varepsilon (t,x)$ of order $m \geq 2$, $0 \leq q \leq m - 1$, is studied; $\varepsilon \in (0,1]$ is a small parameter. The problem possesses a solution on any given finite interval of time provided $\varepsilon $ is sufficiently small. Estimations for Sobolev and $L^\infty $ norms of solutions are derived. Applications to the stability problem for hyperbolic equations, and to justification of the nonlinear geometric optics method for spin glass models, are given.