Problems with Different Time Scales for Nonlinear Partial Differential Equations

Abstract

Consider a system of partial differential equations $u_t ( \varepsilon ^{ - 1} P_0 + P_1 )u$. Here $0 < \varepsilon \ll 1$ is a small constant, $P_0 = P_0 ( \partial/\partial x )$ is a differential operator with constant coefficients and $P_1 = P_1 ( x,t,u,\partial/\partial x )$ is a quasilinear operator. The coefficients of $P_0$, $P_1$ are of order $O( 1 )$. It is shown how to prepare the initial data such that the solutions $u( x,t )$ and their derivatives can be bounded independently of $\varepsilon $. Applications are discussed.