Problems with Different Time Scales for Nonlinear Partial Differential Equations

Consider a system of partial differential equations uₜ ( ε - 1 P₀ + P₁ )u. Here 0 < ε 1 is a small constant, P₀ = P₀ ( ∂/∂ x ) is a differential operator with constant coefficients and P₁ = P₁ ( x,t,u,∂/∂ x ) is a quasilinear operator. The coefficients of P₀, P₁ 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 ε. Applications are discussed.