Time periodic solutions of a class of degenerate parabolic equations

Abstract

This paper is devoted to the time periodic solutions to the degenerate parabolic equations of the form $$\frac{{\partial u}}{{\partial t}} = \vartriangle u^m + u^p (a(x,t) - b(x,t)u) in \Omega \times R$$ under the Dirichlet boundary value condition, wherem>1,p≥0, Ω∈R N is a bounded domain with smooth boundary σΩ anda,b are positive, smooth functions which are periodic int with period ω>0. The existence of nontrivial nonnegative solutions is established provided that 0≤p<m. The existence is also proved in the casep=m but with an additional assumption $$\begin{array}{*{20}c} {\min } \\ {\bar Q} \\ \end{array} a(x,t) > \lambda _1 $$ , where $$\lambda _1 $$ is the first eigenvalue of the operator −Δ under the homogeneous Dirichlet boundary condition.