Some degenerate and quasilinear parabolic systems not in divergence form

Abstract

This paper deals with positive solutions of degenerate and quasilinear parabolic systems not in divergence form: u t = u p ( Δu+ av), v t = v q ( Δv+ bu), with null Dirichlet boundary conditions and positive initial conditions, where p, q, a and b are all positive constants. The local existence and uniqueness of classical solution are proved. Moreover, it will be proved that all solutions exist globally if and only if ab⩽ λ 1 2, where λ 1 is the first eigenvalue of − Δ in Ω with homogeneous Dirichlet boundary condition.