A degenerate and strongly coupled quasilinear parabolic system not in divergence form

Abstract

This paper deals with positive solutions of degenerate and strongly coupled quasi-linear parabolic system not in divergence form: u t =v p (Δu+au), v t =u q (Δv+bv) with null Dirichlet boundary condition and positive initial condition, where p, q, a and b are all positive constants, and p, q ≥ 1. The local existence of positive classical solution is proved. Moreover, it will be proved that: (i) When min {a, b} ≤ λ1 then there exists global positive classical solution, and all positive classical solutions can not blow up in finite time in the meaning of maximum norm (we can not prove the uniqueness result in general); (ii) When min {a, b} > λ1, there is no global positive classical solution (we can not still prove the uniqueness result), if in addition the initial datum (u0v0) satisfies Δu0 + au0 ≥ 0, Δv0+bv0 ≥ 0 in Ω, then the positive classical solution is unique and blows up in finite time, where λ1 is the first eigenvalue of −Δ in Ω with homogeneous Dirichlet boundary condition.