Some degenerate and quasilinear parabolic systems not in divergence form ✩

This paper deals with positive solutions of degenerate and quasilinear parabolic systems not in divergence form: ut = up(Δu + av), vt = vq(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 λ21 , where λ1 is the first eigenvalue of −Δ in Ω with homogeneous Dirichlet boundary condition.  2002 Elsevier Science (USA). All rights reserved.