Global existence and blow-up analysis for some degenerate and quasilinear parabolic systems

This paper deals with positive solutions of some degenerate and quasilinear parabolic systems not in divergence form: u1t = f1(u2)(∆u1 + a1u1), · · ·, u(n−1)t = fn−1(un)(∆un−1 + an−1un−1), unt = fn(u1)(∆un+anun) with homogenous Dirichlet boundary condition and positive initial condition, where ai (i = 1, 2, ···, n) are positive constants and fi (i = 1, 2, ···, n) satisfy some conditions. The local existence and uniqueness of classical solution are proved. Moreover, it will be proved that: (i) when min{a1, · · ·, an} ≤ 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; (ii) when min{a1, · · ·, an} > λ1, and the initial datum (u10, · · ·, un0) satisfies some assumptions, then the positive classical solution is unique and blows up in finite time, where λ1 is the first eigenvalue of −∆ in Ω with homogeneous Dirichlet boundary condition.