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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 posi- tive 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} ≤ 1 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} > 1, and the initial datum (u10,    , un0) satisfies some assumptions, then the positive classical solution is unique and blows up in finite time, where 1 is the first eigenvalue of − in with homogeneous Dirichlet boundary condition.

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