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

Abstract

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.