Quasilinear parabolic systems of several components

In this paper we investigate the blowup criteria of the quasilinear parabolic system with homogeneous Dirichlet boundary conditions on a bounded domain R ^N , where c >0, >0, p 0 (1, n) are constants. Denote by I the identity matrix and P=(p ), which is assumed to be irreducible. That I–P is a singular M-matrix is shown to be the critical case, in which ₁ plays a fundamental role, where ₁ is the first Dirichlet eigenvalue of the Laplacian on . As a result, we give a general answer to the question of Galaktionov and Levine on the porous medium systems. Mathematics Subject Classification (2000):35K50, 35K55, 35K65