Quasilinear parabolic systems of several components

In this paper we investigate the blowup criteria of the quasilinear parabolic system \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{ u_{{\imath t}}=c_{{\imath}}u_{{\imath}}^{{\alpha_{{\imath}}}}(\Delta u_\imath+ \prod_{{\jmath=1}}^n u_\jmath^{{p_{{\imath\jmath}}}}), \imath=1, 2, \cdots, n }}$$\end{document} with homogeneous Dirichlet boundary conditions on a bounded domain Ω⊂RN, where cı>0, αı>0, pı[graphic not available: see fulltext]≥0 (1≤ı, [graphic not available: see fulltext]≤n) are constants. Denote by I the identity matrix and P=(pı[graphic not available: see fulltext]), which is assumed to be irreducible. That I−P is a singular M-matrix is shown to be the critical case, in which λ1 plays a fundamental role, where λ1 is the first Dirichlet eigenvalue of the Laplacian on Ω. As a result, we give a general answer to the question of Galaktionov and Levine on the porous medium systems.