Quasilinear parabolic systems of several components (Q1416727)

The paper deals with blow-up criteria for the homogeneous Cauchy-Dirichlet problem in \(\Omega\times(0,\infty),\) \(\Omega\subset{\mathbb R}^N,\) for the quasilinear parabolic system \[ (u_i)_t=c_i u_i^{\alpha_i} \Biggl(\Delta u_i +\prod_{j=1}^n u_j^{p_{ij}}\Biggr),\quad i=1,2,\ldots,n, \] where \(c_i>0,\) \(\alpha_i>0,\) \(p_{ij}\geq 0,\) \(1\leq i,j\leq n,\) are constants. Set \(I\) for the identity matrix and \(P=(p_{ij})\) which is assumed to be irreducible. The authors show that ``\(I-P\) is singular \(M\)-matrix'' is a critical case in which the first Dirichlet eigenvalue \(\lambda_1\) of the Laplacian in \(\Omega\) plays a fundamental role. This gives various results concerning porous medium systems.