Semilinear reaction–diffusion systems of several components

The homogeneous Dirichlet boundary value problem in a bounded domain Ω RN is considered, where pij 0 (1 i,j n) are constants. Denote by I the identity matrix and P=(pij), which is assumed to be irreducible. We find out that whether or not I−P is a so-called M-matrix plays a fundamental role in the blow-up theorems