Global existence and nonexistence for some degenerate and strongly coupled quasilinear parabolic systems

This paper deals with positive solutions of degenerate and strongly coupled quasilinear parabolic systems u(t) = v(alpha) Delta u + u(a(1) - b(1)u + c(1)v), v(t) = u(beta) Delta v + (a(2) + b(2)u - c(2)v) with null Dirichlet boundary condition and positive initial conditions describing a cooperating two-species Lotka-Volterra model with cross-diffusion, where the constants a(i), b(i), c(i) > 0 for i = 1, 2 and alpha, beta are non-negative. The local existence of positive classical solutions is proved. Moreover, the authors proved that the solutions are global if intra-specific competition of the species are strong, whereas the solutions may blow up if the inter-specific cooperation are strong and alpha, beta <= 1. (C) 2011 Elsevier Inc. All rights reserved.