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 α Δ u + u ( a 1 − b 1 u + c 1 v ) , v t = u β Δ v + 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 α, β 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 α , β ⩽ 1 .