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 ut=vαΔu+u(a1−b1u+c1v), vt=uβΔv+v(a2+b2u−c2v) with null Dirichlet boundary condition and positive initial conditions describing a cooperating two-species Lotka–Volterra model with cross-diffusion, where the constants ai,bi,ci>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.