A degenerate and strongly coupled quasilinear parabolic system with crosswise diffusion for a mutualistic model

This paper deals with positive solutions of degenerate and strongly coupled quasilinear parabolic system ut=vαΔu+u(a1−b1ul+c1vs),vt=uβΔv+v(a2+b2up−c2vq) with null Dirichlet boundary condition describing a cooperating model with crosswise diffusion, where the constants ai,bi,ci>0(i=1,2)ai,bi,ci>0(i=1,2), α,β≥0α,β≥0 and l,s,p,q≥1l,s,p,q≥1. Local existence of positive classical solution is proved. Moreover, it will be proved that the solution is global if intra-specific competitions of the species are strong, whereas the solution may be nonglobal if the inter-specific cooperation is strong and 0<α≤s0<α≤s, 0<β≤p0<β≤p with α,β<2α,β<2.