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

Abstract

This paper deals with positive solutions of degenerate and strongly coupled quasilinear parabolic system u t = v α Δ u + u ( a 1 − b 1 u l + c 1 v s ) , v t = u β Δ v + v ( a 2 + b 2 u p − c 2 v q ) with null Dirichlet boundary condition describing a cooperating model with crosswise diffusion, where the constants a i , b i , c i > 0 ( i = 1 , 2 ) , α , β ≥ 0 and l , 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 ≤ s , 0 < β ≤ p with α , β < 2 .