A Lotka–Volterra cooperating reaction–diffusion system with degenerate density-dependent diffusion

Abstract

In the Lotka–Volterra cooperating reaction–diffusion system if the diffusion coefficients are constants then for a certain set of reaction rates in the reaction function the solution of the system blows up in finite time, and for another set of reaction rates, a unique global solution exists and converges to the trivial solution. However, if the diffusion coefficients are density-dependent then the dynamic behavior of the solution can be quite different. The aim of this paper is to investigate the global existence and the asymptotic behavior of the solution for a class of density-dependent cooperating reaction–diffusion systems where the diffusion coefficients are degenerate. It is shown that the time-dependent problem has a unique bounded global solution, and in addition to the trivial and semi-trivial solutions the corresponding steady-state problem has a positive maximal solution and a positive minimal solution. Moreover, the time-dependent solution converges to the maximal solution for one class of initial functions, and to the minimal solution for another class of initial functions. The above convergence property holds true for any reaction rates in the reaction function. Applications of the above results are given to a porous medium type of reaction–diffusion problem as well as other types of diffusion coefficients, including the finite sum and products of these functions.