Wavefront solutions of degenerate quasilinear reaction–diffusion systems with mixed quasi-monotonicity

Abstract

This paper is concerned with the existence of wavefront solutions to a system of degenerate quasilinear reaction–diffusion equations of mixed quasi-monotone properties in the form ∂ui∕∂t=∇⋅Diui∇ui+fiu−∞<x<∞,t>0for i=1,…,n. The important features of this system are that some of the diffusion coefficients Diui are density dependent and may vanish at certain value of ui, and that each function fi is quasi-monotone increasing for some components of u =u1,…,un and decreasing for other components of u. Such systems model reaction–diffusion processes with density driven diffusion mechanism. Under certain general conditions we prove the existence of a traveling wave solution that is between a pair of coupled upper and lower solutions. A predator–prey model with nonlinear diffusion is used as an illustration of application. The presence of wavefront solutions flowing toward the coexistence states is established by constructing appropriate upper and lower solutions.