On nonlinear reaction-diffusion systems

Abstract

This paper presents a qualitative analysis for a coupled system of two reaction-diffusion equations under various boundary conditions which arises from a number of physical problems. The nonlinear reaction functions are classified into three basic types according to their relative quasi-monotone property. For each type of reaction functions, an existence-comparison theorem, in terms of upper and lower solutions, is established for the time-dependent system as well as some boundary value problems. Three concrete physical systems arising from epidemics, biochemistry and engineering are taken as representatives of the basic types of reacting problems. Through suitable construction of upper and lower solutions, various qualitative properties of the solution for each system are obtained. These include the existence and bounds of time-dependent solutions, asymptotic behavior of the solution, stability and instability of nontrivial steady-state solutions, estimates of stability regions, and finally the blowing-up property of the solution. Special attention is given to the homogeneous Neumann boundary condition.