Systems of Reaction—Diffusion Equations

Abstract

In recent years, systems of reaction-diffusion equations have received a great deal of attention, motivated by both their widespread occurrence in models of chemical and biological phenomena, and by the richness of the structure of their solution sets. In the simplest models, the equations take the form

$$ \frac{{\partial u}}{{\partial t}}{\mkern 1mu} = {\mkern 1mu} D\Delta u{\mkern 1mu} + {\mkern 1mu} f(u),\quad {\mkern 1mu} x \in {\mkern 1mu} \Omega {\mkern 1mu} \subset {\mkern 1mu} {R^k},{\mkern 1mu} t > {\mkern 1mu} 0, $$

(14.1)

where u ∈ Rⁿ, D is an n x n matrix, and f(u) is a smooth function. The combination of diffusion terms together with the nonlinear interaction terms, produces mathematical features that are not predictable from the vantage point of either mechanism alone. Thus, the term DΔu acts in such a way as to “dampen” u, while the nonlinear function f(u) tends to produce large solutions, steep gradients, etc. This leads to the possibility of threshold phenomena, and indeed this is one of the interesting features of this class of equations.