Chapter 4 Singular elliptic and parabolic equations

Abstract

Reaction-diffusion equations have played an important role in the study of many different phenomena related with applications. These applications include, among many others, population dynamics, chemical reactions, combustion, morphogenesis, nerve impulses, and genetics. Very often positive solutions are the only physically meaningful solutions or, at least, the more interesting ones. A very simple, but already interesting model problem is the semilinear parabolic equation, together with an initial condition. One of the main problems that are considered is the asymptotic behavior of solutions to the semilinear parabolic equation. Many different possibilities are available as, for example, traveling waves, but the chapter discusses the situation where the unique positive solution to the parabolic problem tends to one of the steady-state positive solutions, that is, to a solution to the stationary elliptic problem.