The Effect of Varying Coefficients on the Dynamics of a Class of Superlinear Indefinite Reaction-Diffusion Equations

Abstract

In this paper we analyze how the dynamics of a class of superlinear indefinite reaction-diffusion equations varies as the nodal behavior of a coefficient changes. To perform this analysis we use both theoretical and numerical tools. The analysis aids the numerical study, and the numerical study confirms and completes the analysis. The numerics in addition provides us with some further results for which-at-first glance analytical tools are not available yet. Our main analytical result shows that the problem possesses a unique positive solution which is linearly asymptotically stable if the trivial state is linearly unstable and the model admits some positive solution. This result is a relevant feature for superlinear indefinite problems, since our numerical computations show how these models can have an arbitrarily large number of positive solutions if the trivial state is unstable. © 2000 Academic Press.