The uniqueness of the stable positive solution for a class of superlinear indefinite reaction diffusion equations

Abstract

In this paper we characterize the existence and prove the uniqueness of the stable positive steady-state for a general class of superlinear indefinite reaction diffusion equations in the absence of $L_\infty$ a priori bounds. More precisely, it will be shown that the model possesses a linearly stable positive steady-state if, and only if, the trivial solution is linearly unstable and the model possesses some positive steady-state. Moreover, it is unique if it exists. Actually, the minimal positive steady-state provides us with the unique linearly stable positive steady-state of the model. This is an extremely striking result since these problems can have an arbitrarily large number of positive steady-states as a result of having spatial inhomogeneities or varying the geometry of the support domain where the reaction takes place.