Extremal equilibria for reaction-diffusion equations in bounded domains and applications

Aníbal Rodríguez-Bernal*, Alejandro Vidal-López

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

18 Citations (Scopus)

Abstract

We show the existence of two special equilibria, the extremal ones, for a wide class of reaction-diffusion equations in bounded domains with several boundary conditions, including non-linear ones. They give bounds for the asymptotic dynamics and so for the attractor. Some results on the existence and/or uniqueness of positive solutions are also obtained. As a consequence, several well-known results on the existence and/or uniqueness of solutions for elliptic equations are revisited in a unified way obtaining, in addition, information on the dynamics of the associated parabolic problem. Finally, we ilustrate the use of the general results by applying them to the case of logistic equations. In fact, we obtain a detailed picture of the positive dynamics depending on the parameters appearing in the equation.

Original languageEnglish
Pages (from-to)2983-3030
Number of pages48
JournalJournal of Differential Equations
Volume244
Issue number12
DOIs
Publication statusPublished - 15 Jun 2008
Externally publishedYes

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