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

Rodríguez Bernal, Aníbal and Vidal López, Alejandro (2008) Extremal equilibria for reaction-diffusion equations in bounded domains and applications. Journal of Differential Equations, 244 (2008). 2983-3030 . ISSN 0022-0396

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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

Item Type:Article
Uncontrolled Keywords:Reaction-diffusion equation; Extremal equilibria; Attractor; Nonlinear boundary conditions; Dirichlet boundary condition; Robin boundary condition
Subjects:Sciences > Mathematics > Differential equations
ID Code:12868
Deposited On:16 Jun 2011 07:06
Last Modified:06 Feb 2014 09:34

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