Remarks on the Stability of Steady-State Solutions of Reaction-Diffusion Equations

1980 ◽  
pp. 47-56 ◽  
Author(s):  
C. Conley ◽  
J. Smoller
Keyword(s):  
Steady State ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
The Stability ◽  
Steady State Solutions

A mathematical framework for determining the stability of steady states of reaction–diffusion equations with periodic source terms

2019 ◽  
Vol 84 (4) ◽  
pp. 669-678
Author(s):  
Lennon Ó Náraigh ◽  
Khang Ee Pang
Keyword(s):  
Steady State ◽  
A Priori ◽  
Reaction Diffusion ◽  
Spatial Dimension ◽  
Steady States ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Mathematical Framework ◽  
Source Terms ◽  
The Stability

Abstract We develop a mathematical framework for determining the stability of steady states of generic nonlinear reaction–diffusion equations with periodic source terms in one spatial dimension. We formulate an a priori condition for the stability of such steady states, which relies only on the properties of the steady state itself. The mathematical framework is based on Bloch’s theorem and Poincaré’s inequality for mean-zero periodic functions. Our framework can be used for stability analysis to determine the regions in an appropriate parameter space for which steady-state solutions are stable.

scholarly journals Characterization of Cocycle Attractors for Nonautonomous Reaction–Diffusion Equations

2016 ◽  
Vol 26 (08) ◽  
pp. 1650135 ◽  
Author(s):  
C. A. Cardoso ◽  
J. A. Langa ◽  
R. Obaya
Keyword(s):  
Chaotic Dynamics ◽  
Linear Part ◽  
Reaction Diffusion ◽  
Positive Cone ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Minimal Set ◽  
Full Characterization ◽  
The Stability

In this paper, we describe in detail the global and cocycle attractors related to nonautonomous scalar differential equations with diffusion. In particular, we investigate reaction–diffusion equations with almost-periodic coefficients. The associated semiflows are strongly monotone which allow us to give a full characterization of the cocycle attractor. We prove that, when the upper Lyapunov exponent associated to the linear part of the equations is positive, the flow is persistent in the positive cone, and we study the stability and the set of continuity points of the section of each minimal set in the global attractor for the skew product semiflow. We illustrate our result with some nontrivial examples showing the richness of the dynamics on this attractor, which in some situations shows internal chaotic dynamics in the Li–Yorke sense. We also include the sublinear and concave cases in order to go further in the characterization of the attractors, including, for instance, a nonautonomous version of the Chafee–Infante equation. In this last case we can show exponentially forward attraction to the cocycle (pullback) attractors in the positive cone of solutions.

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