scholarly journals Attractors for reaction–diffusion equations on thin domains whose linear part is non-self-adjoint

2004 ◽  
Vol 206 (1) ◽  
pp. 94-126 ◽  
Author(s):  
Thomas Elsken
Keyword(s):  
Linear Part ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Thin Domains

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.

Lattice differential equations embedded into reaction–diffusion systems

2009 ◽  
Vol 139 (1) ◽  
pp. 193-207 ◽  
Author(s):  
Arnd Scheel ◽  
Erik S. Van Vleck
Keyword(s):  
Differential Equations ◽  
Invariant Manifolds ◽  
Linear Part ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Dimensional Manifold ◽  
Lattice Differential Equations ◽  
Reaction Diffusion Systems ◽  
Infinite Dimensional

We show that lattice dynamical systems naturally arise on infinite-dimensional invariant manifolds of reaction–diffusion equations with spatially periodic diffusive fluxes. The result connects wave-pinning phenomena in lattice differential equations and in reaction–diffusion equations in inhomogeneous media. The proof is based on a careful singular perturbation analysis of the linear part, where the infinite-dimensional manifold corresponds to an infinite-dimensional centre eigenspace.

scholarly journals Limiting dynamics for non-autonomous stochastic retarded reaction-diffusion equations on thin domains

2019 ◽  
Vol 39 (7) ◽  
pp. 3717-3747 ◽  
Author(s):  
Dingshi Li ◽  
◽  
Kening Lu ◽  
Bixiang Wang ◽  
Xiaohu Wang ◽  
...  
Keyword(s):  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Thin Domains
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