scholarly journals Upper semicontinuity of the attractor for lattice dynamical systems of partly dissipative reaction diffusion systems

2005 ◽  
Vol 2005 (3) ◽  
pp. 273-288 ◽  
Author(s):  
Ahmed Y. Abdallah
Keyword(s):  
Dynamical System ◽  
Hilbert Space ◽  
Upper Semicontinuity ◽  
Reaction Diffusion ◽  
Dimensional Lattice ◽  
Reaction Diffusion Systems ◽  
Infinite Dimensional ◽  
Diffusion Systems ◽  
Lattice Dynamical System ◽  
Lattice Dynamical Systems

We investigate the existence of a global attractor and its upper semicontinuity for the infinite-dimensional lattice dynamical system of a partly dissipative reaction diffusion system in the Hilbert spacel2×l2. Such a system is similar to the discretized FitzHugh-Nagumo system in neurobiology, which is an adequate justification for its study.

UNIFORM ATTRACTORS OF NONAUTONOMOUS DISCRETE REACTION–DIFFUSION SYSTEMS IN WEIGHTED SPACES

2008 ◽  
Vol 18 (03) ◽  
pp. 695-716 ◽  
Author(s):  
BIXIANG WANG
Keyword(s):  
Upper Semicontinuity ◽  
Reaction Diffusion ◽  
Global Attractors ◽  
Weighted Spaces ◽  
Lattice Systems ◽  
Limiting Behavior ◽  
Reaction Diffusion Systems ◽  
Nonautonomous Systems ◽  
Diffusion Systems ◽  
Uniform Attractors

We study the asymptotic behavior of nonautonomous discrete Reaction–Diffusion systems defined on multidimensional infinite lattices. We show that the nonautonomous systems possess uniform attractors which attract all solutions uniformly with respect to the translations of external terms when time goes to infinity. These attractors are compact subsets of weighted spaces, and contain all bounded solutions of the system. The upper semicontinuity of the uniform attractors is established when an infinite-dimensional reaction–diffusion system is approached by a family of finite-dimensional systems. We also examine the limiting behavior of lattice systems with almost periodic, rapidly oscillating external terms in weighted spaces. In this case, it is proved that the uniform global attractors of nonautonomous systems converge to the global attractor of an averaged autonomous system.

scholarly journals Reaction-diffusion coupled inclusions with variable exponents and large diffusion

2021 ◽  
Vol 41 (4) ◽  
pp. 539-570
Author(s):  
Jacson Simsen ◽  
Mariza Stefanello Simsen ◽  
Petra Wittbold
Keyword(s):  
Initial Conditions ◽  
Upper Semicontinuity ◽  
Reaction Diffusion ◽  
Coupled System ◽  
Global Attractors ◽  
Coupled Systems ◽  
Variable Exponents ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Large Diffusion

This work concerns the study of asymptotic behavior of coupled systems of \(p(x)\)-Laplacian differential inclusions. We obtain that the generalized semiflow generated by the coupled system has a global attractor, we prove continuity of the solutions with respect to initial conditions and a triple of parameters and we prove upper semicontinuity of a family of global attractors for reaction-diffusion systems with spatially variable exponents when the exponents go to constants greater than 2 in the topology of \(L^{\infty}(\Omega)\) and the diffusion coefficients go to infinity.

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