The existence of uniform attractors for non-autonomous reaction-diffusion equations on the whole space

2012 ◽  
Vol 53 (8) ◽  
pp. 082703 ◽  
Author(s):  
Yongqin Xie ◽  
Kaixuan Zhu ◽  
Chunyou Sun
Keyword(s):  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Whole Space ◽  
Uniform Attractors

Asymptotic analysis and homogenization of invariant measures

2019 ◽  
Vol 19 (02) ◽  
pp. 1950015
Author(s):  
Oleksandr Misiats ◽  
Oleksandr Stanzhytskyi ◽  
Nung Kwan Yip
Keyword(s):  
Asymptotic Behavior ◽  
Invariant Measure ◽  
Stationary Solution ◽  
Invariant Measures ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Regular Case ◽  
Limiting Behavior ◽  
Whole Space

In this paper, we study limiting behavior of the invariant measures for reaction–diffusion equations in the whole space [Formula: see text] with regular and singular perturbations. In the regular case, we show the convergence of the unique stationary solution of [Formula: see text] to a stationary solution of the limiting equation [Formula: see text]. We also consider the asymptotic behavior of the stationary solution under the perturbations of spectrum. Finally, for the singular perturbation of homogenization type, we show the weak convergence of invariant measure to its homogenized limit.

scholarly journals Uniform attractors for nonautonomous reaction-diffusion equations with the nonlinearity in a larger symbol space

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Xiangming Zhu ◽  
Chengkui Zhong
Keyword(s):  
Existence Of Solutions ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Uniform Attractor ◽  
Symbol Space ◽  
Uniform Attractors

<p style='text-indent:20px;'>Existence and structure of the uniform attractors for reaction-diffusion equations with the nonlinearity in a weaker topology space are considered. Firstly, a weaker symbol space is defined and an example is given as well, showing that the compactness can be easier obtained in this space. Then the existence of solutions with new symbols is presented. Finally, the existence and structure of the uniform attractor are obtained by proving the <inline-formula><tex-math id="M1">\begin{document}$ (L^{2}\times \Sigma, L^{2}) $\end{document}</tex-math></inline-formula>-continuity of the processes generated by solutions.</p>

scholarly journals Blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients under Neumann boundary conditions

2020 ◽  
Vol 18 (1) ◽  
pp. 1552-1564
Author(s):  
Huimin Tian ◽  
Lingling Zhang
Keyword(s):  
Boundary Conditions ◽  
Blow Up ◽  
Sufficient Conditions ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Neumann Boundary Conditions ◽  
Reaction Diffusion Equations ◽  
Time Dependent ◽  
Diffusion Equations ◽  
Nonlocal Reaction

Abstract In this paper, the blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients are investigated under Neumann boundary conditions. By constructing some suitable auxiliary functions and using differential inequality techniques, we show some sufficient conditions to ensure that the solution u ( x , t ) u(x,t) blows up at a finite time under appropriate measure sense. Furthermore, an upper and a lower bound on blow-up time are derived under some appropriate assumptions. At last, two examples are presented to illustrate the application of our main results.

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