scholarly journals Limiting behavior of trajectory attractors of perturbed reaction-diffusion equations

2017 ◽  
Vol 22 (11) ◽  
pp. 1-22
Author(s):  
Gaocheng Yue ◽  
Keyword(s):  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Limiting Behavior ◽  
Trajectory 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 Lyapunov Functions for Weak Solutions of Reaction-Diffusion Equations with Discontinuous Interaction Functions and its Applications

2015 ◽  
Vol 2 (1) ◽  
Author(s):  
Mark O. Gluzman ◽  
Nataliia V. Gorban ◽  
Pavlo O. Kasyanov
Keyword(s):  
Weak Solutions ◽  
Sufficient Conditions ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Balance Model ◽  
Time Behavior ◽  
Differential Reaction ◽  
Regularity Properties ◽  
Trajectory Attractors

AbstractIn this paper we investigate additional regularity properties for global and trajectory attractors of all globally defined weak solutions of semi-linear parabolic differential reaction-diffusion equations with discontinuous nonlinearities, when initial data uτ ∈ L2(Ω). The main contributions in this paper are: (i) sufficient conditions for the existence of a Lyapunov function for all weak solutions of autonomous differential reaction-diffusion equations with discontinuous and multivalued interaction functions; (ii) convergence results for all weak solutions in the strongest topologies; (iii) new structure and regularity properties for global and trajectory attractors. The obtained results allow investigating the long-time behavior of state functions for the following problems: (a) a model of combustion in porous media; (b) a model of conduction of electrical impulses in nerve axons; (c) a climate energy balance model; (d) a parabolic feedback control problem.

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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