scholarly journals MONOTONE METHODS AND STABILITY RESULTS FOR NONLOCAL REACTION-DIFFUSION EQUATIONS WITH TIME DELAY

2018 ◽  
Vol 8 (5) ◽  
pp. 1342-1368
Author(s):  
Yueding Yuan ◽  
◽  
Zhiming Guo ◽  
◽  
Keyword(s):  
Time Delay ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Monotone Methods ◽  
Nonlocal Reaction ◽  
Stability Results

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.

The Dynamics of Localized Solutions of Nonlocal Reaction-Diffusion Equations

2000 ◽  
Vol 43 (4) ◽  
pp. 477-495
Author(s):  
Michael J. Ward
Keyword(s):  
Materials Science ◽  
Principal Eigenvalue ◽  
Dynamical Behavior ◽  
Reaction Diffusion ◽  
Propagation Model ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Nonlocal Term ◽  
Localized Solutions ◽  
Nonlocal Reaction

AbstractMany classes of singularly perturbed reaction-diffusion equations possess localized solutions where the gradient of the solution is large only in the vicinity of certain points or interfaces in the domain. The problems of this type that are considered are an interface propagation model from materials science and an activator-inhibitor model of morphogenesis. These two models are formulated as nonlocal partial differential equations. Results concerning the existence of equilibria, their stability, and the dynamical behavior of localized structures in the interior and on the boundary of the domain are surveyed for these two models. By examining the spectrum associated with the linearization of these problems around certain canonical solutions, it is shown that the nonlocal term can lead to the existence of an exponentially small principal eigenvalue for the linearized problem. This eigenvalue is then responsible for an exponentially slow, or metastable, motion of the localized structure.

scholarly journals Bistable travelling waves for nonlocal reaction diffusion equations

2014 ◽  
Vol 34 (5) ◽  
pp. 1775-1791 ◽  
Author(s):  
Matthieu Alfaro ◽  
◽  
Jérôme Coville ◽  
Gaël Raoul ◽  
◽  
...  
Keyword(s):  
Travelling Waves ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Nonlocal Reaction
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