scholarly journals Doubly nonlocal reaction–diffusion equations and the emergence of species

2017 ◽  
Vol 42 ◽  
pp. 591-599 ◽  
Author(s):  
M. Banerjee ◽  
V. Vougalter ◽  
V. Volpert
Keyword(s):  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Nonlocal Reaction

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

scholarly journals An iterative method for non-autonomous nonlocal reaction-diffusion equations

2017 ◽  
Vol 2 (1) ◽  
pp. 73-82 ◽  
Author(s):  
Tomás Caraballo ◽  
Marta Herrera-Cobos ◽  
Pedro Marín-Rubio
Keyword(s):  
Auxiliary Problem ◽  
Nonlocal Problem ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Nonlocal Operator ◽  
Diffusion Term ◽  
Linear Diffusion ◽  
Nonlocal Reaction ◽  
Compactness Arguments

AbstractIn this paper we provide a method to prove the existence of weak solutions for a type of non-autonomous nonlocal reaction-diffusion equations. Due to the presence of the nonlocal operator in the diffusion term, we cannot apply the Monotonicity Method directly. To use it, we build an auxiliary problem with linear diffusion and later, through iterations and compactness arguments, we show the existence of solutions for the nonlocal problem.

scholarly journals Preface to the Issue Nonlocal Reaction-Diffusion Equations

2015 ◽  
Vol 10 (6) ◽  
pp. 1-5 ◽  
Author(s):  
M. Alfaro ◽  
N. Apreutesei ◽  
F. Davidson ◽  
V. Volpert
Keyword(s):  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Nonlocal Reaction
Sign in / Sign up

Export Citation Format

Share Document