Uniform blow-up profile and boundary behaviour for a non-local reaction–diffusion equation with critical damping

2004 ◽  
Vol 27 (15) ◽  
pp. 1819-1829 ◽  
Author(s):  
Philippe Souplet
Keyword(s):  
Diffusion Equation ◽  
Blow Up ◽  
Local Reaction ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Boundary Behaviour ◽  
Non Local ◽  
Critical Damping

scholarly journals Traveling Waves Solutions for Delayed Temporally Discrete Non-Local Reaction-Diffusion Equation

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1999
Author(s):  
Hongpeng Guo ◽  
Zhiming Guo
Keyword(s):  
Diffusion Equation ◽  
Traveling Wave ◽  
Local Reaction ◽  
Traveling Wave Solutions ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Birth Function ◽  
Wave Solutions ◽  
Monotonicity Assumption ◽  
Non Local

This paper deals with the existence of traveling wave solutions to a delayed temporally discrete non-local reaction diffusion equation model, which has been derived recently for a single species with age structure. When the birth function satisfies monotonic condition, we obtained the traveling wavefront by using upper and lower solution methods together with monotonic iteration techniques. Otherwise, without the monotonicity assumption for birth function, we constructed two auxiliary equations. By means of the traveling wavefronts of the auxiliary equations, using the Schauder’ fixed point theorem, we proved the existence of a traveling wave solution to the equation under consideration with speed c>c*, where c*>0 is some constant. We found that the delayed temporally discrete non-local reaction diffusion equation possesses the dynamical consistency with its time continuous counterpart at least in the sense of the existence of traveling wave solutions.

Dynamics of a non-local delayed reaction–diffusion equation without quasi-monotonicity

2010 ◽  
Vol 140 (5) ◽  
pp. 1081-1109 ◽  
Author(s):  
Zhi-Cheng Wang ◽  
Wan-Tong Li
Keyword(s):  
Diffusion Equation ◽  
Global Attractivity ◽  
Reaction Diffusion ◽  
Single Species ◽  
Travelling Wave ◽  
Reaction Diffusion Equation ◽  
Positive Equilibrium ◽  
Cauchy Type ◽  
Delayed Reaction ◽  
Non Local

AbstractThis paper is concerned with the dynamics of a non-local delayed reaction–diffusion equation without quasi-monotonicity on an infinite n-dimensional domain, which can be derived from the growth of a stage-structured single-species population. We first prove that solutions of the Cauchy-type problem are positively preserving and bounded if the initial value is non-negative and bounded. Then, by establishing a comparison theorem and a series of comparison arguments, we prove the global attractivity of the positive equilibrium. When there exist no positive equilibria, we prove that the zero equilibrium is globally attractive. In particular, these results are still valid for the non-local delayed reaction–diffusion equation on a bounded domain with the Neumann boundary condition. Finally, we establish the existence of new entire solutions by using the travelling-wave solutions of two auxiliary equations and the global attractivity of the positive equilibrium.

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