scholarly journals Traveling Wavefronts for Delayed Reaction-Diffusion Systems via a Fixed Point Theorem

2001 ◽  
Vol 171 (2) ◽  
pp. 294-314 ◽  
Author(s):  
Shiwang Ma
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Reaction Diffusion ◽  
Delayed Reaction ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems

scholarly journals Travelling fronts in time-delayed reaction–diffusion systems

2019 ◽  
Vol 377 (2153) ◽  
pp. 20180127 ◽  
Author(s):  
Jan Rombouts ◽  
Lendert Gelens ◽  
Thomas Erneux
Keyword(s):  
Reaction Diffusion ◽  
Delayed Feedback ◽  
Delay Systems ◽  
Wave Shape ◽  
Delayed Reaction ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Speed Up ◽  
Travelling Fronts ◽  
Time Delayed Feedback

We review a series of key travelling front problems in reaction–diffusion systems with a time-delayed feedback, appearing in ecology, nonlinear optics and neurobiology. For each problem, we determine asymptotic approximations for the wave shape and its speed. Particular attention is devoted to their validity and all analytical solutions are compared to solutions obtained numerically. We also extend the work by Erneux et al. (Erneux et al. 2010 Phil. Trans. R. Soc. A 368 , 483–493 ( doi:10.1098/rsta.2009.0228 )) by considering the case of a slowly propagating front subject to a weak delayed feedback. The delay may either speed up the front in the same direction or reverse its direction. This article is part of the theme issue ‘Nonlinear dynamics of delay systems’.

scholarly journals An Algebraic Method on the Eigenvalues and Stability of Delayed Reaction-Diffusion Systems

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Jian Ma ◽  
Baodong Zheng
Keyword(s):  
Asymptotic Stability ◽  
Linear Operator ◽  
Reaction Diffusion ◽  
Matrix Pencil ◽  
Algebraic Method ◽  
Delayed Reaction ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
The Matrix ◽  
Pure Imaginary Eigenvalues

The eigenvalues and stability of the delayed reaction-diffusion systems are considered using the algebraic methods. Firstly, new algebraic criteria to determine the pure imaginary eigenvalues are derived by applying the matrix pencil and the linear operator methods. Secondly, a practical checkable criteria for the asymptotic stability are introduced.

scholarly journals A Class of Reaction-Diffusion Systems with Nonlocal Initial Conditions

2015 ◽  
Vol 61 (1) ◽  
pp. 59-78 ◽  
Author(s):  
Monica-Dana Burlică ◽  
Daniela Roşu
Keyword(s):  
Fixed Point ◽  
Initial Conditions ◽  
Sufficient Conditions ◽  
Reaction Diffusion ◽  
Diffusion System ◽  
Reaction Diffusion Systems ◽  
Nonlocal Initial Conditions ◽  
Diffusion Systems ◽  
Compactness Arguments ◽  
Fixed Point Techniques

Abstract We consider an abstract nonlinear multi-valued reaction-diffusion system with delay and, using some compactness arguments coupled with metric fixed point techniques, we prove some sufficient conditions for the existence of at least one C0-solution.

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