scholarly journals Exponential stability of travelling waves for a general reaction-diffusion equation with spatio-temporal delays

ScienceAsia ◽  
2018 ◽  
Vol 44 (6) ◽  
pp. 421
Author(s):  
Rui Yan ◽  
Guirong Liu
Keyword(s):  
Diffusion Equation ◽  
Exponential Stability ◽  
Travelling Waves ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
General Reaction ◽  
Spatio Temporal

Existence of bistable waves for a nonlocal and nonmonotone reaction-diffusion equation

2019 ◽  
Vol 150 (2) ◽  
pp. 721-739
Author(s):  
Sergei Trofimchuk ◽  
Vitaly Volpert
Keyword(s):  
Diffusion Equation ◽  
A Priori Estimates ◽  
Travelling Waves ◽  
A Priori ◽  
Unbounded Domains ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Nonlocal Nonlinearity ◽  
Estimates Of Solutions ◽  
Priori Estimates

AbstractReaction-diffusion equation with a bistable nonlocal nonlinearity is considered in the case where the reaction term is not quasi-monotone. For this equation, the existence of travelling waves is proved by the Leray-Schauder method based on the topological degree for elliptic operators in unbounded domains and a priori estimates of solutions in properly chosen weighted spaces.

Emergence of Waves in a Nonlinear Convection-Reaction-Diffusion Equation

2004 ◽  
Vol 4 (3) ◽  
Author(s):  
Shoshana Kamin ◽  
Philip Rosenau
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Diffusion Equation ◽  
Travelling Waves ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Nonlinear Convection ◽  
The Cauchy Problem ◽  
The Right

AbstractIn this work we prove that for some class of initial data the solution of the Cauchy problemuu(0; x) = uapproaches the travelling solution, spreading either to the right or to the left, or two travelling waves moving in opposite directions.

scholarly journals Amplitude Equations and Chemical Reaction–Diffusion Systems

1997 ◽  
Vol 07 (07) ◽  
pp. 1539-1554 ◽  
Author(s):  
M. Ipsen ◽  
F. Hynne ◽  
P. G. Sørensen
Keyword(s):  
Diffusion Equation ◽  
Chemical Reaction ◽  
Bifurcation Point ◽  
Landau Equation ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Amplitude Equations ◽  
Ginzburg Landau ◽  
Ginzburg Landau Equation ◽  
Spatio Temporal

The paper discusses the use of amplitude equations to describe the spatio-temporal dynamics of a chemical reaction–diffusion system based on an Oregonator model of the Belousov–Zhabotinsky reaction. Sufficiently close to a supercritical Hopf bifurcation the reaction–diffusion equation can be approximated by a complex Ginzburg–Landau equation with parameters determined by the original equation at the point of operation considered. We illustrate the validity of this reduction by comparing numerical spiral wave solutions to the Oregonator reaction–diffusion equation with the corresponding solutions to the complex Ginzburg–Landau equation at finite distances from the bifurcation point. We also compare the solutions at a bifurcation point where the systems develop spatio-temporal chaos. We show that the complex Ginzburg–Landau equation represents the dynamical behavior of the reaction–diffusion equation remarkably well, sufficiently far from the bifurcation point for experimental applications to be feasible.

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