scholarly journals Existence, uniqueness and stability of travelling waves in a discrete reaction–diffusion monostable equation with delay

2005 ◽  
Vol 217 (1) ◽  
pp. 54-87 ◽  
Author(s):  
Shiwang Ma ◽  
Xingfu Zou
Keyword(s):  
Travelling Waves ◽  
Reaction Diffusion ◽  
Monostable Equation ◽  
Equation With Delay

The development of travelling waves in a simple isothermal chemical system III. Cubic and mixed autocatalysis

1990 ◽  
Vol 430 (1880) ◽  
pp. 509-524 ◽  
Keyword(s):  
Travelling Waves ◽  
Reaction Diffusion ◽  
Chemical System ◽  
Mixed System ◽  
Dimensional Parameter ◽  
Diffusion Wave ◽  
Symmetrical Form ◽  
Asymptotic Speed ◽  
Mixed Systems ◽  
Cubic Autocatalysis

Autocatalytic chemical reactions can support isothermal travelling waves of constant speed and form. This paper extends previous studies to cubic autocatalysis and to mixed systems where quadratic and cubic autocatalyses occur concurrently. A + B → 2B, rate = k q ab , (1) A + 2B → 3B, rate = k c ab 2 . (2) For pure cubic autocatalysis the wave has, at large times, a constant asymptotic speed v 0 (where v 0 = 1/√2 in the appropriate dimensionless units). This result is confirmed by numerical investigation of the initial-value problem. Perturbations to this stable wave-speed decay at long times as t -3/2 e -1/8 t . The mixed system is governed by a non-dimensional parameter μ = k q / k c a 0 which measures the relative rates of transformation by quadratic and cubic modes. In the mixed case ( μ ≠ 0) the reaction-diffusion wave has a form appropriate to a purely cubic autocatalysis so long as μ lies between ½ and 0. When μ exceeds ½, the reaction wave loses its symmetrical form, and all its properties steadily approach those of quadratic autocatalysis. The value μ = ½ is the value at which rates of conversion by the two paths are equal.

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.

scholarly journals FKPP equation with impulses on unbounded domain

2017 ◽  
Vol 1 ◽  
pp. 1 ◽  
Author(s):  
Valaire Yatat ◽  
Yves Dumont
Keyword(s):  
Travelling Waves ◽  
Reaction Diffusion ◽  
Bare Soil ◽  
Travelling Wave ◽  
Reaction Diffusion Equations ◽  
Spreading Speed ◽  
Systems Dynamics ◽  
Travelling Wave Solutions ◽  
Minimal Speed ◽  
Wave Solutions

This paper deals with the problem of travelling wave solutions in a scalar impulsive FKPP-like equation. It is a first step of a more general study that aims to address existence of travelling wave solutions for systems of impulsive reaction-diffusion equations that model ecological systems dynamics such as fire-prone savannas. Using results on scalar recursion equations, we show existence of populated vs. extinction travelling waves invasion and compute an explicit expression of their spreading speed (characterized as the minimal speed of such travelling waves). In particular, we find that the spreading speed explicitly depends on the time between two successive impulses. In addition, we carry out a comparison with the case of time-continuous events. We also show that depending on the time between two successive impulses, the spreading speed with pulse events could be lower, equal or greater than the spreading speed in the case of time-continuous events. Finally, we apply our results to a model of fire-prone grasslands and show that pulse fires event may slow down the grassland vs. bare soil invasion speed.

The evolution of travelling waves in reaction-diffusion equations with monotone decreasing diffusivity. II. Abruptly vanishing diffusivity

1995 ◽  
Vol 350 (1694) ◽  
pp. 361-378 ◽  
Keyword(s):  
Travelling Waves ◽  
Reaction Diffusion ◽  
Travelling Wave ◽  
Reaction Diffusion Equations ◽  
Reaction Diffusion Equation ◽  
Wave Form ◽  
Initial Development ◽  
Finite Concentration ◽  
Distinct Process ◽  
And Diffusion

In this paper we continue our study of some of the qualitative features of chemical polymerization processes by considering a reaction-diffusion equation for the chemical concentration in which the diffusivity vanishes abruptly at a finite concentration. The effect of this diffusivity cut-off is to create two distinct process zones; in one there is both reaction and diffusion and in the other there is only reaction. These zones are separated by an interface across which there is a jump in concentration gradient. Our analysis is focused on both the initial development of this interface and the large time evolution of the system into a travelling wave form. Some distinct differences from our previous analysis of smoothly vanishing diffusivity are found.

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