scholarly journals Hyperbolic traveling waves driven by growth

2014 ◽  
Vol 24 (06) ◽  
pp. 1165-1195 ◽  
Author(s):  
Emeric Bouin ◽  
Vincent Calvez ◽  
Grégoire Nadin
Keyword(s):  
Nonlinear Stability ◽  
Traveling Waves ◽  
Hyperbolic Model ◽  
Kpp Equation ◽  
Transition Parameter ◽  
Minimal Speed ◽  
Traveling Fronts ◽  
Traveling Front ◽  
Maximal Speed ◽  
Existence And Stability

We perform the analysis of a hyperbolic model which is the analog of the Fisher-KPP equation. This model accounts for particles that move at maximal speed ϵ-1 (ϵ > 0), and proliferate according to a reaction term of monostable type. We study the existence and stability of traveling fronts. We exhibit a transition depending on the parameter ϵ: for small ϵ the behavior is essentially the same as for the diffusive Fisher-KPP equation. However, for large ϵ the traveling front with minimal speed is discontinuous and travels at the maximal speed ϵ-1. The traveling fronts with minimal speed are linearly stable in weighted L2 spaces. We also prove local nonlinear stability of the traveling front with minimal speed when ϵ is smaller than the transition parameter.

scholarly journals Analytical and numerical investigation of traveling waves for the Allen–Cahn model with relaxation

2016 ◽  
Vol 26 (05) ◽  
pp. 931-985 ◽  
Author(s):  
Corrado Lattanzio ◽  
Corrado Mascia ◽  
Ramon G. Plaza ◽  
Chiara Simeoni
Keyword(s):  
Numerical Investigation ◽  
Nonlinear Stability ◽  
Traveling Waves ◽  
Propagation Speed ◽  
Stable States ◽  
Cahn Equation ◽  
Numerical Studies ◽  
Traveling Fronts ◽  
Parabolic Case ◽  
Existence And Stability

A modification of the parabolic Allen–Cahn equation, determined by the substitution of Fick’s diffusion law with a relaxation relation of Cattaneo–Maxwell type, is considered. The analysis concentrates on traveling fronts connecting the two stable states of the model, investigating both the aspects of existence and stability. The main contribution is the proof of the nonlinear stability of the wave, as a consequence of detailed spectral and linearized analyses. In addition, numerical studies are performed in order to determine the propagation speed, to compare it to the speed for the parabolic case, and to explore the dynamics of large perturbations of the front.

Existence and stability of invasion traveling waves for a competition system with random vs. nonlocal dispersals

2019 ◽  
Vol 12 (01) ◽  
pp. 1950004
Author(s):  
Jiao Wang ◽  
Zhixian Yu ◽  
Yanling Meng
Keyword(s):  
Traveling Waves ◽  
Weighted Spaces ◽  
Nonlocal Dispersal ◽  
Asymptotic Behaviors ◽  
Competition System ◽  
Weighted Energy ◽  
Existence And Stability ◽  
Exponentially Stable ◽  
Exponentially Weighted

The purpose of this paper is to investigate asymptotic behaviors of the solutions for a competition system with random vs. nonlocal dispersal. We first prove the existence of invasion traveling waves via using the theory of asymptotic speeds of spread. Then we prove the invasion traveling waves are exponentially stable as perturbation in some exponentially weighted spaces by using the weighted energy and the squeezing technique.

EVANS FUNCTIONS AND NONLINEAR STABILITY OF TRAVELING WAVES IN NEURONAL NETWORK MODELS

2007 ◽  
Vol 17 (08) ◽  
pp. 2693-2704 ◽  
Author(s):  
BJÖRN SANDSTEDE
Keyword(s):  
Differential Equations ◽  
Neuronal Network ◽  
Nonlinear Stability ◽  
Traveling Waves ◽  
Network Models ◽  
Function Spaces ◽  
Spectral Stability ◽  
Evans Function ◽  
Evans Functions

Modeling networks of synaptically coupled neurons often leads to systems of integro-differential equations. Particularly interesting solutions in this context are traveling waves. We prove here that spectral stability of traveling waves implies their nonlinear stability in appropriate function spaces, and compare several recent Evans-function constructions that are useful tools when analyzing spectral stability.

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