Transition fronts and stretching phenomena for a general class of reaction-dispersion equations

Jimmy Garnier; FranÇois Hamel; Lionel Roques

doi:10.3934/dcds.2017031

Article Contents

2017, Volume 37, Issue 2: 743-756. Doi: 10.3934/dcds.2017031

This issue Previous Article On the asymptotic behavior of symmetric solutions of the Allen-Cahn equation in unbounded domains in $\mathbb{R}^2$ Next Article Stable patterns with jump discontinuity in systems with Turing instability and hysteresis

Transition fronts and stretching phenomena for a general class of reaction-dispersion equations

1.
Université Savoie Mont-Blanc, LAMA, F-73000 Chambéry, France
2.
CNRS, LAMA, F-73000 Chambéry, France
3.
Aix Marseille Univ, CNRS, Centrale Marseille, I2M, Marseille, France
4.
BioSP, INRA, 84914, Avignon, France

^* Corresponding author: François Hamel
^* Corresponding author: François Hamel

Received: March 31, 2015

Revised: August 31, 2015

Published: May 2017

This work has been carried out in the framework of Archimède LabEx (ANR-11-LABX-0033) and of the A*MIDEX project (ANR-11-IDEX-0001-02), funded by the "Investissements d'Avenir" French Government program managed by the French National Research Agency (ANR). The research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP/2007-2013)/ERC Grant Agreement n.321186 -ReaDi -Reaction-Diffusion Equations, Propagation and Modelling, and from the ANR project NONLOCAL (ANR-14-CE25-0013).

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Abstract

We consider a general form of reaction-dispersion equations with non-local or nonlinear dispersal operators and local reaction terms. Under some general conditions, we prove the non-existence of transition fronts, as well as some stretching properties at large time for the solutions of the Cauchy problem. These conditions are satisfied in particular when the reaction is monostable and when the dispersal operator is either the fractional Laplacian, a convolution operator with a fat-tailed kernel or a nonlinear fast diffusion operator.

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Mathematics Subject Classification: Primary:35B40, 35C07, 35K57, 45K05, 47G20;Secondary:35B08.

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Figure 1. Propagation to the right of the solution of the Cauchy problem (2), at successive times $t=0,1,\ldots,20,$ with (a) $\mathcal D u = -(-\Delta_x)^\alpha u$ with $\alpha=0.9$; (b) $\mathcal D u = J*u-u$ with $J(x)=\exp(-\sqrt{|x|})/4$; (c) $\mathcal D u=(u^\gamma)_{xx}$ with $\gamma=1/2$; and (d) $\mathcal D u = u_{xx}$. In all cases, the initial condition was $u_0(x)=\exp(-x^2/100)$ and the function $f$ was of the KPP type $f(u)=u\, (1-u)$.

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Figure 2. Distance $x_{0.4}(t)-x_{0.6}(t)$ between two level sets of the solution of the Cauchy problem (2), for $t\in(0,20)$, with: (solid line) $\mathcal D u = -(-\Delta_x)^\alpha u$ with $\alpha=0.9$; (dashed line) $\mathcal D u = J*u-u$ with $J(x)=\exp(-\sqrt{|x|})/4$; (dash-dot line) $\mathcal D u=(u^\gamma)_{xx}$ with $\gamma=1/2$; and (circles) $\mathcal D u = u_{xx}$. The assumptions on $f$ and $u_0$ are the same as in Fig. 1

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