Propagation phenomena in monostable integro-differential equations: acceleration or not?

TL;DR

This paper investigates how the tail behavior of the dispersion kernel and the degeneracy of the nonlinearity influence wave existence and propagation speed in monostable integro-differential equations, revealing a dichotomy between acceleration and non-acceleration.

Contribution

It establishes a precise separation criterion for the existence of travelling waves and acceleration phenomena based on kernel decay and nonlinearity degeneracy.

Findings

Algebraic decay kernels determine the existence of travelling waves.

A clear separation between acceleration and non-acceleration is identified.

First estimates of level set positions in accelerating regimes are provided.

Abstract

We consider the homogeneous integro-differential equation$\partial \_t u=J*u-u+f(u)$ with a monostable nonlinearity $f$. Our interest is twofold: we investigate the existence/non existence of travelling waves, and the propagation properties of the Cauchy problem.When the dispersion kernel $J$ is exponentially bounded, travelling waves are known to exist and solutions of the Cauchy problem typically propagate at a constant speed \cite{Schumacher1980}, \cite{Weinberger1982}, \cite{Carr2004}, \cite{Coville2007a}, \cite{Coville2008a}, \cite{Yagisita2009}. %When the dispersion kernel $J$ is exponentially bounded, travelling waves are known to exist when $f$ belongs to one of the three main class of non-linearities (bistable, ignition or monostable), and solutions of the Cauchy problem typically propagate at a constant speed \cite{Schumacher1980}, \cite{Wei-82},\cite{Bates1997},\cite{Chen1997}, \cite{Carr2004}, \cite{Coville2007a}, \cite{Coville2008a}, \cite{Yagisita2009,Yagisita2009a}. On the other hand, when the dispersion kernel $J$ has heavy tails and the non-linearity $f$ is non degenerate, i.e $f'(0)\textgreater{}0$, travelling waves do not exist and solutions of the Cauchy problem propagate by accelerating \cite{Medlock2003}, \cite{Yagisita2009}, \cite{Garnier2011}. For a general monostable non-linearity, a dichotomy between these two types of propagation behaviour is still not known. The originality of our work is to provide such dichotomy by studying the interplay between the tails of the dispersion kernel and the Allee effect induced by the degeneracy of $f$, i.e. $f'(0)=0$. First, for algebraic decaying kernels, we prove the exact separation between existence and non existence of travelling waves. This in turn provides the exact separation between non acceleration and acceleration in the Cauchy problem. In the latter case, we provide a first estimate of the position of the level sets of the solution.