Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Travelling waves in non-local reaction-dispersion equations with diffuse levy measures

Emeric Bouin, Jérôme Coville

TL;DR

The paper develops a unified framework for the existence of traveling fronts in nonlocal reaction-dispersion equations with diffuse Levy measures, covering bistable, ignition, and monostable nonlinearities. By approximating the Levy measure with bounded supports, establishing robust a priori estimates, and proving a key convergence lemma, it constructs fronts with positive, often unique speeds and derives regularity and monotonicity properties of the profiles. The approach encompasses fractional Laplacian and convolution-type operators, yielding fronts in broad nonlocal settings and clarifying how tail behavior of the measure and nonlinearities influence existence and propagation speed. The results provide a rigorous foundation for front propagation in diverse applications, including phase transitions, population dynamics, and neural/spread models, and offer a systematic method to obtain fronts across a wide class of nonlocal operators.

Abstract

In this paper, we focus on the existence of propagation fronts, solutions to non-local dispersion reaction models. Our aim is to provide a unified proof of this existence in a very broad framework using simple real analysis tools. In particular, we review the results that already exist in the literature and complete the table. It appears that the most important case is that of a bistable nonlinearity, which we then extend to other classical nonlinearities.

Travelling waves in non-local reaction-dispersion equations with diffuse levy measures

TL;DR

The paper develops a unified framework for the existence of traveling fronts in nonlocal reaction-dispersion equations with diffuse Levy measures, covering bistable, ignition, and monostable nonlinearities. By approximating the Levy measure with bounded supports, establishing robust a priori estimates, and proving a key convergence lemma, it constructs fronts with positive, often unique speeds and derives regularity and monotonicity properties of the profiles. The approach encompasses fractional Laplacian and convolution-type operators, yielding fronts in broad nonlocal settings and clarifying how tail behavior of the measure and nonlinearities influence existence and propagation speed. The results provide a rigorous foundation for front propagation in diverse applications, including phase transitions, population dynamics, and neural/spread models, and offer a systematic method to obtain fronts across a wide class of nonlocal operators.

Abstract

In this paper, we focus on the existence of propagation fronts, solutions to non-local dispersion reaction models. Our aim is to provide a unified proof of this existence in a very broad framework using simple real analysis tools. In particular, we review the results that already exist in the literature and complete the table. It appears that the most important case is that of a bistable nonlinearity, which we then extend to other classical nonlinearities.

Paper Structure

This paper contains 17 sections, 22 theorems, 219 equations, 2 figures.

Table of Contents

  1. Introduction and main results
  2. Estimates and useful formulas
  3. A priori estimates
  4. An integration by parts formula
  5. A key lemma
  6. Proof of \ref{['lem:v-unif-int']}
  7. Bistable type nonlinearities - Proof of \ref{['thm:bi']}
  8. A priori regularity
  9. Upper bound on the speed $c_{\varepsilon}$
  10. A positive uniform lower bound for $c_{\varepsilon}$
  11. Construction of the front
  12. Uniqueness of the speed
  13. Ignition type non-linearities - Proof of \ref{['thm:ign']}
  14. Monostable type nonlinearities - Proof of \ref{['thm:mono']}
  15. Mathematical toolbox
  16. ...and 2 more sections

Key Result

Theorem 1.1

Assume that $f$ is of type eq:bistable and that it is not well balanced, with Then, there exists $(c,u) \in {\mathbb{R}}_+^\star \times \mathsf{L}^{\infty}({\mathbb{R}})$ satisfying eq:TW. There, $c$ is positive and unique, $u$ is monotone decreasing. If $\mu$ is unbounded, then $u$ is at least continuous.

Figures (2)

  • Figure 1: Spreading in \ref{['eq:main']} with a nonlinearity of type \ref{['eq:bistable']}.
  • Figure 2: (Rates of) propagation when $\mu(z)=J(z)\,dz$ with $J\in L^1({\mathbb{R}})$ or $J \propto |\cdot|^{-(1+2s)}$.

Theorems & Definitions (32)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 2.1
  • proof : Proof of \ref{['prop:esti']}
  • Lemma 2.1
  • proof : Proof of \ref{['lem:iden']}
  • Lemma 3.1
  • Lemma 3.2
  • Theorem 3.1: Vitali's Theorem
  • ...and 22 more