[Paper Review] Doubly nonlocal Fisher-KPP equation: Existence and properties of traveling waves

Dmitri Finkelshtein, Yuri Kondratiev+1 more|arXiv (Cornell University)|Apr 26, 2018

Mathematical and Theoretical Epidemiology and Ecology ModelsMedicine3 references4 citations

TL;DR

This paper establishes the existence and properties of monotone traveling wave solutions for a doubly nonlocal Fisher-KPP equation with anisotropic, exponentially integrable diffusion kernels and a combination of local and nonlocal monostable reactions. Using semiflow analysis and integral estimates, it proves the existence of traveling waves with continuous, strictly decreasing profiles that decay exponentially, under conditions ensuring nonlocal diffusion dominance in specific directions.

ABSTRACT

We consider a reaction-diffusion equation with nonlocal anisotropic diffusion and a linear combination of local and nonlocal monostable-type reactions in a space of bounded functions on $\mathbb{R}^d$. Using the properties of the corresponding semiflow, we prove the existence of monotone traveling waves along those directions where the diffusion kernel is exponentially integrable. Among other properties, we prove continuity, strict monotonicity and exponential integrability of the traveling wave profiles.

Motivation & Objective

• To investigate the existence of monotone traveling wave solutions in a reaction-diffusion equation with both nonlocal diffusion and nonlocal reaction terms.
• To analyze the properties of traveling wave profiles, including continuity, strict monotonicity, and exponential integrability.
• To determine under which conditions on the diffusion kernel (specifically, exponential integrability) traveling waves exist in particular spatial directions.
• To unify the analysis of local and nonlocal competition terms in the reaction term, extending previous results on purely local or purely nonlocal cases.
• To establish the existence of traveling waves with specific decay and regularity properties in the context of nonlocal population dynamics.

Proposed method

• Formulates a reaction-diffusion equation with nonlocal diffusion (via convolution with kernel $a^+$) and a reaction term combining local and nonlocal components ($Gu = \varkappa_\ell u + \varkappa_{nl}(a^- * u)$).
• Analyzes the semiflow generated by the equation in the space of bounded, nonnegative functions on $\mathbb{R}^d$.
• Applies integral estimates and comparison arguments to derive bounds on the derivative of the wave profile, using exponential weights to control decay behavior.
• Uses a transformation $w(s) = \psi(s)e^{-\mu s}$ to convert differential inequalities into integral inequalities and derive exponential decay estimates.
• Employs the condition that the diffusion kernel $a^+$ is exponentially integrable in the direction of propagation to ensure existence of waves with specific decay rates.
• Combines a priori estimates and monotonicity arguments to prove continuity, strict decrease, and exponential integrability of the wave profile $\psi$.

Experimental results

Research questions

• RQ1 Under what conditions on the diffusion kernel $a^+$ does a monotone traveling wave solution exist in a given direction $\xi \in S^{d-1}$?
• RQ2 What are the regularity and decay properties of the traveling wave profile $\psi$ connecting the unstable zero state to the stable positive equilibrium $\theta$?
• RQ3 How does the interplay between local and nonlocal reaction terms affect the existence and shape of traveling waves?
• RQ4 Can the wave profile be shown to be strictly decreasing and exponentially integrable under mild integrability assumptions on the kernels?
• RQ5 What role does the exponential integrability of the diffusion kernel play in ensuring the existence of traveling waves?

Key findings

• Traveling wave solutions exist for all directions $\xi \in S^{d-1}$ where the diffusion kernel $a^+$ is exponentially integrable.
• The wave profile $\psi$ is continuous, strictly decreasing, and satisfies $\psi(-\infty) = \theta$, $\psi(+\infty) = 0$, with $\theta = (\varkappa^+ - m)/\varkappa^-$ being the positive equilibrium.
• The profile $\psi$ is exponentially integrable, i.e., $\int_\mathbb{R} \psi(s) e^{\nu s} ds < \infty$ for some $\nu > 0$, ensuring fast decay at infinity.
• The derivative of $\psi$ satisfies $\psi'(s)/\psi(s) > -\nu$ for all $s \in \mathbb{R}$, confirming exponential decay behavior.
• The existence proof relies on a priori estimates and comparison techniques applied to the transformed function $w(s) = \psi(s)e^{-\mu s}$, which leads to a contradiction if exponential decay fails.
• The wave speed $c$ must be positive; no traveling wave exists for $c < 0$ under the given assumptions, due to the monotonicity and decay constraints.