Existence and Uniqueness of Solutions to a Nonlocal Equation with Monostable Nonlinearity

TL;DR

This paper investigates the existence and uniqueness of solutions to a nonlocal equation with monostable nonlinearity, establishing conditions based on the principal eigenvalue and symmetry of the kernel.

Contribution

It provides a rigorous analysis linking the principal eigenvalue to solution existence and proves uniqueness under symmetry assumptions.

Findings

Nontrivial solutions exist if and only if the principal eigenvalue is negative.

Uniqueness of solutions is guaranteed when the kernel is symmetric.

The principal eigenvalue is well-defined for the linearized operator.

Abstract

Let $J \in C(\mathbb{R})$, $J\ge 0$, $\int_{\tiny$\mathbb{R}$} J = 1$ and consider the nonlocal diffusion operator $\mathcal{M}[u] = J \star u - u$. We study the equation $\mathcal{M} u + f(x,u) = 0$, $u \ge 0$, in $\mathbb{R}$, where $f$ is a KPP-type nonlinearity, periodic in $x$. We show that the principal eigenvalue of the linearization around zero is well defined and that a nontrivial solution of the nonlinear problem exists if and only if this eigenvalue is negative. We prove that if, additionally, $J$ is symmetric, then the nontrivial solution is unique.