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

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.

Cited by in corpus (23)

• On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators
• Nonlocal anisotropic dispersal with monostable nonlinearity
• Pulsating fronts for nonlocal dispersion and KPP nonlinearity
• Traveling Wave Solutions of Spatially Periodic Nonlocal Monostable Equations
• Harnack type inequality for positive solution of some integral equation
• Liouville Type Property and Spreading Speeds of KPP Equations in Periodic Media with Localized Spatial Inhomogeneity
• Traveling waves and long-time behavior in a doubly nonlocal Fisher-KPP equation
• Semi-wave, traveling wave and spreading speed for monostable cooperative systems with nonlocal diffusion and free boundaries
• Spreading Speeds and Traveling Waves of Nonlocal Monostable Equations in Time and Space Periodic Habitats
• On Principal Spectrum Points/Principal Eigenvalues of Nonlocal Dispersal Operators and Applications
• Positive Stationary Solutions and Spreading Speeds of KPP Equations in Locally Spatially Inhomogeneous Media
• Can a population survive in a shifting environment using non-local dispersion
• Spreading Speeds and Linear Determinacy for Two Species Competition Systems with Nonlocal Dispersal in Periodic Habitats
• A Lotka-Volterra competition model with nonlocal diffusion and free boundaries
• Persistence and time periodic positive solutions of doubly nonlocal Fisher-KPP equations in time periodic and space heterogeneous media
• Persistence criteria for populations with non-local dispersion
• Nonlinear nonlocal reaction-diffusion problem with local reaction
• Spectral theory for nonlocal dispersal operators with time periodic indefinite weight functions and applications
• Approximations of Random Dispersal Operators/Equations by Nonlocal Dispersal Operators/Equations
• Criteria for the Existence of Principal Eigenvalues of Time Periodic Cooperative Linear Systems with Nonlocal Dispersal
• A Nonlocal Dispersal SIS Epidemic Model in Heterogeneous Environment
• Nonlocal dispersal equations in time-periodic media: principal spectral theory, bifurcation and asymptotic behaviors
• Nonlocal dispersal equations with almost periodic dependence. I. Principal spectral theory