# American Institute of Mathematical Sciences

April  2015, 35(4): 1409-1419. doi: 10.3934/dcds.2015.35.1409

## Finite mass solutions for a nonlocal inhomogeneous dispersal equation

 1 Departamento de Matemáticas, Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Casilla 306, Correo 22. Santiago, Chile 2 Departamento de Matemáticas, Pontificia Universidad Católica de Chile, Santiago 3 Departamento de Análisis Matemático, Universidad de La Laguna, C/. Astrofísico Francisco Sánchez s/n, 38271. La Laguna, Spain 4 Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático, Universidad de Chile, silla 170 Correo 3, Santiago

Received  October 2013 Revised  March 2014 Published  November 2014

In this paper we study the asymptotic behavior of the following nonlocal inhomogeneous dispersal equation $$u_t(x,t) = \int_{\mathbb{R}} J\left(\frac{x-y}{g(y)}\right) \frac{u(y,t)}{g(y)} dy -u(x,t) \qquad x\in \mathbb{R},\ t>0,$$ where $J$ is an even, smooth, probability density, and $g$, which accounts for a dispersal distance, is continuous and positive. We prove that if $g(|y|)\sim a |y|$ as $|y|\to + \infty$ for some $0 < a < 1$, there exists a unique (up to normalization) positive stationary solution, which is in $L^1(\mathbb{R})$. On the other hand, if $g(|y|)\sim |y|^p$, with $p > 2$ there are no positive stationary solutions. We also establish the asymptotic behavior of the solutions of the evolution problem in both cases.
Citation: Carmen Cortázar, Manuel Elgueta, Jorge García-Melián, Salomé Martínez. Finite mass solutions for a nonlocal inhomogeneous dispersal equation. Discrete & Continuous Dynamical Systems - A, 2015, 35 (4) : 1409-1419. doi: 10.3934/dcds.2015.35.1409
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