Finite mass solutions for a nonlocal inhomogeneous dispersal equation

Abstract

In this paper we study the asymptotic behavior of the following nonlocal inhomogeneous dispersal equation

u(t)(x, t) = integral(R) J (x-y/g(y)) u(y, t)/g(y)dy - u(x, t) x is an element of 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(vertical bar y vertical bar) similar to a vertical bar y vertical bar as vertical bar y vertical bar -> +infinity for some 0 < a < 1, there exists a unique (up to normalization) positive stationary solution, which is in L-1(R). On the other hand, if g(vertical bar y vertical bar) similar to vertical bar y vertical bar(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.