An inhomogeneous nonlocal diffusion problem with unbounded steps

Abstract

We consider the following nonlocal equation

integral J (x-y/g(y)) u(y)/g(y) dy - u(x) = 0 x is an element of R,

where J is an even, compactly supported, Holder continuous kernel with unit integral and g is a continuous positive function. Our main concern will be with unbounded functions g, contrary to previous works. More precisely, we study the influence of the growth of g at infinity on the integrability of positive solutions of this equation, therefore determining the asymptotic behavior as t -> +infinity of the solutions to the associated evolution problem in terms of the growth of g.