EXISTENCE AND ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO SOME INHOMOGENEOUS NONLOCAL DIFFUSION PROBLEMS

Abstract

We consider the nonlocal evolution Dirichlet problem u(t)(x, t) = f(Omega) J(x-y/g(y)) u(y, t)/g(y)(N) dy- u(x, t), x is an element of Omega, t > 0; u = 0, x is an element of R-N\Omega, t >= 0; u(x, 0) = u(0)(x), x is an element of R-N; where Omega is a bounded domain in R-N, J is a Holder continuous, nonnegative, compactly supported function with unit integral and g is an element of C((Omega) over bar) is assumed to be positive in Omega. We discuss existence, uniqueness, and asymptotic behavior of solutions as t -> |infinity. Moreover, we prove the existence of a positive stationary solution when the inequality g(x) <= delta(x) holds at every point of Omega, where delta(x) = dist(x, partial derivative Omega). The behavior of positive stationary solutions near the boundary is also analyzed.