Abstract

We consider the nonlocal evolution Dirichlet problem ut(x, t) = ∫?(x-y/g(y)) u(y,t)/g(y)N-u(x,t), x ?, t> 0; u = 0, x ?&Rdbl;N\?, t > 0; u(x, 0) = u0(x), x ?&Rdbl;N; where ? is a bounded domain in RN, J is a Hölder continuous, nonnegative, compactly supported function with unit integral and g ? C(¯) is assumed to be positive in ?. We discuss existence, uniqueness, and asymptotic behavior of solutions as t ? +8. Moreover, we prove the existence of a positive stationary solution when the inequality g(x) = d(x) holds at every point of ?, where d(x)= dist(x, ?). The behavior of positive stationary solutions near the boundary is also analyzed. © 2009 Society for Industrial and Applied Mathematics.