On a long range segregation model

In this work we study the properties of segregation processes modeled by a family of equations L(u(i)) (x) = u(i) (x) F-i (u(1),..., u(K)) (x), i = 1,..., K, where Fi(u(1),..., u(K)) (x) is a non-local factor that takes into consideration the values of the functions u(j) in a full neighborhood of x. We consider as a model problem Delta u(i)(epsilon) (x) = 1/epsilon(2)u(i)(epsilon) (x) Sigma(i not equal j) H(u(j)(epsilon)) (x) where epsilon is a small parameter and H(u(j)(epsilon)) (x) is for instance H(u(j)(epsilon)) (x) = integral(B1(x)) u(j)(epsilon)(y) dy or H(u(j)(epsilon)) (x) = sup (y is an element of B1(x)) u(j)(epsilon) (y). Here B-1(x) is the unit ball centered at x with respect to a smooth, uniformly convex norm rho in R-n. Heuristically, this will force the populations to stay at rho-distance 1 from each other as epsilon -> 0.