Nonlinear Nonlocal Reaction-Diffusion Equations

Let Omega subset of R-N, and J be a nonnegative function defined in Omega x Omega. We consider the problem {u(t)(x,t) = integral(Omega) J(x, y)u(y,t)dy - h(x)u(x, t) + f(x, u(x, t)), x is an element of Omega, t> 0 u(x, 0) = u(0)(x), x is an element of Omega, (1) with h is an element of L-infinity(Omega), u(0) is an element of L-p(Omega) and the function f defined as f : Omega x R -> R, that maps (x, s) into f(x, s). We assume f globally Lipschitz or f locally Lipschitz in the variable s 2 R, uniformly with respect to x is an element of Omega, and f satisfies that there exist C is an element of R and D >= 0 such that f(., s)s <= Cs-2 + D vertical bar s vertical bar, for all s is an element of R. The aim is to study the existence and uniqueness and we give some asymptotic estimates of the norm L-infinity(Omega) of the solution u of the problem (1), following the ideas of [2], and we prove the existence of two ordered extremal equilibria, like in [6], which give some information about the set that attracts the dynamics of the solution of (1), for all u(0) in L-infinity(Omega).