Blow-up time for a nonlocal diffusion problem with dirichlet boundary conditions

This paper concerns the study of the following initial-boundary value problem {u(t) = epsilon(J * u - u) + f(u) in Omega x (0, T), u = 0 in (R-N - Omega) x (0, T), u(x, 0) = u(0) > 0 in Omega, where Omega is a bounded domain in R-N with smooth boundary partial derivative Omega, J * u(x, t) = integral(RN) J(x - y)u(y, t)dy, J: R-N -> R is nonnegative, symmetric (J(z) = J(-z)), bounded and integral(RN) J(z)dz = 1, f(s) is positive, increasing, convex function for positive values of s and integral(infinity) ds/f(s) < infinity. The initial data u(0) is an element of C-1(<(Omega)over bar>). We show that if epsilon is small enough, the solution of the above problem blows up in a finite time and its blow-up time goes to the one of the solution of the following differential equation integral alpha'(t) = f(alpha(t)), t>0, alpha(0) = M, as epsilon goes to zero, where M = sup(x is an element of Omega) u(0)(x). Finally, we give some numerical results to illustrate our analysis.