Global in time unbounded solutions for a non-local thermistor problem

We study the behaviour of solutions of the non-local equation u(t)- u(xx) = lambda f(u)/(integral(1)(-1)f(u)dx)(2), x is an element of (-1,1),t > 0,; with Dirichlet boundary conditions. lambda* is a critical value such that for 0 < lambda < lambda* the corresponding steady-state problem has a unique solution, while for lambda >= lambda* there is no stationary solution. The function f satisfies f > 0; f' < 0 and integral(infinity)(0)f(s)d(s) < infinity. We show, by using comparison methods, that (a) u* (x, t) exists for all t > 0, (b) u* (x,t) -> infinity for every x is an element of (-1,1) as t -> infinity.