Some interesting special cases of a non-local problem modelling ohmic heating with variable thermal conductivity

The non-local equation u(t) = (u(3)u(x))x + lambdaf(u)/(integral (1/)(-1)f(u)dx)(2) is considered, subject to some initial and Dirichlet boundary conditions. H ere f is taken to be either exp(-s(4)) or H(1 - s) with H the Heaviside fun ction, which are both decreasing. It is found that there exists a critical value lambda* = 2, so that for lambda > lambda* there is no stationary solu tion and u 'blows up' (in some sense). If 0 < lambda < lambda*, there is a unique stationary solution which is asymptotically stable and the solution of the IBVP is global in time.