Global existence and divergence of critical solutions of a non-local parabolic problem in Ohmic heating process

We investigate the behaviour of some critical solutions of a non-local initial-boundary value problem for the equation u _t=?u+?f(u)/(?Of(u)dx) ²,O?R^N,N=1,2. Under specific conditions on f, there exists a ?^* such that for each 0<?<?^* there corresponds a unique steady-state solution and u=u(x,t;?) is a global in time-bounded solution, which tends to the unique steady-state solution as t?8 uniformly in x. Whereas for ?=?^* there is no steady state and if ?>?^* then u blows up globally. Here, we show that when (a) N=1,O=(-1,1) and f(s)>0,f'(s)<0,s=0, or (b) N=2,O=B(0,1) and f(s)=e ^-s, the solution u^*=u(x,t;? ^*) is global in time and diverges in the sense ||u ^*(·,t)||₈?8, as t?8. Moreover, it is proved that this divergence is global i.e. u^*(x,t)?8 as t?8 for all x?O. The asymptotic form of divergence is also discussed for some special cases. © 2004 Elsevier Ltd. All rights reserved.