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

Abstract

We investigate the behaviour of some critical solutions of a non-local initial-boundary value problem for the equation u t=Δu+λf(u)/(∫Ωf(u)dx) 2,Ω⊂RN,N=1,2. Under specific conditions on f, there exists a λ* such that for each 0 λ* then u blows up globally. Here, we show that when (a) N=1,Ω=(-1,1) and f(s)>0,f′(s)<0,s≥0, or (b) N=2,Ω=B(0,1) and f(s)=e -s, the solution u*=u(x,t;λ *) is global in time and diverges in the sense ||u *(·,t)||∞→∞, as t→∞. Moreover, it is proved that this divergence is global i.e. u*(x,t)→∞ as t→∞ for all x∈Ω. The asymptotic form of divergence is also discussed for some special cases. © 2004 Elsevier Ltd. All rights reserved.