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

We investigate the behaviour of some critical solutions of a non-local initial-boundary value problem for the equation View the MathML source. Under specific conditions on f, there exists a λ∗ such that for each 0<λ<λ∗ there corresponds a unique steady-state solution and u=u(x,t;λ) is a global in time-bounded solution, which tends to the unique steady-state solution as t→∞ uniformly in x. Whereas for λ⩾λ∗ there is no steady state and if λ>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 View the MathML source, 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.