We consider the non-local problemwhich models the temperature when an electric current flows through a material with temperature dependent electrical resistivity f(u) > 0, subject to a fixed potential difference. It is found that for some special cases where f is decreasing andso the problem can be scaled to makethen:(a) for λ < 8 there is a unique steady state which is globally asymptotically stable: (b) for λ = 8 there is no steady state and u is unbounded; (c) for λ > 8 there is no steady state and u blows up for all x, – 1 < x < 1.

We consider the non-local problemIt is found that for the case of decreasingfthen: (i) forthere is a unique steady state which is globally asymptotically stable; (ii) forthen the problem can be scaled so thatin which case: (a) for λ < 8 there is a unique steady state which is globally asymptotically stable; (b) for λ = 8 there is no steady state anduis unbounded; (c) for λ > 8 there is no steady state andublows up for allx, −1 <x, < 1. Some formal asymptotic estimates for the local behaviour ofuas it blows up are obtained.

No abstract

No abstract

No abstract