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.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.