Blow-up of radially symmetric solutions of a non-local problem modelling Ohmic heating (Q2778475)

A complete study of the asymptotic behaviour of the radially symmetric case of the non-local problem modelling Ohmic heating NEWLINE\[NEWLINE u_t = \Delta u + \lambda f(u) / \Big(\int_\Omega f(u) dx\Big)^2, x\in\Omega\subset \mathbb{R}^2, t\geq 0 NEWLINE\]NEWLINE is achieved in this article for positive and decreasing functions \(f\), under various boundary conditions. NEWLINENEWLINENEWLINEThe introduction contains an extensive presentation of the derivation of the mathematical model and its physical interpretation. The existence and uniqueness of the solution of the general non-local problem is briefly treated in the second paragraph. The first results are obtained for the Heaviside function and for the exponential function in the third and fourth paragraphs respectively, two of the most representative examples concerning the behaviour of the model. NEWLINENEWLINENEWLINEThe main structure of the study employed in paragraph 4 is used to obtain the complete results for general decreasing functions in the next paragraph. NEWLINENEWLINENEWLINEInitially, the behaviour of the stationary solutions is treated, in order to use a form similar to the steady-state to obtain proper decreasing (increasing)-in-time upper (lower) solutions to the initial boundary value problem and consequently study the stability and blow-up of its time-dependent solutions. NEWLINENEWLINENEWLINEIn the same paragraph the Robin problem and the stability of its solutions is treated and the section concludes with the study of the blow-up of unbounded solutions and a short note concerning the Neumann problem. NEWLINENEWLINENEWLINEAn interesting discussion with the open question concerning the possibility of obtaining similar behaviour for asymmetric problems for dimensions strictly higher than one concludes the paper.