Blow-up of Radially Symmetric Solutions of a Non-local Problem Modelling Ohmic Heating

Abstract

We consider a non-local initial boundary-value problem for the equation (Figure Presented) where u represents a temperature and / is a positive and decreasing function. It is shown that for the radially symmetric case, if f0∞ f(s) ds < ∝ then there exists a critical value λ* > 0 such that for λ > λ* there is no stationary solution and u blows up, whereas for λ < λ* there exists at least one stationary solution. Moreover, for the Dirichlet problem with - sf′(s) < f(s) there exists a unique stationary solution which is asymptotically stable. For the Robin problem, if λ < λ* then there are at least two solutions, while if λ = λ* at least one solution. Stability and blow-up of these solutions are examined in this article. © 2002 Southwest Texas State University.