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

We consider a non-local initial boundary-value problem for the equation u(t) = Delta u + lambda f(u)=/(integral f(u)dx)(2); x x is an element of subset of R-2, t > 0; where u represents a temperature and f is a positive and decreasing function. It is shown that for the radially symmetric case, if integral(infinity)(0) > 0 f( s) ds < infinity then there exists a critical value lambda* > 0 such that for lambda > lambda* there is no stationary solution and u blows up, whereas for there exists at least one stationary solution. Moreover, for the Dirichlet problem with s f0( s) < f( s) there exists a unique stationary solution which is asymptotically stable. For the Robin problem, if lambda < lambda here are at least two solutions, while if at least one solution. Stability and blow-up of these solutions are examined in this article.