Electronic Journal of Differential Equations,
Vol. 2002(2002), No. 11, pp. 1-26.

Title: Blow-up of radially symmetric solutions of a non-local problem
       modelling Ohmic heating
   
Author: Dimitrios E. Tzanetis (National Technical Univ. of Athens, Greece)

Abstract:  
  We consider a non-local initial boundary-value problem
  for the equation
   $$ u_t=\Delta u+\lambda f(u)/\Big(\int_{\Omega}f(u)\,dx\Big)^2 ,\quad
      x \in \Omega \subset \mathbb{R}^2 ,\,\;t>0,
   $$
  where $u$ represents a temperature and $f$  is a positive and
  decreasing function. It is shown that for the radially symmetric
  case, if $\int_{0}^{\infty}f(s)\,ds <\infty $
  then there exists a critical value $\lambda^{\ast}>0$
  such that  for $\lambda>\lambda^{\ast}$ there is no stationary
  solution and $u$  blows up, whereas for $\lambda<\lambda^{\ast}$
  there exists at least one stationary solution.
  Moreover, for the Dirichlet problem with
  $-s\,f'(s)<f(s)$ there exists a unique stationary
  solution which is asymptotically stable. For the Robin problem,
  if $\lambda<\lambda^{\ast}$ then there are at least two solutions,
  while if $\lambda=\lambda^{\ast}$ at least one solution.
  Stability and blow-up of these solutions are examined
  in this article.
  
Submitted October 2, 2001. Published February 1, 2002.
Math Subject Classifications: 35B30, 35B40, 35K20, 35K55, 35K99.
Key Words: Nonlocal parabolic equations; blow-up; global existence;
           steady states.