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)