Electron. J. Diff. Eqns., Vol. 2002(2002), No. 11, pp. 1-26.

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

Dimitrios E. Tzanetis

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 $\int_{0}^{\infty}f(s)\,ds$ less than $\infty $ then there exists a critical value $\lambda^{\ast}$
greater than $0$ such that for $\lambda$ greater than $\lambda^{\ast}$ there is no stationary solution and $u$ blows up, whereas for $\lambda$ less than $\lambda^{\ast}$ there exists at least one stationary solution. Moreover, for the Dirichlet problem with $-s\,f'(s)$ less than $f(s)$ there exists a unique stationary solution which is asymptotically stable. For the Robin problem, if $\lambda$ less than $\lambda^{\ast}$ 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.

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Dimitrios E. Tzanetis
Department of Mathematics,
Faculty of Applied Sciences,
National Technical University of Athens,
Zografou Campus, 157 80 Athens, Greece
e-mail: dtzan@math.ntua.gr

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