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 $$ u_t=Delta u+lambda f(u)/Big(int_{Omega}f(u),dxBig)^2 ,quad x in Omega subset mathbb{R}^2 ,,;t>0, $$ where $u$ represents a temperature and $f$ is a positive and decreasing function. It is shown that for the radially symmetric case, if $int_{0}^{infty}f(s),ds 0$ 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.

Keywords

• Nonlocal parabolic equations
• blow-up
• global existence
• steady states.