Some interesting special cases of a nonlocal problem modelling Ohmic heating with variable thermal conductivity (Q2758149)

The nonlocal equation NEWLINE\[NEWLINEu_t= (u^3u_x)_x+ \frac{\lambda f(u)} {\bigl( \int_{-1}^1 f(u) dx\bigr)^2}NEWLINE\]NEWLINE is considered, subject to some initial and Dirichlet boundary conditions. Here \(f\) is taken to be either \(\exp(-s^4)\) or \(H(1-s)\) with \(H\) the Heaviside function, which are both decreasing. It is found that there exists a critical value \(\lambda^*=2\), so that for \(\lambda> \lambda^*\) there is no stationary solution and \(u\) `blows up' (in some sense). If \(0< \lambda< \lambda^*\), there is a unique stationary solution which is asymptotically stable and the solution of the IBVP is global in time.