Some interesting special cases of a non-local problem modelling ohmic heating with variable thermal conductivity

Abstract

The non-local equation ut = (u3ux)x + λf(u)/∫-11f(u)dx)2 is considered, subject to some initial and Dirichlet boundary conditions. Here f is taken to be either exp(-s4) or H (1 - s) with H the Heaviside function, which are both decreasing. It is found that there exists a critical value λ* = 2, so that for λ > λ* there is no stationary solution and u 'blows up' (in some sense). If 0 < λ < λ*, there is a unique stationary solution which is asymptotically stable and the solution of the IBVP is global in time.