On a nonlocal problem modelling Ohmic heating in planar domains

In this paper, we consider the nonlocal problem of the form ut..u=.e.u/(..e.udx)2,x..,t>0 and the associated nonlocal stationary problem ..v=.e.u/(..e.vdx)2,x.., where . is a positive parameter. For . to be an annulus, we prove that the nonlocal stationary problem has a unique solution if and only if . < 2|..|2, and for . = 2|..|2, the solution of the nonlocal parabolic problem grows up globally to infinity as t...