On a nonlocal problem modelling Ohmic heating in planar domains

Abstract

In this paper, we consider the nonlocal problem of the form $$u_t - \Delta u = \frac{{\lambda e^{ - u} }} {{\left( {\int_\Omega {e^{ - u} dx} } \right)^2 }}, x \in \Omega , t > 0$$ and the associated nonlocal stationary problem $$- \Delta v = \frac{{\lambda e^{ - u} }} {{\left( {\int_\Omega {e^{ - v} dx} } \right)^2 }}, x \in \Omega ,$$ where λ is a positive parameter. For Ω to be an annulus, we prove that the nonlocal stationary problem has a unique solution if and only if λ < 2|∂Ω|2, and for λ = 2|∂Ω|2, the solution of the nonlocal parabolic problem grows up globally to infinity as t→∞.