A nonlocal parabolic problem in an annulus for the Heaviside function in Ohmic heating

In this paper, we consider the nonlocal parabolic equation u_t=Delta u+frac{lambda H(1-u)}{big(int_{A_{rho, R}} H(1-u)dxbig)^2}, xin A_{rho, R} subset R^2, t 0, with a homogeneous Dirichlet boundary condition, where lambda is a positive parameter, H is the Heaviside function and A_{rho, R} is an annulus. It is shown for the radial symmetric case that: there exist two critical values lambda_* and lambda^*, so that for 0 lambda lambda_*, u(x,t) is global in time and the unique stationary solution is globally asymptotically stable; for lambda_* lambda lambda^* there also exists a steady state and u(x,t) is global in time; while for lambda lambda^* there is no steady state and u(x,t) ``blows up (in some sense) for any appropriate (u_0(x)leq1) initial data.