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

Authors: FEI LIANG, HONGJUN GAO, CHARLES BU

Abstract: 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)dx\big)^2}, x\in 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.

Keywords: Nonlocal parabolic equation, steady state, stability, blow-up

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