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A Nonlocal Parabolic Problem In An Annulus For The Heaviside Function In Ohmic Heating, Fei Liang, Hongjun Gao, Charles Bu

Turkish Journal of Mathematics

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