Asymptotic behavior for an Ohmic heating model in thermal electricity

In this paper, we consider the asymptotic behavior for a nonlocal parabolic problem with Dirichlet boundary condition, which is raised from the thermal-electricity and is so-called an Ohmic heating model. It models the system of the temperature of a conductor in the device, which is connected in series with another conductor with constant. The electrical resistivity of the one of the conductors depends on the temperature and the other one remains constant. An analysis of the nonlinear problem shows that the solution exists global and the unique stationary solution is globally asymptotically stable. The results assert that the temperature of the conductor remains bounded and the system converges asymptotically to the unique equilibrium.