Asymptotic stability for a nonlocal parabolic problem

We consider the asymptotic behavior of the solution of the nonlocal parabolic equationut=uxx+f(u)a+b∫-11f(u)dxp,(x,t)∈(-1,1)×(0,∞),with homogeneous Dirichlet boundary condition. When p=2p=2, the equation is so-called Ohmic-heating model, which comes from thermal electricity in this paper, u   and f(u)f(u) represent the temperature of the conductor and the electrical conductivity. The model prescribes the dimensionless temperature when the electric current flows through two conductors, subject to a fixed electric potential difference. The global existence and uniform boundedness of the solution to the problem, indicates that the temperature of conductor remains uniformly bounded. Furthermore, the asymptotic stability of the global solution is obtained.