Long-time asymptotic behavior of nonlinear Fokker-Planck type equations with periodic boundary conditions

In this paper, we study the asymptotic behavior of a class of nonlinear Fokker-Planck type equations in a bounded domain with periodic boundary conditions. The system is motivated by our study of grain boundary dynamics, especially under the non-isothermal environments. To obtain the long time behavior of the solutions, in particular, the exponential decay, the kinematic structures of the systems are investigated using novel reinterpretation of the classical entropy method.