Long-time behavior of free energy in the nonlinear Fokker-Planck equation

We study the asymptotic behavior of Fokker-Planck equations with spatially inhomogeneous nonlinear diffusion, based on the energy dissipation law. First, we consider the Fokker-Planck equation with porous-medium-type nonlinear diffusion that satisfies the energy dissipation law by introducing spatial inhomogeneity into the free energy. We obtain a result on the long-time behavior of the dissipation function for sufficiently large diffusion coefficients by extending the entropy dissipation method to the case of inhomogeneous diffusion.