The evolution to equilibrium of solutions to nonlinear Fokker-Planck equation

One proves the $H$-theorem for mild solutions to a nondegenerate, nonlinear Fokker-Planck equation $$ u_t-\Delta\beta(u)+{\rm div}(D(x)b(u)u)=0, t\geq0, x\in\mathbb{R}^d,\qquad (1)$$ and under appropriate hypotheses on $\beta,$ $D$ $b$ convergence in $L^1_\textrm{loc}(\mathbb{R}^d)$, $L^1(\mathbb{R}^d)$, respectively, some $t_n\to\infty$ of solution $u(t_n)$ an equilibrium state large set nonnegative initial data $L^1$. These results are new literature equations arising mean field theory also relevant stochastic differential equations. As matter fact, by above result, it follows that McKean-Vlasov corresponding (1), which is distorted Brownian motion, has this as its unique invariant measure. Keywords: equation, $m$-accretive operator, probability density, Lyapunov function, $H$-theorem, motion. 2010 Mathematics Subject Classification: 35B40, 35Q84, 60H10.