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 $$ u_t-\Delta\beta(u)+\mathrm{ div}(D(x)b(u)u)=0, \ t\ge0, \ x\in\mathbb{R}^d,\hspace{1cm} (1)$$ and under appropriate hypotheses on $\beta,$ $D$ and $b$ the convergence in $L^1_\mathrm{ loc}(\mathbb{R}^d)$, $L^1(\mathbb{R}^d)$, respectively, for some $t_n\rightarrow\infty$ of the solution $u(t_n)$ to an equi\-li\-brium state of the equation for a large set of nonnegative initial data in $L^1$. Furthermore, the solution to the McKean--Vlasov stochastic differential equation corresponding to (1), which is a {\it nonlinear distorted Brownian motion}, is shown to have this equilibrium state as its unique invariant measure.\smallskip\\ {\bf Keywords:} Fokker-Planck equation, $m$-accretive operator, probability density, Lyapunov function, $H$-theorem, McKean--Vlasov stochastic differential equation, nonlinear distorted Brownian motion.\\ {\bf 2010 Mathematics Subject Classification:} 35B40, 35Q84, 60H10.