Existence and Blow-Up for Nonlinear Fokker-Planck with Controlled Drift Divergence

Abstract

We extend the global existence results for critical parameters of [Bianchini and Leccese, J. Math. Anal. Appl. (2024)] to the multidimensional problem \begin{equation*} \partial_t u + \text{div } (b(t,x) u^{1+k}) = Δu \end{equation*} for $b:\mathbb{R}\times {\mathbb{R}}^d\to {\mathbb{R}}^d$ non-autonomous fields, in the case \begin{equation*} (\text{div } b)_-\in L^\infty_{\mathrm{loc}} \big([0,\infty);L^{p,\infty}({\mathbb R}^d)\big), \qquad p>d. \end{equation*} The proof of global existence follows the parabolic symmetrization approach of [Bandle, J. Analyse Math. (1976)] and is based on the study of the function \begin{equation*} m(t,s)=\int_0^s u^*(t,r)\,dr. \end{equation*}