Existence and Blow-Up for Non-linear Fokker-Planck with Controlled Drift

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 $b:(0,\mathrm{\infty })\times {\mathbb{R}}^d\to {\mathbb{R}}^d$ non-autonomous fields, in the case \begin{equation*} b\in L^\infty_{$\mathrm{l}\mathrm{o}\mathrm{c}$} \bigl((0,\infty),L^{p,\infty}($\mathbb{R}$^d)\bigr), \qquad p>d\ge 2. \end{equation*} For the critical case, we study the difference between two ordered solutions whose conserved small mass absorbs the drift. Finitely many increments then reach an arbitrary solution.