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 $b:(0,\infty)\times\mathbb{R}^d\to\mathbb{R}^d$ non-autonomo