Let $\Omega\subset \mathbb{R}^d$ be a compact set with smooth boundary and $V$ be a (strongly) convex, smooth function. For the equation $$ \partial_t\rho_t = \nabla\cdot(\rho_t\nabla(\log\rho_t + V)) $$ with no-flux boundary condition $\rho_tn\cdot\nabla(\log\rho_t +V) = 0$ on $\partial\Omega$, let $\rho_t$ be a weak solution starting from $\rho_0$, where $m < \rho_0(x) < M$ for some constant $m, M > 0$.
My questions are, is $\rho_t$ differentiable w.r.t $t$? Is $\nabla\log\rho_t$ uniformly bounded on $\Omega$? And under what conditions, there are constants $\alpha, \tau > 0$ s.t. $$ \Big|\Big|\nabla\log\frac{\rho_t}{\rho_0}\Big|\Big|_{\infty} \lesssim t^{\alpha} $$ for all $0\leq t\leq\tau$?