Immediate smoothing and global solutions for initial data in $L^1\times W^{1,2}$ in a Keller-Segel system with logistic terms in 2D

This article deals with the logistic Keller-Segel model $$\begin{cases} u_t = \Delta u - \chi \nabla\cdot(u\nabla v) + \kappa u - \mu u^2, \\ \\ v_t = \Delta v - v + u \end{cases}$$ in bounded two-dimensional domains (with homogeneous Neumann boundary conditions and for parameters $\chi, \kappa\in \mathbb{R}$ and $\mu>0$), and shows that any nonnegative initial data $(u_0,v_0)\in L^1\times W^{1,2}$ lead to global solutions that are smooth in $\bar{\Omega}\times(0,\infty)$.