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 $$\{\begin{array}{l}{u}_t=\mathrm{\Delta }u-\chi \mathrm{\nabla }\cdot (u\mathrm{\nabla }v)+\kappa u-\mu u^2,\\ \\ v_t=\mathrm{\Delta }v-v+u\end{array}$$ 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 $\bar{\mathrm{\Omega }}\times (0,\mathrm{\infty })$.