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 $\begin{cases} u_t = Δu - χ\nabla\cdot(u\nabla v) + κu - μu^2, \\ \\ v_t = Δv - v + u \end{cases}$ in bounded two-dimensional domains (with homogeneous Neumann boundary conditions and for parameters $χ, κ\in \mathbb{R}$ and $μ>0$), 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 $\barΩ\times(0,\infty)$.

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