Global existence and large time behavior of a 2D Keller-Segel system in logarithmic Lebesgue spaces (Q1756983)

The authors study existence of solutions for the parabolic-parabolic Keller-Segel system of chemotaxis in the whole plane. They construct global-in-time solutions for small initial data \((u_0,v_0)\) in \(L^1\times L^\infty\), as well as for data such that \(\sup_{t>0}\|\log(\mathrm{e}+t)\mathrm{e}^{t\Delta}u_0\|_\infty<\infty\) and \(v_0\in L^\infty\), both small enough. The large time behavior of such solutions is determined.