Keller-Segel model with Logarithmic Interaction and nonlocal reaction term

We investigate the global existence and blow-up of solutions to the Keller-Segel model with nonlocal reaction term $u\left(M_0-\int_{\R^2} u dx\right)$ in dimension two. By introducing a transformation in terms of the total mass of the populations to deal with the lack of mass conservation, we exhibit that the qualitative behavior of solutions is decided by a critical value $8\pi$ for the growth parameter $M_0$ and the initial mass $m_0$. For general solutions, if both $m_0$ and $M_0$ are less than $8\pi$, solutions exist globally in time using the energy inequality, whereas there are finite time blow-up solutions for $M_0>8\pi$ (It involves the case $m_0<8\pi$) with any initial data and $M_0<8\pi<m_0$ with small initial second moment. We also show the infinite time blow-up for the critical case $M_0=8 \pi.$ Moreover, in the radial context, we show that if the initial data $u_0(r)<\frac{m_0}{M_0} \frac{8 \lambda}{(r^2+\lambda)^2}$ for some $\lambda>0$, then all the radially symmetric solutions are vanishing in $L_{loc}^1(\R^2)$ as $t \to \infty$. If the initial data $u_0(r)>\frac{m_0}{M_0} \frac{8 \lambda}{(r^2+\lambda)^2}$ for some $\lambda>0$, then there could exist a radially symmetric solution satisfying a mass concentration at the origin as $t \to \infty.$