Eventual smoothness and asymptotic behavior in a three-dimensional Keller-Segel system with subquadratic degradation

The Keller-Segel system with subquadratic degradation $$\begin{cases}u_{t}=\Delta u-\chi\nabla\cdot(u\nabla v)+ru-\mu u^{\alpha}\\ v_{t}=\Delta v-v+u,\end{cases}$$ is considered under homogeneous Neumann boundary conditions in a smoothly bounded domain $\Omega\subset\mathbb{R}^{3}$, where $\chi>0$, $r\in\mathbb{R}$, $\mu>0$ and $\alpha\in(1,2)$. Some previous studies have asserted that for all reasonably mild initial data, the no-flux/no-flux initial-boundary value problem for (*) possesses at least one global generalized solution whenever $\alpha\in(1,2)$, and that such solutions converge to the spatially homogeneous steady state $((\frac{r}{\mu})^{\frac{1}{\alpha-1}},(\frac{r}{\mu})^{\frac{1}{\alpha-1}})$ in the large-time limit if $r>0$, $\alpha\in[\frac{4}{3},2)$ and $\mu>0$ is appropriately large. However, the knowledge on the regularity properties of solutions remains limited to fairly basic integrability properties, and the convergence rates of such solutions are still unknown. The present work first establishes that these generalized solutions eventually become smooth and bounded if $\alpha\in(\frac{5}{3},2)$ and $r\le\eta\cdot min\{\mu^{\frac{2}{4\alpha-\alpha^{2}-1}},\mu^{\frac{2}{3-\alpha}}\}$ with some $\eta=\eta(\chi,\alpha,\Omega)>0$, and second shows that these solutions stabilize exponentially, algebraically and exponentially toward $((\frac{r_{+}}{\mu})^{\frac{1}{\alpha-1}},(\frac{r_{+}}{\mu})^{\frac{1}{\alpha-1}})$ for the cases $r < 0$, $r=0$ and $r>0$, respectively, provided that $\mu>0$ is suitably large when $r>0$. Our results imply that any type of infinite-time blow-up and persistent oscillatory behavior of solutions to the three-dimensional system (*) with the considered subquadratic degradation can be excluded in scenarios where the underlying population does not proliferate spontaneously, or proliferates at a sufficiently small rate.