Hyperbolic Keller-Segel equations with sensitivity adjustment in Besov spaces: Global well-posedness and blow-up criteria

Abstract

This paper focuses on the initial value problem for the hyperbolic Keller-Segel (HKS) equation with sensitivity adjustment in Besov sapces over $\mathbb{R}^d$: $\partial_t u + \nabla \cdot \left(\varrho (t) u (1 - u) \nabla S \right)= 0$, $\Delta S = S - u$, where $\varrho (t) (1 - u)$ denotes the adjustment of the classical chemotaxis sensitively by virtue of a time-dependent function $\varrho$. In the case of $\varrho\equiv 1$, existed results [Zhou et al., \emph{J. Differ. Equ.}, 302 (2021) 662-679] and [Zhang et al., \emph{J. Differ. Equ.}, 334 (2022) 451-489] have shown that the HKS equation admits local-in-time Besov solution, whereas the global theory in Besov space still remains an unsolved problem. The main novelty of our observation lies in the fact that if the chemotaxis sensitivity is adjusted by the function $\varrho(t)$ with suitable integrability over $[0,\infty)$, then the associated HKS equation possesses a unique global-in-time Besov solution. As an application, we conclude that the HKS equation with weakly dissipation $-\lambda u$ (i.e., a nonzero interaction-related source term) can also be globally solved in the framework of Besov spaces. Moreover, we derive two types of blow-up criteria for strong solutions in both critical and non-critical Besov spaces, and explicitly characterize the lower bound of blow-up time. These findings reveals how time-dependent parameters (especially, the dissipation parameter $\lambda$) affect the global existence of solutions.