Improvement of conditions for global solvability in a chemotaxis system with signal-dependent motility and generalized logistic source

Abstract

This paper deals with a chemotaxis system with signal-dependent motility $\left(\begin{array}{cc}{u}_t=\nabla \cdot (\gamma \left(v\right)\nabla u)-\nabla \cdot (\chi \left(v\right)u\nabla v)+\lambda u-\mu u^l& x\in \Omega ,t>0,\\ v_t=\Delta v-v+u& x\in \Omega ,t>0,\\ {\partial }_{\nu }u={\partial }_{\nu }v=0& x\in \partial \Omega ,t>0,\\ u(x,0)=u_0\left(x\right)\ge 0,v(x,0)=v_0\left(x\right)\ge 0& x\in \Omega ,\end{array}\right)$ under homogeneous Neumann boundary conditions in a bounded domain $\Omega \subset R^n$ $(n>2)$. If $\lambda \in R$ and $\mu >0$ are constants, we prove that this problem possesses a global classical solution that is uniformly bounded under the conditions that $l>min\left(3,\frac{n+2}{2}\right).$ This result partially improved the work of Lv and Wang (Proc Roy Soc Edinburgh Sect A. 2021, 151 (2): 821-841), in which, the global boundedness of solution is established for $l>\frac{n+2}{2}$. Our results show that there is a consistent decay rate that effectively rules out the occurrence of blow-up phenomena across all spatial dimensions in the system.