Global boundedness of solutions to a parabolic-parabolic chemotaxis system with local sensing in higher dimensions

Abstract

This paper deals with classical solutions to the parabolic-parabolic system \begin{align*} \begin{cases} u_t=\Delta (\gamma (v) u ) &\mathrm{in}\ \Omega\times(0,\infty), \\[1mm] v_t=\Delta v - v + u &\mathrm{in}\ \Omega\times(0,\infty), \\[1mm] \displaystyle \frac{\partial u}{\partial \nu} = \frac{\partial v}{\partial \nu} = 0 &\mathrm{on}\ \partial\Omega \times (0,\infty), \\[1mm] u(\cdot,0)=u_0, \ v(\cdot,0)=v_0 &\mathrm{in}\ \Omega, \end{cases} \end{align*} where $\Omega$ is a smooth bounded domain in $\mathbf{R}^n$($n \geq 3$), $\gamma (v)=v^{-k}$ ($k>0$) and the initial data $(u_0,v_0)$ is positive and regular. This system has striking features similar to those of the logarithmic Keller--Segel system. It is established that classical solutions of the system exist globally in time and remain uniformly bounded in time if $k \in (0,n/(n-2))$, independently the magnitude of mass. This constant $n/(n-2)$ is conjectured as the optimal range guaranteeing global existence and boundedness in the corresponding logarithmic Keller--Segel system. We will derive sufficient estimates for solutions through some single evolution equation that some auxiliary function satisfies. The cornerstone of the analysis is the refined comparison estimate for solutions, which enables us to control the nonlinearity of the auxiliary equation.