Boundedness in a chemotaxis system with nonlinear diffusion and singular sensitivity

Abstract

This paper studies the following chemotaxis system with nonlinear diffusion and singular sensitivity $\begin{cases} u_t = \nabla \cdot \left( (u+a)^{m-1} \nabla u \right) - χ\nabla \cdot \left( u \dfrac{\nabla v}{v^k} \right), \\[4pt] v_t = Δv - v + u, \end{cases}$ in a smoothly bounded domain $Ω\subset \mathbb{R}^n$, $n \geq 2$, subject to no-flux boundary conditions, where $χ,k,a,m>0$. For suitably regular initial data, we prove global existence and uniform-in-time boundedness of classical solutions if one of the following conditions holds: $\begin{aligned} & n=2,\quad k\in(0,\infty),\quad m>1;\\ & n\ge 3,\quad k\in\left(0,\tfrac12\right),\quad m>1+\frac{(1-k)(n-2)}{n};\\ & n\ge 3,\quad k\in\left[\tfrac12,1\right),\quad m>\frac{3n-2}{2n};\\ & n\ge 3,\quad k\in[1,\infty),\quad m>\max\left\{1,\frac{3n-2}{2n+2}\right\}. \end{aligned}$ The main contribution is the physically relevant three-dimensional case. In particular, for every $χ>0$, $a>0$, $m>1$ and $k\ge 1$, the system admits a global bounded classical solution. Thus, in dimension three, nonlinear diffusion of porous-medium type removes the smallness restriction on the chemotactic sensitivity $χ$ that earlier results require in the case $m=1$ and $k\geq 1$.