Boundedness and finite-time blow-up in a Keller–Segel chemotaxis-growth system with flux limitation

This paper deals with a parabolic–elliptic Keller–Segel chemotaxis-growth system with flux limitation \(\begin{aligned} \left\{ \begin{aligned} u_t&=\nabla \cdot ((u+1)^{m-1}\nabla u)- \nabla \cdot (uf(|\nabla v|^{2})\nabla v)+\lambda u-\mu u^k,&\quad x\in \Omega ,t>0,\\ 0&=\Delta v-M(t)+u,&\quad x\in \Omega ,t>0, \end{aligned} \right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{ccc}\hfill u_t& =\mathrm{\nabla }·({(u+1)}^{m-1}\mathrm{\nabla }u)-\mathrm{\nabla }·{(uf(|\mathrm{\nabla }v|}^2)\mathrm{\nabla }v)+\lambda u-\mu u^k,\hfill & \hfill x\in \mathrm{\Omega },t>0,\\ \hfill 0& =\mathrm{\Delta }v-M(t)+u,\hfill & \hfill x\in \mathrm{\Omega },t>0,\end{array}\right)\end{array}$$ under homogeneous Neumann boundary conditions, where \(\Omega \subset {\mathbb {R}}^N\) $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is a smoothly bounded domain, \(m\in {\mathbb {R}}\) $m\in \mathbb{R}$ , \(\lambda>0, \mu >0\) $\lambda >0,\mu >0$ , \(k>1\) $k>1$ , \(M(t):=\frac{1}{|\Omega |} \mathop {\int }\limits _{\Omega } u(x, t) d x\) $M(t):=\frac{1}{|\mathrm{\Omega }|}\underset{\mathrm{\Omega }}{\int }u(x,t)dx$ , \(f\left( |\nabla v|^2\right) =(1+|\nabla v|^2)^{-\alpha }, \alpha \in {\mathbb {R}}\) $f\left({|\mathrm{\nabla }v|}^2\right)={(1+|\mathrm{\nabla }v|}^2{)}^{-\alpha },\alpha \in \mathbb{R}$ . In this framework, it is shown that when \(N\ge 2, m+k>2, k>1, k\ge m\) $N\ge 2,m+k>2,k>1,k\ge m$ and \(\begin{aligned} \alpha >\frac{4N-(m+k)N-2}{4(N-1)}, \end{aligned}\) $$\begin{array}{c}\hfill \alpha >\frac{4N-(m+k)N-2}{4(N-1)},\end{array}$$ then for all nonnegative initial data, the solution is global and bounded in time. Moreover, when \(\Omega \subset {\mathbb {R}}^N\) $\mathrm{\Omega }\subset {\mathbb{R}}^N$ \((N\ge 5)\) $(N\ge 5)$ is a ball, if \(1<m<\min \left\{ \frac{2N-4}{N},1-\frac{1}{N}+\frac{1}{N}\sqrt{N^2-4N+1}\right\} \) $1<m<min\left\{\frac{2N-4}{N},,,1,-,\frac{1}{N},+,\frac{1}{N},\sqrt{N^2-4N+1}\right\}$ and the parameters \(\alpha \) $\alpha$ and k satisfy suitable conditions, there exist some initial data \(u_{0}\) $u_0$ such that the solution u(x, t) blows up at finite time \(T_{\max }\) $T_{max}$ in \(L^{\infty }\) $L^{\infty }$ -norm sense.