Finite time blow-up in a parabolic–elliptic Keller–Segel system with flux dependent chemotactic coefficient

A parabolic–elliptic Keller–Segel system $\left\{\begin{array}{c}{u}_t=\Delta u-\chi \nabla \cdot (uf\left(|\nabla v|\right)\nabla v),\hfill \\ 0=\Delta v-M+u,\hfill \end{array}\right.$with homogeneous Neumann boundary condition is considered in a radially symmetric domain $\Omega =B_R\left(0\right)\subset {\mathbb{R}}^N(N\ge 3),$ where $f\left(\xi \right)=\left({\xi }^{p-2}{\left(1+{\xi }^p\right)}^{\frac{q-p}{p}}\right),\xi \ge 0,p\ge 2,1<q\le p<\infty ,$and $B_R\left(0\right)$ is a $N$-dimensional ball of radius $R>0.$ We assert that under a condition on the initial data, radial weak solutions blow-up in finite time when $\frac{N}{N-1}<q<2.$     

Boundedness and large time behavior for a chemotaxis system with signal-dependent motility and indirect signal consumption

In this paper, we study the following chemotaxis system with signal-dependent motility, indirect signal consumption and logistic source $\left\{\begin{array}{cc}{u}_t=\Delta \left(u\gamma \left(v\right)\right)+\rho u-\mu u^l,\hfill & x\in \Omega ,t>0,\hfill \\ v_t=\Delta v-vw,\hfill & x\in \Omega ,t>0,\hfill \\ w_t=-\delta w+u,\hfill & x\in \Omega ,t>0\hfill \end{array}\right.$under homogeneous Neumann boundary conditions in a smooth bounded domain $\Omega \subset {\mathbb{R}}^n$, where the motility function $\gamma \left(v\right)\in C^3\left((0,+\infty )\right),\gamma \left(v\right)>0,{\gamma }^{\prime }\left(v\right)<0$ on $(0,+\infty )$, ${lim}_{v\to \infty }\gamma \left(v\right)=0$, $\rho >0,\mu >0,l>1$ and $\delta >0$. The purpose of this paper is to prove that if $l>max\{1,\frac{n}{2}\}$, then the system possesses a global solution. In addition, if $l$ satisfies $l\left\{\begin{array}{cc}\ge 2,\hfill & \text{if }n\le 3,\hfill \\ >\frac{n}{2},\hfill & \text{if }n\ge 4,\hfill \end{array}\right.$then the solution $(u,v,w)$ satisfies ${6u(\cdot ,t)-{\left(\frac{\rho }{\mu }\right)}^{\frac{1}{l-1}}6}_{L^{\infty }\left(\Omega \right)}+{6v(\cdot ,t)6}_{L^{\infty }\left(\Omega \right)}+{6w(\cdot ,t)-\frac{1}{\delta }{\left(\frac{\rho }{\mu }\right)}^{\frac{1}{l-1}}6}_{L^{\infty }\left(\Omega \right)}$$\to 0ast\to \infty .$