Boundedness in a fully parabolic attraction–repulsion chemotaxis system with nonlinear diffusion and signal-dependent sensitivity

This paper deals with the quasilinear fully parabolic attraction–repulsion chemotaxis system $\left\{\begin{array}{cc}{u}_t=\nabla \cdot (D\left(u\right)\nabla u)-\nabla \cdot (G\left(u\right)\chi \left(v\right)\nabla v)+\nabla \cdot (H\left(u\right)\xi \left(w\right)\nabla w),\hfill & x\in \Omega ,t>0,\hfill \\ v_t=d_1\Delta v+\alpha u-\beta v,\hfill & x\in \Omega ,t>0,\hfill \\ w_t=d_2\Delta w+\gamma u-\delta w,\hfill & x\in \Omega ,t>0,\hfill \end{array}\right.$under homogeneous Neumann boundary conditions and initial conditions, where $\Omega \subset {\mathbb{R}}^n$ $(n\ge 1)$ is a bounded domain with smooth boundary, $d_1,d_2,\alpha ,\beta ,\gamma ,\delta >0$ are constants. Also, $D,G,H\in C^2\left([0,\infty )\right)$ fulfill that $a_0{(s+1)}^{m-1}\le D\left(s\right)\le a_1{(s+1)}^{m-1}$ with $a_0,a_1>0$ and $m\in \mathbb{R}$; $G\left(0\right)=0$, $0\le G\left(s\right)\le b_0{(s+1)}^{q-1}$ with $b_0>0$ and $q<min\{2,m+1\}$; $H\left(0\right)=0$, $0\le H\left(s\right)\le c_0{(s+1)}^{r-1}$ with $c_0>0$ and $r<min\{2,m+1\}$, and $\chi ,\xi$ satisfy that $0\le \chi \left(s\right)\le \frac{{\chi }_0}{s^{k_1}}$ with ${\chi }_0>0$ and $k_1>1$; $0\le \xi \left(s\right)\le \frac{{\xi }_0}{s^{k_2}}$ with ${\xi }_0>0$ and $k_2>1$. Global existence and boundedness in the case that $w=0$ were proved by Ding (2018). However, there is no work on the above fully parabolic attraction–repulsion chemotaxis system with nonlinear diffusion and signal-dependent sensitivity. This paper develops global existence and boundedness of classical solutions to the above system.     

Dynamical behavior of solutions of a reaction–diffusion model in river network

In this paper, we study the long-time behavior of solutions of a reaction–diffusion model in a one-dimensional river network, where the river network has two branches, and the water flow speeds in each branch are the same constant $\beta$. We show the existence of two critical values $c_0$ and 2 with $0<c_0<2$, and prove that when $-c_0\le \beta <2$, the population density in every branch of the river goes to 1 as time goes to infinity; when $-2<\beta <-c_0$, then, as time goes to infinity, the population density in every river branch converges to a positive steady state strictly below 1; when $\left|\beta \right|\ge 2$, the species will be washed down the stream, and so locally the population density converges to 0. Our result indicates that only if the water-flow speed is suitably small (i.e., $\left|\beta \right|<2$), the species will survive in the long run.     

A generalization of Alon–Aydinian–Huang theorem

Let $S_{n,k}^m$ be the collection of sets of real numbers of size n, in which every subset of size larger than k has a sum less than m, where $n\ge k+1$, and m is some real number. Denote by $a_{n,k}^m$ the maximum number of nonempty subsets of a set in $S_{n,k}^m$ with a sum at least m. In particular, when $m=0$, Alon, Aydinian, Huang ((2014) [1]) proved that $a_{n,k}^0={\sum }_{i=0}^{k?1}\left(\begin{array}{c}n?1\\ i\end{array}\right)$, where two technical proofs, based on a weighted version of Hall's theorem and an extension of the nonuniform Erd?s–Ko–Rado theorem, were presented. In this note, we extend their elegant result from $m=0$ to any real number m, and show that $a_{n,k}^m=\{\begin{array}{cc}{\sum }_{i=0}^{k?1}\left(\begin{array}{c}n?1\\ i\end{array}\right)\hfill & \text{ if }m\ge 0\hfill \\ {\sum }_{i=1}^k\left(\begin{array}{c}n\\ i\end{array}\right)\hfill & \text{ if }m<0\hfill \end{array}$. Our proof is obtained by exploring the recurrence relation and initial conditions of $a_{n,k}^m$.