Symmetric results of a Hénon type elliptic equation

In this paper, we study the following weighted elliptic system $\left\{\begin{array}{cc}\hfill & -\Delta u+u={\lambda }_1{\left|x\right|}^{\alpha }{u}^3+\mu {\left|x\right|}^{\beta }u{v}^2,x\in B,\hfill \\ \hfill & -\Delta v+v={\lambda }_2{\left|x\right|}^{\alpha }{v}^3+\mu {\left|x\right|}^{\beta }{u}^{2}v,x\in B,\hfill \\ \hfill & u,v>0,x\in B,u=v=0,x\in \partial B,\hfill \end{array}\right.$where $B\subset {\mathbb{R}}^N(N=2,3)$ is the unit ball centered at the origin, ${\lambda }_1,{\lambda }_2>0$, $\mu >0$, $\beta >0$, $\alpha >0$. By virtue of variational approaches and rescaling methods, the system has a nontrivial ground state solution with $\alpha >\beta >0$, moreover, by reduction methods, the ground state solution is radial symmetry if $\beta >0$ small enough.     

Large time behavior in a fractional chemotaxis–Navier–Stokes system with logistic source

This paper deals with a coupled chemotaxis–Navier–Stokes system with logistic source and a fractional diffusion of order $\alpha \in (\frac{1}{2},1)$ $\left\{\begin{array}{c}{n}_t+u\cdot \nabla n=-{(-\Delta )}^{\alpha }n-\chi \nabla \cdot (n\nabla c)+an-b{n}^2,\hfill \\ c_t+u\cdot \nabla c=\Delta c-nc,\hfill \\ u_t+(u\cdot \nabla )u=\Delta u+\nabla P+n\nabla \varphi +f,\nabla \cdot u=0\hfill \end{array}\right.$on three dimensional periodic torus ${\mathbb{T}}^3$. Since there is no classical solution in the three-dimensional full Navier–Stokes equations, our main purpose of this paper is to investigate the global existence of weak solutions to the above system in the case of a weaker diffusion, and after some waiting time, the weak solutions in fact become smooth and converge to the semi-trivial steady state $(\frac{a}{b},0,0).$     

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 .$