arXiv subjects

Jiashan Zheng

Publications and source records attributed to Jiashan Zheng.

23 records · Page 2Linked to original sources

arXiv2017

A new result for global existence and boundedness of solutions to a parabolic--parabolic Keller--Segel system with logistic source

We consider the following fully parabolic Keller--Segel system with logistic source $$ \left\{\begin{array}{ll} u_t=Δu-χ\nabla\cdot(u\nabla v)+ au-μu^2,\quad x\in Ω, t>0, \disp{v_t=Δv- v +u},\quad x\in Ω, t>0, \end{array}\right.\eqno(KS) $$ over a bounded domain $Ω\subset\mathbb{R}^N(N\geq1)$, with smooth boundary $\partialΩ$, the parameters $a\in \mathbb{R}, μ>0, χ>0$. It is proved that if $μ>0$, then $(KS)$ admits a global weak solution, while if $μ>\frac{(N-2)_{+}}{N}χC^{\frac{1}{\frac{N}{2}+1}}_{\frac{N}{2}+1}$, then $(KS)$ possesses a global classical solution which is bounded, where $C^{\frac{1}{\frac{N} {2}+1}}_{\frac{N}{2}+1}$ is a positive constant which is corresponding to the maximal Sobolev regularity. Apart from this, we also show that if $a = 0$ and $μ>\frac{(N-2)_{+}}{N}χC^{\frac{1}{\frac{N}{2}+1}}_{\frac{N}{2}+1}$, then both $u(\cdot, t)$ and $v(\cdot, t)$ decay to zero with respect to the norm in $L^\infty(Ω)$ as $t\rightarrow\infty$.

math.AP↗

arXiv2017

A note for global existence of a two-dimensional chemotaxis-haptotaxis model with remodeling of non-diffusible attractant

In this paper, we study the following the coupled chemotaxis--haptotaxis model with remodeling of non-diffusible attractant $$ \left\{\begin{array}{ll} u_t = Δu-χ\nabla\cdot(u\nabla v)- ξ\nabla\cdot(u\nabla w)+μu(1- u-w), \displaystyle{v_t=Δv- v +u},\quad \displaystyle{w_t=- vw+ηw(1-u-w)},\quad \end{array}\right. $$ in a bounded smooth domain $\mathbb{R}^2$ with zero-flux boundary conditions, where $χ$, $ξ$ and $η$ are positive parameters. Under appropriate regularity assumptions on the initial data $(u_0, v_0, w_0)$, by develops some $L^p$-estimate techniques, we prove the global existence and uniqueness of classical solutions when $μ>0$ (where $μ$ is the logistic growth rate of cancer cells). Here we use an approach based on maximal Sobolev regularity and the variation-of-constants formula remove the restrictions $μ$ is sufficiently large, which required in \cite{PangPang1}.

math.AP↗

arXiv2017

Global existence to a $3D$ chemotaxis-Navier-stokes system with nonlinear diffusion and rotation

This paper is concerned with the following quasilinear chemotaxis--Navier--Stokes system with nonlinear diffusion and rotation $$ \left\{ \begin{array}{l} n_t+u\cdot\nabla n=Δn^m-\nabla\cdot(nS(x,n,c)\cdot\nabla c),\quad x\in Ω, t>0, c_t+u\cdot\nabla c=Δc-nc,\quad x\in Ω, t>0,\\ u_t+κ(u \cdot \nabla)u+\nabla P=Δu+n\nabla ϕ,\quad x\in Ω, t>0,\\ \nabla\cdot u=0,\quad x\in Ω, t>0 \end{array}\right.\eqno(CNF) $$ is considered under the no-flux boundary conditions for $n, c$ and the Dirichlet boundary condition for $u$ in a three-dimensional convex domain $Ω\subseteq \mathbb{R}^3$ with smooth boundary, which describes the motion of oxygen-driven bacteria in a fluid. Here % $Ω\subseteq \mathbb{R}^3$ is a , $κ\in \mathbb{R}$ and $S$ denotes the strength of nonlinear fluid convection and a given tensor-valued function, respectively. Assume $m>\frac{10}{9}$ and $S$ fulfills $|S(x,n,c)| \leq S_0(c)$ for all $(x,n,c)\in \barΩ \times [0, \infty)\times[0, \infty)$ with $S_0(c)$ nondecreasing on $[0,\infty)$, then for any reasonably regular initial data, the corresponding initial-boundary problem $(CNF)$ admits at least one global weak solution.

math.AP↗

arXiv2017

Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system with nonlinear diffusion

The coupled quasilinear Keller-Segel-Navier-Stokes system is considered under Neumann boundary conditions for $n$ and $c$ and no-slip boundary conditions for $u$ in three-dimensional bounded domains $Ω\subseteq \mathbb{R}^3$ with smooth boundary, where $m>0,κ\in \mathbb{R}$ are given constants, $ϕ\in W^{1,\infty}(Ω)$. If $ m> 2$, then for all reasonably regular initial data, a corresponding initial-boundary value problem for $(KSNF)$ possesses a globally defined weak solution.

math.AP↗

arXiv2017

A new approach toward locally bounded global solutions to a $3D$ chemotaxis-stokes system with nonlinear diffusion and rotation

We consider a degenerate quasilinear chemotaxis--Stokes type involving rotation in the aggregative term, \begin{equation} \left\{ \begin{array}{l} n_t+u\cdot\nabla n=Δn^m-\nabla\cdot(nS(x,n,c)\cdot\nabla c),\quad x\in Ω, t>0, c_t+u\cdot\nabla c=Δc-nc, x\in Ω, t>0,\\ u_t+\nabla P=Δu+n\nabla ϕ,x\in Ω, t>0,\\ \nabla\cdot u=0, x\in Ω, t>0, \end{array}\right. \end{equation} where $Ω\subseteq \mathbb{R}^3$ is a bounded convex domain with smooth boundary. Here $ S\in C^2(\barΩ\times[0,\infty)^2;\mathbb{R}^{3\times3})$ is a matrix with $s_{i,j}\in C^1( \barΩ \times [0, \infty)\times[0, \infty)).$ Moreover, $|S(x,n,c)| \leq S_0(c)$ for all $(x,n,c)\in \barΩ \times [0, \infty)\times[0, \infty)$ with $S_0(c)$ nondecreasing on $[0,\infty)$. If $$m>\frac{9}{8}, $$ then for all reasonably regular initial data, a corresponding initial-boundary value problem for $(0.1)$ possesses a globally defined weak solution $(n,c,u)$. Moreover, for any fixed $T > 0$ this solution is bounded in $Ω\times (0,T)$ in the sense that $$ \|u(\cdot,t)\|_{L^\infty(Ω)} +\|c(\cdot,t)\|_{W^{1,\infty}(Ω)}+\|n(\cdot,t)\|_{L^\infty(Ω)} \leq C ~~\mbox{for all}~~ t\in(0,T) $$ is valid with some $C(T) > 0$.

math.AP↗