arXiv subjects

Masaaki Mizukami

Publications and source records attributed to Masaaki Mizukami.

25 records · Page 2Linked to original sources

arXiv2017

Global existence and asymptotic behavior of classical solutions for a 3D two-species Keller--Segel-Stokes system with competitive kinetics

This paper deals with the two-species Keller--Segel-Stokes system with competitive kinetics $(n_1)_t + u\cdot\nabla n_1 =Δn_1 - χ_1\nabla\cdot(n_1\nabla c)+ μ_1n_1(1- n_1 - a_1n_2)$, $(n_2)_t + u\cdot\nabla n_2 =Δn_2 - χ_2\nabla\cdot(n_2\nabla c) + μ_2n_2(1- a_2n_1 - n_2), c_t + u\cdot\nabla c =Δc - c + αn_1 +βn_2$, $u_t= Δu + \nabla P+ (γn_1 + δn_2)\nablaϕ$, $ \nabla\cdot u = 0$ under homogeneous Neumann boundary conditions in a bounded domain $Ω\subset \mathbb{R}^3$ with smooth boundary. Many mathematicians study chemotaxis-fluid systems and two-species chemotaxis systems with competitive kinetics. However, there are not many results on coupled two-species chemotaxis-fluid systems which have difficulties of the chemotaxis effect, the competitive kinetics and the fluid influence. Recently, in the two-species chemotaxis-Stokes system, where $-c+αn_1+βn_2$ is replaced with $-(αn_1+βn_2)c$ in the above system, global existence and asymptotic behavior of classical solutions were obtained in the 3-dimensional case under the condition that $μ_1,μ_2$ are sufficiently large. Nevertheless, the above system has not been studied yet; we cannot apply the same argument as in the previous works because of lacking the $L^\infty$-information of $c$. The main purpose of this paper is to obtain global existence and stabilization of classical solutions to the above system in the 3-dimensional case under the largeness conditions for $μ_1,μ_2$.

math.AP↗

arXiv2017

Improvement of conditions for asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity

This paper deals with the two-species chemotaxis-competition system. About the problem, Bai--Winkler first obtained asymptotic stability under some conditions. Recently, the conditions assumed in the previous work were improved; however, there is a gap between the conditions assumed in the previous works. The purpose of this work is to improve the conditions assumed in the previous works for asymptotic behavior.

math.AP↗

arXiv2017

Boundedness and stabilization in a two-species chemotaxis-competition system of parabolic-parabolic-elliptic type

This paper deals with the two-species chemotaxis-competition system $u_t = d_1 Δu - χ_1 \nabla \cdot (u \nabla w) + μ_1 u(1 - u - a_1 v)$, $v_t = d_2 Δv - χ_2 \nabla \cdot (v \nabla w) + μ_2 v(1 - a_2 u - v)$, $0 = d_3 Δw + αu + βv - γw$, where $Ω$ is a bounded domain in $\mathbb{R}^n$ with smooth boundary, $n\ge 2$; $χ_i$ and $μ_i$ are constants satisfying some conditions. The above system was studied in the cases that $a_1,a_2\in (0,1)$ and $a_1>1>a_2$, and it was proved that global existence and asymptotic stability hold when $\frac{χ_i}{μ_i}$ are small. However, the conditions in the above two cases strongly depend on $a_1,a_2$, and have not been obtained in the case that $a_1,a_2\ge 1$. Moreover, convergence rates in the cases that $a_1,a_2\in (0,1)$ and $a_1 > 1 > a_2$ have not been studied. The purpose of this work is to construct conditions which derive global existence of classical bounded solutions for all $a_1,a_2>0$ which covers the case that $a_1,a_2 \ge 1$, and lead to convergence rates for solutions of the above system in the cases that $a_1,a_2\in (0,1)$ and $a_1\ge 1 >a_2$.

math.AP↗

arXiv2017

Global existence and asymptotic behavior of classical solutions for a 3D two-species chemotaxis-Stokes system with competitive kinetics

This paper considers the two-species chemotaxis-Stokes system with competitive kinetics under homogeneous Neumann boundary conditions in a three-dimensional bounded domain with smooth boundary. Both chemotaxis-fluid systems and two-species chemotaxis systems with competitive terms are studied by many mathematicians. However, there has not been rich results on coupled two-species-fluid systems. Recently, global existence and asymptotic stability in this problem with convection term in the fluid equation of the above system were established in the 2-dimensional case. The purpose of this paper is to give results for global existence, boundedness and stabilization of solutions to this system in the 3-dimensional case.

math.AP↗

arXiv2017

A unified method for boundedness in fully parabolic chemotaxis systems with signal-dependent sensitivity

This paper deals with the Keller--Segel system with signal-dependent sensitivity \begin{equation*} u_t=Δu - \nabla \cdot (u χ(v)\nabla v), \quad v_t=Δv + u - v, \quad x\inΩ,\ t>0, \end{equation*} where $Ω$ is a bounded domain in $\mathbb{R}^n$, $n\geq 2$; $χ$ is a function satisfying $χ(s)\leq K(a+s)^{-k}$ for some $k\geq 1$ and $a\geq 0$. In the case that $k=1$, Fujie (J. Math. Anal. Appl.; 2015; 424; 675--684) established global existence of bounded solutions under the condition $K<\sqrt{\frac{2}{n}}$. On the other hand, when $k>1$, Winkler (Math. Nachr.; 2010; 283; 1664--1673) asserted global existence of bounded solutions for arbitrary $K>0$. However, there is a gap in the proof. Recently, Fujie tried modifying the proof; nevertheless it also has a gap. It seems to be difficult to show global existence of bounded solutions for arbitrary $K>0$. Moreover, the condition for $K$ when $k>1$ cannot connect to the condition when $k=1$. The purpose of the present paper is to obtain global existence and boundedness under more natural and proper condition for $χ$ and to build a mathematical bridge between the cases $k=1$ and $k>1$.

math.AP↗

arXiv2016

How far does small chemotactic interaction perturb the Fisher-KPP dynamics?

This paper deals with nonnegative solutions of the Neumann initial-boundary value problem for the fully parabolic chemotaxis-growth system $ (u_{\varepsilon})_t$ $=Δu_{\varepsilon} - \varepsilon \nabla \cdot ( u_\varepsilon \nabla v_\varepsilon) + μu_\varepsilon(1 - u_\varepsilon)$, $ (v_{\varepsilon})_t=Δv_\varepsilon -v_\varepsilon+u_\varepsilon,$ with positive small parameter $\varepsilon>0$ in a bounded convex domain $Ω\subset\mathbb{R}^n$ ($n\geq 1$) with smooth boundary. The solutions converge to the solution $u$ to the Fisher-KPP equation as $\varepsilon\to 0$. It is shown that for all $μ>0$ and any suitably regular nonnegative initial data $(u_{init},v_{init})$ there are some constants $\varepsilon_0>0$ and $C>0$ such that \[ \sup_{t>0}\|u_\varepsilon(\cdot,t)-u(\cdot,t)\|_{L^\infty(Ω)} \leq C\varepsilon \quad for\ all\ \varepsilon\in(0,\varepsilon_0). \]

math.AP↗

arXiv2016

On the weakly competitive case in a two-species chemotaxis model

In this article we investigate a parabolic-parabolic-elliptic two-species chemotaxis system with weak competition and show global asymptotic stability of the coexistence steady state under a smallness condition on the chemotactic strengths, which seems more natural than the condition previously known. For the proof we rely on the method of eventual comparison, which thereby is shown to be a useful tool even in the presence of chemotactic terms.

math.AP↗