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Masaaki Mizukami

Publications and source records attributed to Masaaki Mizukami.

At least 19 recordsLinked to original sources
arXiv2026

Boundedness and asymptotic stability in a model for tuberculosis granuloma formation

This paper deals with a problem which describes tuberculosis granuloma formation \begin{align*} \begin{cases} u_t = Δu - \nabla \cdot (u \nabla v) - uv - u + β, &x \in Ω,\ t>0, \\ v_t = Δv + v -uv + μw, &x \in Ω,\ t>0, \\ w_t = Δw + uv - wz - w, &x \in Ω,\ t>0, \\ z_t = Δz - \nabla \cdot (z \nabla w) + f(w)z -z, &x \in Ω,\ t>0 \end{cases} \end{align*} under homogeneous Neumann boundary conditions and initial conditions, where $Ω\subset \mathbb{R}^n$ ($n\ge 2$) is a smooth bounded domain, $β,μ>0$ and $f$ is some function, and shows that if initial data are small in some sense then the solution $(u,v,w,z)$ of the problem exists globally and convergences to $(β,0,0,0)$ exponentially when $β>1$ and the reproduction number $R_0 := \frac{μβ+ 1}β$ satisfies $R_0<1$.

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arXiv2024

Global solvability of a model for tuberculosis granuloma formation

We discuss a nonlinear system of partial differential equations modelling the formation of granuloma during tuberculosis infections and prove the global solvability of the homogeneous Neumann problem for \begin{align*} \begin{cases} u_t = D_u Δu - χ_u \nabla \cdot (u \nabla v) - γ_u uv - δ_u u + β_u, \\ v_t = D_v Δv + ρ_v v - γ_v uv + μ_v w,\\ w_t = D_w Δw + γ_w uv - α_w wz - μ_w w,\\ z_t = D_z Δz - χ_z \nabla \cdot (z \nabla w) + α_z f(w)z - δ_z z \end{cases} \end{align*} in bounded domains in the classical and weak sense in the two- and three-dimensional setting, respectively. In order to derive suitable a~priori estimates, we study the evolution of the well-known energy functional for the chemotaxis-consumption system both for the $(u, v)$- and the $(z, w)$-subsystem. A key challenge compared to "pure" consumption systems consists of overcoming the difficulties raised by the additional, in part positive, terms in the second and third equations. This is inter alia achieved by utilising a dissipative term of the (quasi-)energy functional, which may just be discarded in simpler consumption systems.

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arXiv2023

Global existence and boundedness in a chemotaxis-convection model with sensitivity functions for tumor angiogenesis

This paper deals with the fully parabolic chemotaxis-convection model with sensitivity functions for tumor angiogenesis, \begin{align*} \begin{cases} u_t=Δu-\nabla \cdot (uχ_1(v)\nabla v) +\nabla \cdot (uχ_2(w)\nabla w), &x \in Ω,\ t>0, \\[1.05mm] v_t=Δv+\nabla \cdot (vξ(w)\nabla w)+αu-βv, &x \in Ω,\ t>0, \\[1.05mm] w_t=Δw+γu-δw, &x \in Ω,\ t>0 \end{cases} \end{align*} under homogeneous Neumann boundary conditions and initial conditions, where $Ω\subset \mathbb{R}^n$ $(n \le 3)$ is a bounded domain with smooth boundary, $χ_1, χ_2, ξ$ are functions satisfying some conditions and $α, β, γ, δ>0$ are constants. The purpose of this paper is to establish global existence and boundedness in this system.

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arXiv2022

Possible points of blow-up in chemotaxis systems with spatially heterogeneous logistic source

We discuss the influence of possible spatial inhomogeneities in the coefficients of logistic source terms in parabolic-elliptic chemotaxis-growth systems of the form \begin{align*} u_t &= Δu - \nabla\cdot(u\nabla v) + κ(x)u-μ(x)u^2, 0 &= Δv - v + u \end{align*} in smoothly bounded domains $Ω\subset\mathbb{R}^2$. Assuming that the coefficient functions satisfy $κ,μ\in C^0(\overlineΩ)$ with $μ\geq0$ we prove that finite-time blow-up of the classical solution can only occur in points where $μ$ is zero, i.e.\ that the blow-up set $\mathcal{B}$ is contained in \begin{align*} \big\{x\in\overlineΩ\midμ(x)=0\big\}. \end{align*} Moreover, we show that whenever $μ(x_0)>0$ for some $x_0\in\overlineΩ$, then one can find an open neighbourhood $U$ of $x_0$ in $\overlineΩ$ such that $u$ remains bounded in $U$ throughout evolution.

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arXiv2022

Can chemotactic effects lead to blow-up or not in two-species chemotaxis-competition models?

This paper deals with the two-species chemotaxis-competition models \begin{align*} \begin{cases} u_t = d_1 Δu - χ_1 \nabla \cdot (u \nabla w) + μ_1 u (1- u^{κ_1-1} - a_1 v^{λ_1-1}), &\quad x \in Ω,\ t>0,\\ % v_t = d_2 Δv - χ_2 \nabla \cdot (v \nabla w) + μ_2 v (1- a_2 u^{λ_2-1} - v^{κ_2-1}), &\quad x \in Ω,\ t>0,\\ % 0 = d_3 Δw + αu + βv - h(u,v,w), &\quad x \in Ω,\ t>0, \end{cases} \end{align*} where $Ω\subset \mathbb{R}^n$ $(n\ge2)$ is a bounded domain with smooth boundary, and $h=γw$ or $h=\frac{1}{|Ω|}\int_Ω(αu+ βv)\,dx$. In the case that $κ_1=λ_1=κ_2=λ_2=2$ and $h=γw$, it is known that smallness conditions for the chemotacic effects lead to boundedness of solutions (Math.\ Methods Appl.\ Sci.; 2018; 41; 234--249). However, the case that the chemotactic effects are large seems not to have been studied yet; therefore it remains to consider the question whether the solution is bounded also in the case that the chemotactic effects are large. The purpose of this paper is to give a negative answer to this question.

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arXiv2021

Global existence and boundedness in a fully parabolic attraction-repulsion chemotaxis system with signal-dependent sensitivities without logistic source

This paper deals with the fully parabolic attraction-repulsion chemotaxis system with signal-dependent sensitivities, \begin{align*} \begin{cases} u_t=Δu-\nabla \cdot (uχ(v)\nabla v) +\nabla \cdot (uξ(w)\nabla w), &x \in Ω,\ t>0,\\[1.05mm] v_t=Δv-v+u, &x \in Ω,\ t>0,\\[1.05mm] w_t=Δw-w+u, &x \in Ω,\ t>0 \end{cases} \end{align*} under homogeneous Neumann boundary conditions and initial conditions, where $Ω\subset \mathbb{R}^n$ $(n \ge 2)$ is a bounded domain with smooth boundary, $χ, ξ$ are functions satisfying some conditions. Global existence and boundedness of classical solutions to the system with logistic source have already been obtained by taking advantage of the effect of logistic dampening (J. Math. Anal. Appl.; 2020;489;124153). This paper establishes existence of global bounded classical solutions despite the loss of logistic dampening.

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arXiv2020

Long-term behaviour in a parabolic-elliptic chemotaxis-consumption model

Global existence and boundedness of classical solutions of the chemotaxis--consumption system \begin{align*} n_t &= Δn - \nabla \cdot (n \nabla c), \\ 0 &= Δc - nc, \end{align*} under no-flux boundary conditions for $n$ and Robin-type boundary conditions \[ \partial_ν c = (γ-c) g \] for $c$ (with $γ>0$ and $C^{1+β}(\partialΩ) \ni g > 0$ for some $β\in(0,1)$) are established in bounded domains $Ω\subset\mathbb{R}^{N}$, $N\ge 1$. Under a smallness condition on $γ$, moreover, we show convergence to the stationary solution.

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arXiv2019

Extensibility criterion ruling out gradient blow-up in a quasilinear degenerate chemotaxis system with flux limitation

This paper deals with the quasilinear degenerate chemotaxis system with flux limitation \begin{equation*} \begin{cases} u_t = \nabla\cdot\left(\dfrac{u^p \nabla u}{\sqrt{u^2 + |\nabla u|^2}} \right) -χ\nabla\cdot\left(\dfrac{u^q\nabla v}{\sqrt{1 + |\nabla v|^2}}\right), \\[1mm] 0 = Δv - μ+ u \end{cases}\end{equation*} under no-flux boundary conditions in balls $Ω\subset\mathbb{R}^n$, and the initial condition $u|_{t=0}=u_0$ for a radially symmetric and positive initial data $u_0\in C^3(\overlineΩ)$, where $χ>0$ and $μ:=\frac{1}{|Ω|}\int_Ωu_0$. Bellomo--Winkler (Comm.\ Partial Differential Equations;2017;42;436--473) proved local existence of unique classical solutions and extensibility criterion ruling out gradient blow-up as well as global existence and boundedness of solutions when $p=q=1$ under some conditions for $χ$ and $\int_Ωu_0$. This paper derives local existence and extensibility criterion ruling out gradient blow-up when $p,q\geq 1$, and moreover shows global existence and boundedness of solutions when $p>q+1-\frac{1}{n}$.

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arXiv2019

Finite-time blow-up in a quasilinear degenerate chemotaxis system with flux limitation

This paper deals with the quasilinear degenerate chemotaxis system with flux limitation \begin{align*} \begin{cases} u_t = \nabla\cdot\left(\dfrac{u^p \nabla u}{\sqrt{u^2 + |\nabla u|^2}} \right) -χ\nabla\cdot\left( \dfrac{u^q\nabla v}{\sqrt{1 + |\nabla v|^2}}\right), &x\in Ω,\ t>0, \\[1mm] 0 = Δv - μ+ u, &x\in Ω,\ t>0, \end{cases} \end{align*} where $Ω:= B_R(0) \subset \mathbb{R}^n$ ($n \in \mathbb{N}$) is a ball with some $R>0$, and $χ>0$, $p,q\geq1$, $μ:= \frac 1{|Ω|} \int_Ωu_0$ and $u_0$ is an initial data of an unknown function $u$. Bellomo--Winkler (Trans.\ Amer.\ Math.\ Soc.\ Ser.\ B;2017;4;31--67) established existence of an initial data such that the corresponding solution blows up in finite time when $p=q=1$. This paper gives existence of blow-up solutions under some condition for $χ$ and $u_0$ when $1\leq p\leq q$.

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arXiv2019

How strongly does diffusion or logistic-type degradation affect existence of global weak solutions in a chemotaxis-Navier--Stokes system?

This paper considers the chemotaxis-Navier--Stokes system with nonlinear diffusion and logistic-type degradation term \begin{align*} \begin{cases} n_t + u\cdot\nabla n = \nabla \cdot(D(n)\nabla n) - \nabla\cdot(n χ(c) \nabla c) + κn - μn^α, & x\in Ω,\ t>0, \\ c_t + u\cdot\nabla c = Δc - nf(c), & x \in Ω,\ t>0, \\ u_t + (u\cdot\nabla)u = Δu + \nabla P + n\nablaΦ+ g, \ \nabla\cdot u = 0, & x \in Ω,\ t>0, \end{cases} \end{align*} where $Ω\subset \mathbb{R}^3$ is a bounded smooth domain; $D \ge 0$ is a given smooth function such that $D_1 s^{m-1} \le D(s) \le D_2 s^{m-1}$ for all $s\ge 0$ with some $D_2 \ge D_1 > 0$ and some $m > 0$; $χ,f$ are given functions satisfying some conditions; $κ\in \mathbb{R},μ\ge0,α>1$ are constants. This paper shows existence of global weak solutions to the above system under the condition that \begin{align*} m >\frac{2}{3},\quad μ\ge 0 \quad \mbox{and}\quad α>1 \end{align*} hold, or that \begin{align*} m> 0, \quad μ>0 \quad \mbox{and} \quad α> \frac{4}{3} \end{align*} hold. This result asserts that `strong' diffusion effect or `strong' logistic damping derives existence of global weak solutions even though the other effect is `weak', and can include previous works.

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arXiv2018

Stabilization in the Keller--Segel system with signal-dependent sensitivity

This paper deals with the Keller--Segel system with signal-dependent sensitivity \begin{align*} &u_t = Δu - χ\nabla \cdot (uS(v)\nabla v), &v_t = Δv - v + u, \end{align*} where $χ>0$ and $S$ is a given function generalizing the sensitivity $S(s)=\frac{1}{(a+s)^{k}}$, $k>1$, $a\ge 0$, and shows exponential convergence of global classical solutions under an additional smallness condition condition for $χ>0$.

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arXiv2018

The fast signal diffusion limit in a chemotaxis system with strong signal sensitivity

This paper gives a first insight into making a mathematical bridge between the parabolic-parabolic signal-dependent chemotaxis system and its parabolic-elliptic version. To be more precise, this paper deals with convergence of a solution for the parabolic-parabolic chemotaxis system with strong signal sensitivity $$ (u_λ)_t = Δu_λ- \nabla \cdot (u_λχ(v_λ)\nabla u_λ), \quad λ(v_λ)_t = Δv_λ- v_λ+u_λ\quad \mbox{in} \ Ω\times (0,\infty) $$ to that for the parabolic-elliptic chemotaxis system $$ u_t = Δu -\nabla \cdot (uχ(v)\nabla v), \quad 0= Δv -v +u \quad \mbox{in} \ Ω\times (0,\infty), $$ where $Ω$ is a bounded domain in $\mathbb{R}^n$ ($n\in\mathbb{N}$) with smooth boundary, $λ>0$ is a constant and $χ$ is a function generalizing $$ χ(v) = \frac{χ_0}{(1+v)^k} \quad (χ_0>0,\ k>1).$$ In chemotaxis systems parabolic-elliptic systems often provided some guide to methods and results for parabolic-parabolic systems. However, the relation between parabolic-elliptic systems and parabolic-parabolic systems has not been studied. Namely, it still remains to analyze on the following question: Does a solution of the parabolic-parabolic system converge to that of the parabolic-elliptic system as $λ\searrow 0$? This paper gives some positive answer in the chemotaxis system with strong signal sensitivity.

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arXiv2018

The fast signal diffusion limit in a Keller-Segel system

This paper deals with convergence of a solution for the parabolic-parabolic Keller-Segel system \[ (u_λ)_t = Δu_λ- χ\nabla \cdot (u_λ\nabla v_λ), \quad λ(v_λ)_t = Δv_λ- v_λ+ u_λ\quad \mbox{in} \ Ω\times (0,\infty) \] to that for the parabolic-elliptic Keller-Segel system \[ u_t = Δu - χ\nabla \cdot (u \nabla v), \quad 0= Δv -v +u \quad \mbox{in} \ Ω\times (0,\infty) \] as $λ\searrow 0$, where $Ω$ is a bounded domain in $\mathbb{R}^n$ ($n\ge 2$) with smooth boundary, $χ, λ>0$ are constants. In chemotaxis systems parabolic-elliptic systems often provided some guide to methods and results for parabolic-parabolic systems. However, there have not been rich results on the relation between parabolic-elliptic systems and parabolic-parabolic systems. Namely, it still remains to analyze on the following question except some cases: Does a solution of the parabolic-parabolic system converge to that of the parabolic-elliptic system as $λ\searrow 0$? In the case that $Ω$ is the whole space $\mathbb{R}^n$, or $Ω$ is a bounded domain and $χ$ is a strong signal sensitivity, some positive answers were shown in the previous works. Therefore, one can expect a positive answer to this question also in the Keller-Segel system in a bounded domain $Ω$ in some cases. This paper gives some positive answer in the 2-dimensional and the higher-dimensional Keller-Segel system.

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arXiv2018

A Keller-Segel-fluid system with singular sensitivity: Generalized solutions

In bounded smooth domains $Ω\subset\mathbb{R}^N$, $N\in\{2,3\}$, we consider the Keller-Segel-Stokes system \begin{align*} n_t + u\cdot \nabla n &= Δn - χ\nabla \cdot(\frac{n}{c}\nabla c),\\ c_t + u\cdot \nabla c &= Δc - c + n,\\ u_t &= Δu + \nabla P + n\nabla ϕ, \qquad \nabla \cdot u=0, \end{align*} and prove global existence of generalized solutions if \[ χ<\begin{cases} \infty,&N=2,\\ \frac{5}{3},&N=3. \end{cases} \] These solutions are such that blow-up into a persistent Dirac-type singularity is excluded.

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arXiv2018

Global weak solutions to a 3-dimensional degenerate and singular chemotaxis-Navier--Stokes system with logistic source

This paper considers the degenerate and singular chemotaxis-Navier--Stokes system with logistic term $n_t + u\cdot\nabla n =Δn^m - χ\nabla\cdot(n\nabla c) + κn -μn^2$, $x \in Ω,\ t>0$, $c_t + u\cdot\nabla c = Δc - nc$, $x \in Ω,\ t>0$, $u_t + (u\cdot\nabla)u = Δu + \nabla P + n\nablaΦ, \quad \nabla\cdot u = 0$, $x \in Ω,\ t>0$, where $Ω\subset \mathbb{R}^3$ is a bounded domain and $χ,κ\ge 0$ and $m, μ>0$. In the above system without fluid environment Jin (J. Differential Equations, 2017) showed existence and boundedness of global weak solutions. On the other hand, in the above system with $m=1$, Lankeit (Math.\ Models Methods Appl. Sci., 2016) established global existence of weak solutions. However, the above system with $m>0$ has not been studied yet. The purpose of this talk is to establish global existence of weak solutions in the chemotaxis-Navier--Stokes system with degenerate diffusion and logistic term.

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arXiv2017

Boundedness and stabilization in a three-dimensional two-species chemotaxis-Navier--Stokes system with competitive kinetics

This paper is concerned with the 3-dimensional two-species chemotaxis-Navier--Stokes system with Lotka--Volterra competitive kinetics under homogeneous Neumann boundary conditions and initial conditions. Recently, in the 2-dimensional setting, global existence and stabilization of classical solutions to the above system were first established. However, the 3-dimensional case has not been studied: Because of difficulties in the Navier--Stokes system, we can not expect existence of classical solutions to the above system. The purpose of this paper is to obtain global existence of weak solutions to the above system, and their eventual smoothness and stabilization.

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arXiv2017

Singular sensitivity in a Keller-Segel-fluid system

In bounded smooth domains $Ω\subset\mathbb{R}^N$, $N\in\{2,3\}$, considering the chemotaxis--fluid system \[ \begin{cases} \begin{split} & n_t + u\cdot \nabla n &= Δn - χ\nabla \cdot(\frac{n}{c}\nabla c) &\\ & c_t + u\cdot \nabla c &= Δc - c + n &\\ & u_t + κ(u\cdot \nabla) u &= Δu + \nabla P + n\nabla Φ& \end{split}\end{cases} \] with singular sensitivity, we prove global existence of classical solutions for given $Φ\in C^2(\barΩ)$, for $κ=0$ (Stokes-fluid) if $N=3$ and $κ\in\{0,1\}$ (Stokes- or Navier--Stokes fluid) if $N=2$ and under the condition that \[ 0<χ<\sqrt{\frac{2}{N}}. \]

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