In this paper, we study the initial-boundary value problem and its asymptotic behavior for a repulsive chemotaxis model with logarithmic sensitivity and logistic growth. We establish global well-posedness of strong solutions for large initial data with Neumann boundary conditions and, moreover, establish the qualitative result that both the population density and chemical concentration asymptotically converge to constant states with the population density specifically converging to its carrying capacity. We additionally prove that the vanishing chemical diffusivity limit holds in this regime. Lastly, we provide numerical confirmation of the rigorous qualitative results, as well as numerical simulations that demonstrate a separation of scales phenomenon.
This paper is concerned with a parabolic-elliptic chemotaxis model with density-suppressed motility and general logistic source in an $n$-dimensional smooth bounded domain with Neumann boundary conditions. Under the minimal conditions for the density-suppressed motility function, we explore how strong the logistic damping can warrant the global boundedness of solutions, and further establish the asymptotic behavior of solutions on top of the conditions.
It is well-known that the Neumann initial-boundary value problem for the minimal-chemotaxis-logistic system in a 2D bounded smooth domain has no blow-up for any choice of parameters. Here, for a large class of kinetic terms including sub-logistic sources, we show that the corresponding 2D Neumann initial-boundary value problems do not possess any blow-up. This illustrates a new phenomenon that even a class of sub-logistic sources can prevent blow-up for the 2D problem, indicating that logistic damping is not the weakest damping to guarantee uniform-in-time boundedness for the 2D minimal Keller…
We consider the Neumann problem for a coupled chemotaxis-haptotaxis model of cancer invasion with/without kinetic source in a 2D bounded and smooth domain. For a large class of cell kinetic sources including zero source and sub-logistic sources, we detect an explicit condition involving the chemotactic strength, the asymptotic "damping" rate, and the initial mass of cells to ensure uniform-in-time boundedness for the corresponding Neumann problem. Our finding significantly improves existing 2D global existence and boundedness in related chemotaxis-/haptotaxis systems.
In this paper, we study the minimal Keller-Segel model with a logistic source and obtain quantitative and qualitative descriptions of the competition between logistic damping and other ingredient, especially, chemotactic aggregation to guarantee boundedness and convergence. More specifically, we establish how precisely strong a logistic source can prevent blow-up, and then we obtain an explicit relationship between logistic damping and other ingredient, especially, chemotactic aggregation so that convergences are ensured and their respective convergence rates are explicitly calculated out. Kno…
We prove existence of global weak solutions to the chemotaxis system $ u_t=Δu - \nabla\cdot (u\nabla v) +κu -μu^2 $ $ v_t=Δv-v+u $ under homogeneous Neumann boundary conditions in a smooth bounded convex domain $Ω\subset R^n$, for arbitrarily small values of $μ>0$. Additionally, we show that in the three-dimensional setting, after some time, these solutions become classical solutions, provided that $κ$ is not too large. In this case, we also consider their large-time behaviour: We prove decay if $κ\leq 0$ and the existence of an absorbing set if $κ>0$ is sufficiently small. Keywords: chemotaxi…
Introducing a suitable solution concept, we show that in bounded smooth domains $Ω\subset \mathbb{R}^n$, $n\ge 1$, the initial boundary value problem for the chemotaxis system \begin{align*} u_t&=Δu -χ\nabla\cdot\left(\frac{u}{v}\nabla v\right)+κu -μu^2,\\ v_t&=Δv -uv, \end{align*} with homogeneous Neumann boundary conditions and widely arbitrary initial data has a generalized global solution for any $μ, κ, χ>0$.
We study the following Keller-Segel chemotaxis system with logistic source and nonlinear secretion: \begin{align*} u_t=Δu- \nabla\cdot(u\nabla v)+κ(|x|)u-μ(|x|)u^p\quad\text{and}\quad 0=Δv-v+u^γ, \end{align*} where $κ(\cdot),~μ(\cdot):[0,R]\rightarrow [0,\infty)$, $γ\in (1,\infty)$, $p\in(γ+1,\infty)$ and $Ω\subset \mathbb{R}^n, n\geq 2$. For this system, we prove the global existence of solutions under suitable assumptions on the initial condition and the functions $κ(\cdot)$ and $μ(\cdot).$
We consider the coupled chemotaxis Navier-Stokes model with logistic source terms \[ n_t + u\cdot \nabla n = Δn - χ\nabla \cdot (n \nabla c) + κn - μn^2\] \[ c_t + u\cdot \nabla c = Δc - nc\] \[ u_t + (u\cdot \nabla)u = Δu +\nabla P + n\nabla Φ+ f, \quad\qquad \nabla \cdot u=0 \] in a bounded, smooth domain $Ω\subset \mathbb{R}^3$ under homogeneous Neumann boundary conditions for $n$ and $c$ and homogeneous Dirichlet boundary conditions for $u$ and with given functions $f\in L^\infty(Ω\times(0,\infty))$ satisfying certain decay conditions and $Φ\in C^{1+β}(\barΩ)$ for some $β\in(0,1)$. We cons…
We study the stationary Keller--Segel chemotaxis models with logistic cellular growth over a one-dimensional region subject to the Neumann boundary condition. We show that nonconstant solutions emerge in the sense of Turing's instability as the chemotaxis rate $χ$ surpasses a threshold number. By taking the chemotaxis rate as the bifurcation parameter, we carry out bifurcation analysis on the system to obtain the explicit formulas of bifurcation values and small amplitude nonconstant positive solutions. Moreover we show that solutions stay strictly positive in the continuum of each branch. The…
In this paper, we study chemotaxis effect vs logistic dampening on boundedness for the two-dimensional minimal Keller-Segel model with logistic source in a 2-D smooth and bounded domain. It is well-known that this model allows only for global and uniform-in-time bounded solutions for any chemotactic strength and logistic dampening. Here, we carefully employ a simple and new method to regain its boundedness and, with particular attention to how boundedness depends qualitatively on the coefficient of chemotactic strength and logistic dampening rate. Up to a scaling constant depending only on ini…
We consider an attraction-repulsion chemotaxis model coupled with the Navier-Stokes system. This model describes the interaction between a type of cells (e.g., bacteria), which proliferate following a logistic law, and two chemical signals produced by the cells themselves that degraded at a constant rate. Also, it is considered that the chemoattractant is consumed with a rate proportional to the amount of organisms. The cells and chemical substances are transported by a viscous incompressible fluid under the influence of a force due to the aggregation of cells. We prove the existence of global…
Assuming that $0<χ<\sqrt{\frac{2}n}$, $κ\ge 0$ and $μ>\frac{n-2}{n}$, we prove global existence of classical solutions to a chemotaxis system slightly generalizing \[ \begin{split} u_t &= Δu - χ\nabla\cdot ( \frac{u}{v} \nabla v ) + κu -μu^2\\ v_t &= Δv - u v \end{split} \] in a bounded domain $Ω\subset \mathbb{R}^n$, with homogeneous Neumann boundary conditions and for widely arbitrary positive initial data. In the spatially one-dimensional setting, we prove global existence and, moreover, boundedness of the solution for any $χ>0$, $μ>0$, $κ\ge 0$.
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 syst…
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…
This paper is concerned with the attraction-repulsion chemotaxis system with superlinear logistic degradation, \begin{align*} \begin{cases} u_t = Δu - χ\nabla\cdot(u \nabla v) + ξ\nabla\cdot (u \nabla w) + λu - μu^k, \quad &x \in Ω,\ t>0,\\[1.05mm] 0= Δv + αu - βv, \quad &x \in Ω,\ t>0,\\[1.05mm] 0= Δw + γu - δw, \quad &x \in Ω,\ t>0, \end{cases} \end{align*} under homogeneous Neumann boundary conditions, in a ball $Ω\subset \mathbb{R}^n$ ($n \ge 3$), with constant parameters $λ\in \mathbb{R}$, $k>1$, $μ, χ, ξ, α, β, γ, δ>0$. Blow-up phenomena in the system have been well investigated in the c…
This paper is concerned with the well-posedness and large-time behavior of a two dimensional PDE-ODE hybrid chemotaxis system describing the initiation of tumor angiogenesis. We first transform the system via a Cole-Hopf type transformation into a parabolic-hyperbolic system and show that the solution of the transformed system converges to a constant equilibrium state as time tends to infinity. Finally we reverse the Cole-Hopf transformation and obtain the relevant results for the pre-transformed PDE-ODE hybrid system. In contrast to the existing related results, where continuous initial datum…
We consider a parabolic-elliptic system of partial differential equations with chemotaxis and logistic growth given by the system $$ \left\{ \begin{array}{l} u_t -Δ(u γ(v)= μu(1-u), \\ - Δv +v=u, \end{array} \right. $$ under Neumann boundary conditions and appropriate initial data in a bounded and regular domain $Ω$ of $\R^N$ (for $N \geq 1)$, where $γ\in C^3([0, \infty))$ and satisfies the assumptions $γ(s) > 0$, $γ^{\prime}(s) \leq 0$, $γ^{\prime \prime} (s) \geq 0$, $γ^{\prime \prime \prime}(s) \leq 0$ for any $s \geq 0$ $$-2 γ^{\prime}(s) + γ^{\prime \prime}(s)s \leq μ_0< μ$$ $$\frac{[γ^{\…
In present paper, we consider a chemotaxis consumption system with density-signal governed sensitivity and logistic source: $u_t=Δu-\nabla\cdot(\frac{S(u)}{v}\nabla v)+ru-μu^2$, $v_t=Δv-uv$ in a smooth bounded domain $Ω\subset\mathbb{R}^n$ $(n\ge2)$, where parameters $r,μ>0$ and density governed sensitivity fulfills $ S(u) \simeq u(u+1)^{β-1}$ for all $u\ge0$ with $β\in \mathbb{R}$. It is proved that for any $r,μ>0$, there exists a global classical solution if $β<1$ and $n\ge2$. Moreover, the global boundedness and the asymptotic behavior of the classical solution are determined for the case $…
This paper is concerned with the asymptotic stability of the initial-boundary value problem of a singular PDE-ODE hybrid chemotaxis system in the half space $\R_+=[0, \infty)$. We show that when the non-zero flux boundary condition at $x=0$ is prescribed and the initial data are suitably chosen, the solution of the initial-boundary value problem converges, as time tend to infinity, to a shifted traveling wavefront restricted in the half space $[0,\infty)$ where the wave profile and speed are uniquely selected by the boundary flux data. The results are proved by a Cole-Hopf type transformation …
This paper is concerned with traveling wave solutions of the following full parabolic Keller-Segel chemotaxis system with logistic source, \begin{equation} \begin{cases} u_t=Δu -χ\nabla\cdot(u\nabla v)+u(a-bu),\quad x\in\mathbb{R}^N \cr τv_t=Δv-λv +μu,\quad x\in \mathbb{R}^N, \end{cases}(1) \end{equation} where $χ, μ,λ,a,$ and $b$ are positive numbers, and $τ\ge 0$. Among others, it is proved that if $b>2χμ$ and $τ\geq \frac{1}{2}(1-\fracλ{a})_{+} ,$ then for every $c\ge 2\sqrt{a}$, (1) has a traveling wave solution $(u,v)(t,x)=(U^{τ,c}(x\cdotξ-ct),V^{τ,c}(x\cdotξ-ct))$ ($\forall\, ξ\in\mathbb…
The current paper is concerned with pointwise persistence in full chemotaxis models with local as well as nonlocal time and space dependent logistic source in bounded domains. We first prove the global existence and boundedness of nonnegative classical solutions under some conditions on the coefficients in the models. Next, under the same conditions on the coefficients, we show that pointwise { persistence} occurs, that is, any globally defined positive solution is bounded below by a positive constant independent of its initial condition when the time is large enough. It should be pointed out …
In this paper, we study the parabolic-elliptic Keller-Segel system with singular sensitivity and logistic-type source: $ u_t=Δu-χ\nabla\cdot(\frac{u}{v}\nabla v)+ru-μu^k$, $0=Δv-v+u$ under the non-flux boundary conditions in a smooth bounded convex domain $Ω\subset\mathbb{R}^n$, $χ,r,μ>0$, $k>1$ and $n\ge 2$. It is shown that the system possesses a globally bounded classical solution if $k>\frac{3n-2}{n}$, and $r>\frac{χ^2}{4}$ for $0<χ\le 2$, or $r> χ-1$ for $χ>2$. In addition, under the same condition for $r,χ$, the system admits a global generalized solution when $k\in(2-\frac{1}{n},\frac{3…
In this paper we investigate pattern formation in Keller--Segel chemotaxis models over a multi--dimensional bounded domain subject to homogeneous Neumann boundary conditions. It is shown that the positive homogeneous steady state loses its stability as chemoattraction rate $χ$ increases. Then using Crandall--Rabinowitz local theory with $χ$ being the bifurcation parameter, we obtain the existence of nonhomogeneous steady states of the system which bifurcate from this homogeneous steady state. Stability of the bifurcating solutions is also established through rigorous and detailed calculations.…
In this paper, we study the spatial propagation dynamics of a parabolic-elliptic chemotaxis system with logistic source which reduces to the well-known Fisher-KPP equation without chemotaxis. It is known that for fast decaying initial functions, this system has a finite spreading speed. For slowly decaying initial functions, we show that the accelerating propagation will occur and chemotaxis does not affect the propagation mode determined by slowly decaying initial functions if the logistic damping is strong, that is, the system has the same upper and lower bounds of the accelerating propagati…
This article deals with an initial-boundary value problem for the coupled chemotaxis-haptotaxis system with nonlinear diffusion $$\left\{\begin{array}{ll} u_t=\nabla\cdot( D(u)\nabla u)-χ\nabla\cdot(u\nabla v)- ξ\nabla\cdot(u\nabla w)+μu(1- u-w), x\in Ω, t>0,\\ τv_t=Δv- v +u,\quad x\in Ω, t>0,\\ w_t=- vw,\quad x\in Ω, t>0, \end{array}\right.$$ under homogeneous Neumann boundary conditions in a smooth bounded domain $Ω\subset\mathbb{R}^N(N\geq1)$, where $τ\in\{0,1\}$ and $χ$, $ξ$ and $μ$ are given nonnegative parameters. As far as we know, this situation provides the first {\bf rigorous} result…
This paper deals with the quasilinear parabolic-elliptic chemotaxis system with logistic source and nonlinear production, \begin{equation*} \begin{cases} u_t=\nabla \cdot (D(u) \nabla u) - \nabla \cdot (S(u)\nabla v) + λu - μu^κ, & x\inΩ,\ t>0, \\[1mm] 0=Δv - \overline{M_f}(t) + f(u), & x\inΩ,\ t>0, \end{cases} \end{equation*} where $λ>0$, $μ>0$, $κ>1$ and $\overline{M_f}(t):=\frac{1}{|Ω|}\int_Ω f(u(x,t))\,dx$, and $D$, $S$ and $f$ are functions generalizing the prototypes \begin{align*} D(u)=(u+1)^{m-1},\quad S(u)=u(u+1)^{α-1}\quad\mbox{and}\quad f(u)=u^\ell \end{align*} with $m\in\mathbb{R}$…
This paper is devoted to investigate the pattern formation of a volume-filling chemotaxis model with logistic cell growth. We first apply the local stability analysis to establish sufficient conditions of destabilization for uniform steady-state solution. Then, weakly nonlinear analysis with multi-scales is used to deal with the emerging process of patterns near the bifurcation point. For the single unstable mode case, we derive the Stuart-Landau equations describing the evolution of the amplitude, and thus the asymptotic expressions of patterns are obtained in both supercritical case and subc…
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 som…
The current paper is concerned with the spreading speeds of the following parabolic-parabolic chemotaxis model with logistic source on $\mathbb{R}^{N}$, \begin{equation} \begin{cases} u_t=Δu-χ\nabla\cdot ( u\nabla v) + u(a-bu),\quad x\in\mathbb{R}^{N},\cr {v_t}=Δv -λv+μu,\quad x\in \mathbb{R}^{N}. \end{cases}(1) \end{equation} where $χ, \ a,\ b,\ λ,\ μ$ are positive constants. Assume $b>\frac{Nμχ}{4}$. Among others, it is proved that $2\sqrt{a}$ is the spreading speed of the global classical solutions of (1) with nonempty compactly supported initial functions, that is, $$ \lim_{t\to\infty}\sup…
Uniqueness and nonlinear stability of positive entire solutions in parabolic-parabolic chemotaxis models with logistic source on bounded heterogeneous environments
The chemotaxis--Navier--Stokes system \begin{equation*}\label{0.1} \left\{\begin{array}{ll} n_t+u\cdot \nabla n=\triangle n-χ\nabla\cdotp \left(\displaystyle\frac n {c}\nabla c\right)+n(r-μn), c_t+u\cdot \nabla c=\triangle c-nc, u_t+ (u\cdot \nabla) u=Δu+\nabla P+n\nablaϕ, \nabla\cdot u=0, \end{array}\right. \end{equation*} is considered in a bounded smooth domain $Ω\subset \mathbb{R}^2$, where $ϕ\in W^{1,\infty}(Ω)$, $χ>0$, $r\in \mathbb{R}$ and $μ> 0$ are given parameters. It is shown that there exists a value $μ_*(Ω,χ, r)\geq 0$ such that whenever $ μ>μ_*(Ω,χ, r)$, the global-in-time classi…
In this paper, we study the following chemotaxis--haptotaxis system with (generalized) logistic source $$ \left\{\begin{array}{ll} u_t=Δu-χ\nabla\cdot(u\nabla v)- ξ\nabla\cdot(u\nabla w)+u(a-μu^{r-1}-w), \displaystyle{v_t=Δv- v +u},\quad \\ \displaystyle{w_t=- vw},\quad\\ \displaystyle{\frac{\partial u}{\partial ν}=\frac{\partial v}{\partial ν}=\frac{\partial w}{\partial ν}=0},\quad x\in \partialΩ, t>0,\\ \displaystyle{u(x,0)=u_0(x)},v(x,0)=v_0(x),w(x,0)=w_0(x),\quad x\in Ω, \end{array}\right.\eqno(0.1) $$ %under homogeneous Neumann boundary conditions in a smooth bounded domain $\mathbb{R}^N(…
A semilinear version of parabolic-elliptic Keller-Segel system with the \emph{critical} nonlocal diffusion is considered in one space dimension. We show boundedness of weak solutions under very general conditions on our semilinearity. It can degenerate, but has to provide a stronger dissipation for large values of a solution than in the critical linear case or we need to assume certain (explicit) data smallness. Moreover, when one considers a logistic term with a parameter $r$, we obtain our results even for diffusions slightly weaker than the critical linear one and for arbitrarily large init…
In this paper we study the zero-flux chemotaxis-system \begin{equation*} \begin{cases} u_{ t}=\nabla \cdot ((u+1)^{m-1} \nabla u-(u+1)^αχ(v)\nabla v) + ku-μu^2 & x\in Ω, t>0, \\ v_{t} = Δv-vu & x\in Ω, t>0,\\ \end{cases} \end{equation*} $Ω$ being a bounded and smooth domain of $\mathbb{R}^n$, $n\geq 1$, and where $m,k \in \mathbb{R}$, $μ>0$ and $α< \frac{m+1}{2}$. For any $v\geq 0$ the chemotactic sensitivity function is assumed to behave as the prototype $χ(v) = \frac{χ_0}{(1+av)^2}$, with $a\geq 0$ and $χ_0>0$. We prove that for nonnegative and sufficiently regular initial data $u(x,0)$ and …
In the current paper, we consider the following parabolic-parabolic chemotaxis system with logistic source on $\mathbb{R}^{N}$, \begin{equation} \begin{cases} u_t=Δu-χ\nabla\cdot ( u\nabla v) + u(a-bu),\quad x\in\mathbb{R}^{N}\,\,\, t>0\cr {v_t}=Δv -λv+μu,\quad x\in \mathbb{R}^{N}\,\,\, t>0 \end{cases}(1) \end{equation} where $χ, \ a,\ b,\ λ,\ μ$ are positive constants and $N$ is a positive integer. We investigate the persistence and convergence in (1). To this end, we first prove, under the assumption $b>\frac{Nχμ}{4}$, the global existence of a unique classical solution $(u(x,t;u_0, v_0),v(x…
This work is the second of the series of three papers devoted to the study of asymptotic dynamics in the chemotaxis system with space and time dependent logistic source,$$\partial_tu=Δu-χ\nabla\cdot(u\nabla v)+u(a(x,t)-ub(x,t)),\quad 0=Δv-λv+μu ,\ x\in\mathbb{R}^N, \ (1) $$where $N\ge1$ is a positive integer, $χ,λ,μ>0$, and the functions $a(x,t), b(x,t)$ are positive and bounded. In the first of the series, we studied the phenomena of pointwise and uniform persistence, and asymptotic spreading for solutions with compactly supported or front like initials. In the second of the series, we invest…
Global boundedness, absorbing sets and mass persistene in two-dimensional chemotaxis-Navier-Stokes systems with weakly singular sensitivity and sub-logistic sources
Institute for Theoretical Sciences, Westlake University（西湖大学理论科学研究所）
In this work we analyse a PDE-ODE problem modelling the evolution of a Glioblastoma, which includes chemotaxis term directed to vasculature. First, we obtain some a priori estimates for the (possible) solutions of the model. In particular, under some conditions on the parameters, we obtain that the system does not develop blow-up at finite time. In addition, we design a fully discrete finite element scheme for the model which preserves some pointwise estimates of the continuous problem. Later, we make an adimensional study in order to reduce the number of parameters. Finally, we detect the mai…
