In this work we extend a recent result to chemotaxis fluid systems which include matrix-valued sensitivity functions $S(x,n,c):Ω\times[0,\infty)^2\to\mathbb{R}^{3\times3}$ in addition to the porous medium type diffusion, which were discussed in the previous work. Namely, we will consider the system \begin{align*} \left\{ \begin{array}{r@{\,}c@{\,}c@{\ }l@{\quad}l@{\quad}l@{\,}c} n_{t}&+&u\cdot\!\nabla n&=Δn^m-\nabla\!\cdot(nS(x,n,c)\nabla c),\ &x\inΩ,& t>0,\\ c_{t}&+&u\cdot\!\nabla c&=Δc-c+n,\ &x\inΩ,& t>0,\\ u_{t}&+&(u\cdot\nabla)u&=Δu+\nabla P+n\nablaϕ,\ &x\inΩ,& t>0,\\ &&\nabla\cdot u&=0,\ &x\inΩ,& t>0, \end{array}\right. \end{align*} in a bounded domain $Ω\subset\mathbb{R}^3$ with smooth boundary. Assuming that $m\geq1$, $α\geq0$ satisfy $m+α>\frac43$, that the matrix-valued function $S(x,n,c):Ω\times[0,\infty)^2\to\mathbb{R}^{3\times3}$ satisfies $|S(x,n,c)|\leq\frac{S_0}{(1+n)^α}$ for some $S_0>0$ and suitably regular nonnegative initial data, we show that the corresponding no-flux-Dirichlet boundary value problem emits at least one global very weak solution. Upon comparison with results for the fluid-free system this condition appears to be optimal. Moreover, imposing a stronger condition for the exponents $m$ and $α$, i.e. $m+2α>\frac{5}{3}$, we will establish the existence of at least one global weak solution in the standard sense.
In this work, we study the Neumann initial-boundary value problem for a two-species chemotaxis system with Lotka-Volterra competition and signal production. Under a rather weak and clean condition of sub-quadratic type damping and merely integrable initial data, we establish the global existence of generalized solutions in an N dimensional bounded and smooth domain.
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 $μ, κ, χ>0$.
In this work, we study global existence, eventual smoothness and asymptotical behavior of positive solutions for a two-species chemotaxis consumption model in a bounded smooth but not necessarily convex domain $Ω\subset \mathbb{R}^n (n=2,3,4,5)$ with nonnegative initial data and homogeneous Neumann boundary data Under a smallness condition, boundedness of classical solutions and stabilization to constant equilibrium is known. Here, without any smallness condition, we show global existence and uniform-in-time boundedness of classical solutions in 2D and global existence, eventual smoothness and…
Does indirectness of signal production reduce the explosion-supporting potential in chemotaxis-haptotaxis systems? Global classical solvability in a class of models for cancer invasion (and more)
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…
We study a chemotaxis-Stokes system with signal consumption and logistic source terms of the form \noindent \begin{align*} \left\{ \begin{array}{r@{\ }l@{\quad}l@{\quad}l@{\,}c} n_{t}+u\cdot\!\nabla n&=Δn-\nabla\!\cdot(n\nabla c)+κn-μn^{2},\ &x\inΩ,& t>0,\\ c_{t}+u\cdot\!\nabla c&=Δc-nc,\ &x\inΩ,& t>0,\\ u_{t}&=Δu+\nabla P+n\nablaϕ,\ &x\inΩ,& t>0,\\ \nabla\cdot u&=0,\ &x\inΩ,& t>0,\\ \big(\nabla n-n\nabla c\big)\cdotν&=0,\quad c=c_{\star}(x),\quad u=0, &x\in\partialΩ,& t>0, \end{array}\right. \end{align*} where $κ\geq0$, $μ>0$ and, in contrast to the commonly investigated variants of chemotaxi…
A chemotaxis system possibly containing rotational components of the cross-diffusive flux is studied under no-flux boundary conditions in a bounded domain $Ω\subset R^n$, $n\ge 1$, with smooth boundary, where the evolution of the signal is determined by consumption through cells. In contrast to related Keller-Segel-type problems with scalar sensitivities, in presence of such tensor-valued sensitivities this system in general apparently does not possess any useful gradient-like structure. Accordingly, its analysis needs to be based on new types of a priori bounds. Using a spatio-temporal $L^2$ …
We study a new nonlocal approach to the mathematical modelling of the Chemotaxis problem, which describes the random motion of a certain population due a substance concentration. Considering the initial-boundary value problem for the fractional hyperbolic Keller-Segel model, we prove the solvability of the problem. The solvability result relies mostly on fractional calculus and kinetic formulation of scalar conservation laws.
In this paper, we study a nutrient-taxis model with porous medium slow diffusion \begin{align*} \left\{ \begin{aligned} &u_t=Δu^m-χ\nabla\cdot(u\nabla v)+ξuv-ρu, \\ &v_t-Δv=-vu+μv(1-v), \end{aligned}\right. \end{align*} in a bounded domain $Ω\subset \mathbb R^3$ with zero-flux boundary condition. It is shown that for any $m>\frac{11}4-\sqrt 3$,the problem admits a global weak solution for any large initial datum. We divide the study into three cases,(i) $ξμ=0, ρ\ge 0$; (ii) $ξμρ>0$; (iii) $ξμ>0$, $ρ=0$. In particular, for Case (i) and Case (ii), the global solutions are uniformly bounded. Subs…
In this paper, we investigate the global solvability of the chemotaxis-Navier-Stokes system on a three-dimensional moving domain of finite depth, bounded below by a rigid flat bottom and bounded above by the free surface. Completing the system with boundary conditions that match the boundary descriptions in the experiments and numerical simulations, we establish the global existence and uniqueness of solutions near a constant state $(0,\hat{c},\mathbf{0})$, where $\hat{c}$ is the saturation value of the oxygen on the free surface. To the best of our knowledge, this is the first analytical work…
We consider a chemotaxis-fluid system involving nonlinear cell diffusion of porous medium type, signal consumption by cells, and rather general, possibly matrix-valued, chemotactic sensitivities. It is shown that if the corresponding diffusion exponent $m$ satisfies $m>7/6$, then for all reasonably regaular initial data an associated initial-boundary value problem in smoothly bounded three-dimensional domains possesses a globally defined weak solution which is bounded. Under a mild additional assumption on the signal consumption rate, it is moreover shown that any nontrivial of these solutions…
We show global existence and boundedness of classical solutions to a virus infection model with chemotaxis in bounded smooth domains of arbitrary dimension and for any sufficiently regular nonnegative initial data and homogeneous Neumann boundary conditions. More precisely, the system considered is \[ \begin{cases}\begin{split} & u_t=Δu - \nabla\cdot(\frac{u}{(1+u)^α}\nabla v) - uw + κ- u, \\ & v_t=Δv + uw - v, \\ & w_t=Δw - w + v, \end{split}\end{cases} \] with $κ\ge 0$, and solvability and boundedness of the solution are shown under the condition that \[ \begin{cases} α> \frac 12 + \frac{n^2…
In this paper we consider the zero-flux chemotaxis-system \begin{equation*} \begin{cases} u_{t}=Δu-\nabla \cdot (u χ(v)\nabla v) & \textrm{in}\quad Ω\times (0,\infty), \\ 0=Δv-v+g(u) & \textrm{in}\quad Ω\times (0,\infty),\\ \end{equation*} in a smooth and bounded domain $Ω$ of $\mathbb{R}^2$. The chemotactic sensitivity $χ$ is a general nonnegative function from $C^1((0,\infty))$ whilst $g$, the production of the chemical signal $v$, belongs to $C^1([0,\infty))$ and satisfies $λ_1\leq g(s)\leq λ_2(1+s)^β$, for all $s\geq 0$, $0\leqβ\leq \frac{1}{2}$ and $0<λ_1\leq λ_2.$ It is established that …
In this study, we consider the following extended attraction chemotaxis system of two species parabolic-parabolic-elliptic type with nonlocal terms \[ \begin{cases} u_t=d_1Δu-χ_1\nabla (u\cdot \nabla w)+u\left(a_0-a_1u-a_2v-a_3\int_Ωu-a_4\int_Ωv\right),\quad x\in Ω\quad\cr v_t=d_2Δv-χ_2\nabla (v\cdot \nabla w)+v\left(b_0-b_1u-b_2v-b_3\int_Ωu-b_4\int_Ωv\right),\quad x\in Ω\quad\cr 0=d_3Δw+k u+lv-λw,\quad x\in Ω\quad\cr \end{cases} \] under homogeneous Neuman boundary conditions in a bounded domain $Ω\subset \mathbb{R}^n(n\ge1)$ with smooth boundary, where $a_0,b_0,\, \,a_1,$ and $ b_2$ are posi…
This paper deals with the quasilinear fully parabolic attraction-repulsion chemotaxis system \begin{align*} u_t=\nabla \cdot (D(u)\nabla u) -\nabla \cdot (G(u)χ(v)\nabla v) +\nabla\cdot(H(u)ξ(w)\nabla w), \quad v_t=d_1Δv+αu-βv, \quad w_t=d_2Δw+γu-δw, \quad x \in Ω,\ t>0, \end{align*} under homogeneous Neumann boundary conditions and initial conditions, where $Ω\subset \mathbb{R}^n$ $(n \ge 1)$ is a bounded domain with smooth boundary, $d_1, d_2, α, β, γ, δ>0$ are constants. Also, the diffusivity $D$, the density-dependent sensitivities $G, H$ fulfill $D(s)=a_0(s+1)^{m-1}$ with $a_0>0$ and $m \…
The chemotaxis system \begin{align*} u_t &= Δu - \nabla \cdot (u\nabla v), \\ v_t &= Δv - uv, \end{align*} is considered under the boundary conditions $\frac{\partial u}{\partialν}- u\frac{\partial v}{\partialν}=0$ and $v=v_\star$ on $\partialΩ$, where $Ω\subset\mathbb{R}^n$ is a ball and $v_\star$ is a given positive constant. In the setting of radially symmetric and suitably regular initial data, a result on global existence of bounded classical solutions is derived in the case $n=2$, while global weak solutions are constructed when $n\in \{3,4,5\}$. This is achieved by analyzing an energy-t…
The chemotaxis system \[ \left\{ \begin{array}{l} u_t = Δu - χ\nabla \cdot (\frac{u}{v}\nabla v), v_t=Δv - v+u, \end{array} \right. \] is considered in a bounded domain $Ω\subset \mathbb{R}^n$ with smooth boundary, where $χ>0$. An apparently novel type of generalized solution framework is introduced within which an extension of previously known ranges for the key parameter $χ$ with regard to global solvability is achieved. In particular, it is shown that under the hypothesis that\[ χ< \left\{ \begin{array}{ll} \infty \qquad & \mbox{if } n=2, \sqrt{8} \qquad & \mbox{if } n=3, \frac{n}{n-2} \qqu…
The chemotaxis-Navier-Stokes system linking the chemotaxis equations \[ n_t + u\cdot\nabla n = Δn - \nabla \cdot (nχ(c)\nabla c) \] and \[ c_t + u\cdot\nabla c = Δc-nf(c) \] to the incompressible Navier-Stokes equations, \[ u_t + (u\cdot\nabla)u = Δu +\nabla P + n \nabla Φ, \qquad \nabla \cdot u = 0, \] is considered under homogeneous boundary conditions of Neumann type for $n$ and $c$, and of Dirichlet type for $u$, in a bounded convex domain $Ω\subset R^3$ with smooth boundary, where $Φ\in W^{1,\infty}(Ω)$, and where $f\in C^1([0,\infty))$ and $χ\in C^2([0,\infty))$ are nonnegative with $f(0…
In this work, we study a chemotaxis-Navier-Stokes model in a two-dimensional setting as below, \begin{eqnarray} \left\{ \begin{array}{llll} \displaystyle n_{t}+\mathbf{u}\cdot\nabla n=Δn-\nabla \cdot(n\nabla c)+f(n), &&x\inΩ,\,t>0,\\ \displaystyle c_{t}+\mathbf{u}\cdot\nabla c=Δc - c+ n, &&x\inΩ,\,t>0,\\ \displaystyle \mathbf{u}_{t}+κ(\mathbf{u}\cdot\nabla)\mathbf{u}=Δ\mathbf{u} +\nabla P+ n\nablaϕ, &&x\inΩ,\,t>0,\\ \displaystyle \nabla\cdot\mathbf{u}=0,&&x\inΩ,\,t>0.\\ \end{array} \right. \end{eqnarray} Motivated by a recent work due to Winkler, we aim at investigating generalized solvability…
We consider the chemotaxis-fluid system \begin{align}\label{star}\tag{$\diamondsuit$} \left\{ \begin{array}{r@{\,}c@{\,}c@{\ }l@{\quad}l@{\quad}l@{\,}c} n_{t}&+&u\cdot\!\nabla n&=Δn^m-\nabla\!\cdot(n\nabla c),\ &x\inΩ,& t>0,\\ c_{t}&+&u\cdot\!\nabla c&=Δc-c+n,\ &x\inΩ,& t>0,\\ u_{t}&+&(u\cdot\nabla)u&=Δu+\nabla P+n\nablaϕ,\ &x\inΩ,& t>0,\\ &&\nabla\cdot u&=0,\ &x\inΩ,& t>0, \end{array}\right. \end{align} in a bounded domain $Ω\subset\mathbb{R}^3$ with smooth boundary and $m>1$. Assuming $m>\frac{4}{3}$ and sufficiently regular nonnegative initial data, we ensure the existence of global solutio…
We study the chemotaxis-Navier-Stokes system \[\left\{\; \begin{aligned} n_t + u\cdot\nabla n &=Δn - \nabla\cdot (nS(x,n,c)\nabla c), &&x\inΩ, t > 0, \\ c_t + u\cdot\nabla c &=Δc - n f(c), && x\in Ω, t > 0, \\ u_t + (u\cdot\nabla) u &=Δu + \nabla P + n \nabla ϕ, \;\;\;\; \nabla\cdot u = 0, && x\inΩ, t > 0 \end{aligned}\right.\tag{$\star$} \] with no-flux boundary conditions for $n$, $c$ in a bounded, convex domain $Ω\subseteq\mathbb{R}^2$ with a smooth boundary, which is motivated by recent modeling approaches from biology for aerobic bacteria suspended in a sessile water drop. We further do n…
Computational Oncology of Chemotaxis-Driven Tumour--Immune Spatial Patterning and Stability
Fujian Medical University（福建医科大学）；Boston University（波士顿大学）；Chinese Academy of Sciences（中国科学院）；University of Chinese Academy of Sciences（中国科学院大学）；Northeastern University（东北大学）
Chemotaxis is the process by which cells behave in a way that follows the chemical gradient. Applications to bacteria growth, tissue inflammation, and vascular tumors provide a focus on optimization strategies. Experiments can characterize the form of possible chemotactic sensitivities. This paper addresses the recovery of the chemotactic sensitivity from these experiments while allowing for nonlinear dependence of the parameter on the state variables. The existence of solutions to the forward problem is analyzed. The identification of a chemotactic parameter is determined by inverse problem t…
We consider the system \[ \left\{ \begin{aligned} u_t &= Δu - χ\nabla \cdot ( \tfrac{u}{v} \nabla v) - uv + ρu - μu^2, \\ v_t &= Δv - v + u v \end{aligned} \right. \tag{$\star$} \] with $ρ\in \mathbb{R}, μ> 0, χ> 0$ in a bounded domain $Ω\subseteq \mathbb{R}^2$ with smooth boundary. While very similar to chemotaxis models from biology, this system is in fact inspired by recent modeling approaches in criminology to analyze the formation of crime hot spots in cities. The key addition here in comparison to similar models is the logistic source term. The central complication this system then prese…
