On the global existence solution for a chemotaxis model
Farhad Hatami, Mohammad Bagher Ghaemi
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This paper has been withdrawn by the authors. Please withdrow this immediately. we have a big great scientific problem in hole of paper. please delet it immediately
This paper has been withdrawn by the authors. We consider the attraction-repulsion chemotaxis system (3 complicated PDEs system) under homogeneous Neumann boundary conditions in a bounded domain Ω with smooth boundary, then the classical solutions to the system are uniformly-in-time bounded. After the local existence and uniqueness of solutions was proved, some priory estimates and proves will be established for the global existence of solutions (see the complete abstract in the PDF version of paper).
We consider a system of reaction-diffusion equations including chemotaxis terms and coming out of the modeling of multiple sclerosis. The global existence of strong solutions to this system in any dimension is proved, and it is also shown that the solution is bounded uniformly in time. Finally, a nonlinear stability result is obtained when the chemotaxis term is not too big. We also perform numerical simulations to show the appearance of Turing patterns when the chemotaxis term is large.
Global existence and boundedness of classical solutions are shown for a parabolic-elliptic chemotaxis system with local sensing when the motility function is assumed to be unbounded at infinity. The cornerstone of the proof is the derivation of $L^\infty$-estimates on the second component of the system and is achieved by various comparison arguments.
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.
In this article, we study a mathematical system which models the dynamic of the collective behaviour of oxygen-driven swimming bacteria in an aquatic fluid flowing in a two dimensional bounded domain under stochastic perturbation. This model can be seen as a stochastic version of Chemotaxis-Navier-Stokes model. We prove the existence of a unique (probabilistic) strong solution. In addition, we establish some properties of the strong solution. More precisely, we prove that the unique solution is non-negative and satisfies the mass conservation property and an energy inequality.
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 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 $μ(\cdot).$
In this paper, we study the consumption-chemotaxis-Stokes model with porous medium slow diffusion in a three dimensional bounded domain with zero-flux boundary conditions and no-slip boundary condition. In recent ten years, many efforts have been made to find the global bounded solutions of chemotaxis-Stokes system in three dimensional space. Although some important progress has been carried out in some papers, as mentioned by some authors, the question of identifying an optimal condition on m ensuring global boundedness in the three-dimensional framework remains an open challenge. In the pres…
In this work, we study the Neumann initial boundary value problem for a three-component chemotaxis model in any dimensional bounded and smooth domains; this model is used to describe the branching of capillary sprouts during angiogenesis. First, we find three qualitatively simple sufficient conditions for qualitative global boundedness, and then, we establish two types of global stability for bounded solutions in qualitative ways. As a consequence of our findings, the underlying system without chemotaxis and the effect of ECs mitosis can not give rise to pattern formations. Our findings quanti…
Global existence is established for classical solutions to a chemotaxis model with signal-dependent motility for a general class of motility functions $γ$ which may in particular decay in an arbitrary way at infinity. Assuming further that $γ$ is non-increasing and decays sufficiently slowly at infinity, in the sense that $γ(s)\sim s^{-k}$ as $s\to\infty$ for some $k\in (0,N/(N-2)_+)$, it is also shown that global solutions are uniformly bounded with respect to time. The admissible decay of $γ$ at infinity here is higher than in previous works.
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…
This paper studies the dynamical behavior of classical solutions to a hyperbolic system of balance laws, derived from a chemotaxis model with logarithmic sensitivity, subject to time-dependent boundary conditions. It is shown that under suitable assumptions on the boundary data, solutions starting in $H^2$-space exist globally in time and the differences between the solutions and their corresponding boundary data converge to zero, as time goes to infinity. There is no smallness restriction on the magnitude of initial perturbations. Moreover, numerical simulations show that the assumptions on t…
This paper is devoted to global existence of weak solutions to the following degenerate kinetic model of chemotaxis \begin{equation} \begin{cases}\label{chemo0} u_t=Δ(γ(v)u) τv_{t}=Δv-v+u \end{cases} \end{equation}in a smooth bounded domain with no-flux boundary conditions. The problem features a positive signal-dependent motility function $γ(\cdot)$ which may vanish as $v$ becomes unbounded. In this paper, we first modify the comparison approach developed recently in \cite{GM,GM2} to derive the upper bounds of $v$ under weakened assumptions on $γ(\cdot)$. Then by introducing a suitable approx…
Global existence for a kinetic model of chemotaxis via dispersion and Strichartz estimates
我们进一步研究了趋化性动力学模型解的存在性。这些是具有二次非线性的非线性传输散射方程，自80年代以来，当实验观察表明它们通过一系列“运行和翻滚”移动时，这些方程已被用于描述细菌的运动。解的存在性已经在几篇论文中得到[Chalub et al. Monatsh. 142，123- 141（2004），Hwang等人，SIAM J.Math.Anal. 36，1177- 1199（2005）]使用直接和强色散效应。在这里，我们使用[Castella et al. C. R. Acad. Sci. Paris 322，535- 540（1996）]证明了依赖于转动核的各种情况下的全局存在性。在最困难的情况下，当翻滚前后的速度都出现时，用已知的方法，只有在小的假设下，哈茨估计才能给出结果。
In this paper, we investigate a chemotaxis-fluid system involving both the effect of potential force on cells and the effect of chemotactic force on fluid: \begin{equation*} \left\{ \begin{split} \partial_t n + \mathbf{u}\cdot\nabla n & = Δn - \nabla\cdot\left(χ(c)n\nabla c\right) + \nabla\cdot(n\nablaϕ), \\ \partial_t c + \mathbf{u}\cdot\nabla c &= Δc - nf(c), \\ \partial_t\mathbf{u} + κ(\mathbf{u}\cdot\nabla)\mathbf{u} + \nabla P & = Δ\mathbf{u} - n\nablaϕ+ χ(c)n\nabla c, \\ \nabla\cdot\mathbf{u} &= 0 \end{split} \right. \end{equation*} in $\mathbb{R}^d\times(0,T)\, (d=2,3)$. One of the nove…
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…
Global existence and boundedness for a degenerate chemotaxis system with indirect signal production via minimizing movement schemes
Osaka Metropolitan University（大阪公立大学）；Institut de Mathématiques de Toulouse (IMT) UMR5219, Université Toulouse III - Paul Sabatier（图卢兹第三大学-保罗·萨巴蒂尔大学）；Université Savoie Mont Blanc, CNRS（萨瓦-蒙布朗大学）
This paper deals with the fully parabolic chemotaxis system of local sensing in higher dimensions. Despite the striking similarity between this system and the Keller--Segel system, we prove the absence of finite-time blow-up phenomenon in this system even in the supercritical case. It means that for any regular initial data, independently of the magnitude of mass, the classical solution exists globally in time in the higher dimensional setting. Moreover, for the exponential decaying motility case, it is established that solutions may blow up at infinite time for any magnitude of mass. In order…
In this paper, we consider two species chemotaxis systems with Lotka-Volterra competition reaction terms. Under appropriate conditions on the parameters in such a system, we establish the existence of traveling wave solutions of the system connecting two spatially homogeneous equilibrium solutions with wave speed greater than some critical number c*. We also show the non-existence of such traveling waves with speed less than some critical number c*_0, which is independent of the chemotaxis. Moreover, under suitable hypotheses on the coefficients of the reaction terms, we obtain explicit range …
