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Special Session 81: Reaction-(cross-)diffusion models in mathematical biology


Critical mass in quasilinear Keller-Segel systems

Xinru Cao

Donghua University
Peoples Rep of China

Co-Author(s):

Abstract:

We present some results in quasilinear Keller-Segel systems.

Shrinking vs. expanding: the evolution of spatial support in degenerate Keller-Segel systems

Mario Fuest

Leibniz University Hannover
Germany

Co-Author(s): Frederic Heihoff

Abstract:

We consider radially symmetric solutions to a degenerate parabolic--elliptic Keller--Segel system in bounded balls with initial data having compact support. Our main result shows that the initial evolution of the positivity set is essentially completely determined by the flatness/steepness of the initial data near a boundary point

$x_0$ of the support. If they are sufficiently flat (respectively, steep), the support shrinks (respectively, expands) near

$x_0$ . We give concrete conditions for both behaviors and in particular show that there is a critical exponent and a critical parameter distinguishing between these cases. The proof is based on constructing suitable sub- and supersolutions to a transformed problem.

Long time dynamics for the Cauchy problem of the predator-prey model with cross-diffusion

Chunhua Jin

South China Normal University
Peoples Rep of China

Co-Author(s): Yifu Wang

Abstract:

In this talk, we focuses on the long-time dynamic behavior of the Cauchy problem related to a pursuit-evasion predator-prey model in

$N$ -dimensional spaces with

$1\le N\le 3$ . The system clearly adheres to the law of mass conservation, as evidenced by the fact that the

$L^1$ - norm remains constant. Our findings reveal that any global strong solution of this system converges to to the heat kernel in the sense of

$L^p$ -norm for any

$1\le p\le \infty$ . We also provide estimates on the decay rate of the solution, and obtain estimates on the decay rate of the solution that are consistent with those of the heat equation in

$\mathbb R^N$ (

$N=2, 3$ ), indicating their optimality. However, unfortunately, for one-dimensional case, despite our attempts to provide decay rate estimates, it is evident that this rate is not optimal. Additionally, as a supplementary result, we also verify the global existence and long-time asymptotic behavior of strong solutions for small initial values.

Taxis models on an ecological scale

Johannes Lankeit

Leibniz University Hannover
Germany

Co-Author(s):

Abstract:

In this talk, I will present systems with chemotaxis-like cross-diffusion, whose interpretation stems from ecology. One system of particular interest to this talk involves so-called `alarm-taxis`. I will discuss questions of solvability (in different senses).

Traveling waves to a logarithmic chemotaxis model with fast diffusion and singularities

Jingyu Li

Northeast Normal University
Peoples Rep of China

Co-Author(s): Xiaowen Li, Dongfang Li, Ming Mei

Abstract:

We are concerned with a chemotaxis model with logarithmic sensitivity and fast diffusion, which possesses strong singularities for the sensitivity at zero-concentration of chemical signal, and for the diffusion at zero-population of cells, respectively. The main purpose is to show the existence of traveling waves connecting the singular zero-end-state, and particularly, to show the asymptotic stability of these traveling waves. The challenge of the problem is the interaction of two kinds of singularities involved in the model: one is the logarithmic singularity of the sensitivity; and the other is the power-law singularity of the diffusivity. To overcome the singularities for the wave stability, some new techniques of weighted energy method are introduced artfully.

Critical blow-up exponent in a nonlinear chemotaxis system with indirect signal production

Yuxiang Li

School of Mathematics, Southeast University
Peoples Rep of China

Co-Author(s): Taian Jin, Jianlu Yan

Abstract:

In this talk, we investigate the Neumann initial-boundary value problem for the following nonlinear chemotaxis system with indirect signal production:

$\label{0}\tag{$\star$} \begin{cases} u_t = \Delta u - \nabla \cdot\left(f(u) \nabla v\right), \ 0 = \Delta v - \mu(t) + w, \ w_t + w = u \end{cases}$ in

$\Omega \subset \mathbb{R}^n$ for

$n \geq 2$ . Here,

$\mu(t) := \fint_{\Omega} w(x, t) \,\mathrm{d}x$ and

$f \in C^2([0,\infty))$ is a nonnegative function. We establish the following results: \begin{itemize} \item If

$\Omega = B_R(0)$ for some

$R > 0$ and

$f(s) \geq k s^p$ for all

$s \geq 1$ with constants

$k > 0$ and

$p > \frac{2}{n}$ , then there exist radially symmetric initial data for which the corresponding solution blows up in finite time, regardless of the mass level

$m := \int_{\Omega} u_0 \,\mathrm{d}x > 0$ . \item If

$f(s) \leq K (s + 1)^p$ for all

$s \geq 0$ with constants

$K > 0$ and

$p < \frac{2}{n}$ , then for any appropriately regular initial data, the corresponding solution exists globally and remains bounded. \end{itemize} Our findings extend the results of Tao and Winkler (2017) regarding blow-up phenomena for

$\eqref{0}$ with

$f(u) = u$ in the two-dimensional setting.

Global classical solutions to a triply haptotactic cross-diffusion system modeling oncolytic virotherapy

Suying Liu

Northwestern Polytechnical University
Peoples Rep of China

Co-Author(s): Xueli Bai, Fang Li, Jiale Shi

Abstract:

In this talk, we consider a triply haptotactic cross-diffusion system, which is proposed to describe the interaction among uninfected cancer cells, infected cancer cells, extracellular matrix (ECM) and oncolytic virus particles in oncolytic virotherapy. Our main result asserts that, in the two dimensional domain, an associated initial-boundary value problem has a unique classical solution, which exists globally and is uniformly bounded under suitable assumptions on the parameters and initial data of the system.

The Keller-Segel-Navier-Stokes system in bounded Lipschitz domains

Patrick Tolksdorf

Karlsruhe Institute of Technology
Germany

Co-Author(s): Matthias Hieber, Hideo Kozono, Sylvie Monniaux

Abstract:

We study the coupled Keller-Segel-Navier-Stokes system in bounded Lipschitz domains. It is shown that the system admits local strong as well as global strong solutions for small data in the setting of critical Besov spaces. Moreover, non-trivial equilibria are shown to be exponentially stable. For smoother data, these solutions are shown to be globally bounded and to preserve positivity properties. The approach is based on optimal

$L^q$ -regularity properties of the Neumann Laplacian and the Stokes operator on bounded Lipschitz domains.

The qualitative analysis to a doubly degenerate chemotaxis-consumption system on non-convex domain

Duan Wu

Paderborn university
Peoples Rep of China

Co-Author(s): Tobias Black, Shohei Kohatsu

Abstract:

In this talk, we consider a doubly degenerate chemotaxis-consumption system in non-convex domain when

$N\ge 2$ . We prove that for any suitably regular initial data, this system admits global bounded weak solutions. Furthermore, we also study the large time behavior for the solutions we obtained by using the Moser iteration technique and obtaining a new Harnack-type inequality.

Global existence and stabilization of solutions to a Keller-Segel-(Navier-)Stokes system with prescribed signal concentration on the boundary

Zhaoyin Xiang

University of Electronic Science and Technology of China
Peoples Rep of China

Co-Author(s): Yifei Sun, Yu Tian

Abstract:

In this talk, we focus on the Keller-Segel-(Navier-)Stokes system with tensor-valued sensitivity and logistic source in a bounded domain

$\Omega\subset\mathbb{R}^N$ subject to no-flux/Dirichlet/Dirichlet boundary conditions for cells/signal/fluid, respectively. We will show that when

$N=2$ , the Keller-Segel-Navier-Stokes system admits a global bounded classical solution for any regular initial data. When

$N=3$ similar conclusion holds for the Keller-Segel-Stokes system provided that the logistic damping is strong enough in some sense. We will also give some stabilization analysis under some additional assumptions. This is a joint work with Dr Yifei Sun and Dr Yu Tian.

Boundedness and finite-time blow-up in a repulsion-consumption system with nonlinear chemotactic sensitivity

Ziyue Zeng

School of Mathematics, Southeast University
Peoples Rep of China

Co-Author(s): Yuxiang Li

Abstract:

In this presentation, we will discuss recent progress for a repulsion-consumption system with nonlinear chemotactic sensitivity. The consumption system differs from Keller-Segel production systems, and the literature on solutions for consumption systems primarily focuses on global existence. Inspired by [J. Ahn and M. Winkler, Calc. Var. 64 (2023).] and [Y. Wang and M. Winkler, Proc. Roy. Soc. Edinburgh Sect. A, 153 (2023).], we investigate the effect of the nonlinear chemotactic sensitivity

$S(u)=(1+u)^\beta$ on the occurrence of blow-up phenomenon for the repulsion-consumption parabolic-elliptic system and establish the boundedness of solutions for the repulsion-consumption system to find the critical exponent. Under radially symmetric assumptions, we prove that 1) The signal consumption equation is elliptic and

$n=2$ . For

$\beta>1$ and the boundary signal level large enough, the corresponding radially symmetric solution blows up in finite time. 2) When the signal consumption equation is elliptic or parabolic and

$n \geq 2$ . For

$\beta \in\left(0, \frac{n+2}{2 n}\right)$ the problem

$(\star)$ possesses a global bounded classical solution.

Global well-posedness for the 2D chemotaxis-Euler system with logistic source for large initial data

Qian Zhang

Hebei University
Peoples Rep of China

Co-Author(s): Peiguang Wang

Abstract:

In this paper, the two-dimensional incompressible chemotaxis-Euler system with logical source is studied as following:

$\nonumber\,\, \left\{ \begin{aligned} &n_{t}+u\cdot\nabla n=\Delta n-\nabla\cdot(n\nabla c)+n-n^3,\ &c_{t}+u\cdot\nabla c=\Delta c-nc,\ &u_{t}+u\cdot\nabla u+\nabla P=-n\nabla\phi,\ &\nabla\cdot u=0. \end{aligned} \right.$ By taking advantage of a coupling structure of the equations and using a scale decomposition technique, the global existence and uniqueness of weak solutions to the above system for large initial data is obtained.

Localization in space and Cauchy problem of chemotaxis system with logistic source

Xiaoxin Zheng

Beihang University
Peoples Rep of China

Co-Author(s): Yao Nie

Abstract:

In this talk, we consider Cauchy problem of chemotaxis system with logistic source. In terms of the nonlocal effect of the logistic source and maximal regularity for the heat kernel, we establish the global-in-time bound of smooth solution.