Special Session 1: Analysis of parabolic models for chemotaxis

Solution behaviors in chemotaxis-consumption systems with Dirichlet boundary conditions

Jaewook Ahn Dongguk University Korea Co-Author(s):

Abstract: Bacillus subtilis swim toward oxygen-rich air-water interfaces in water droplets and form large clusters near the boundary. To describe such pattern formation, chemotaxis systems with signal consumption have been proposed, which in numerical studies have exhibited various patterns similar to those observed in actual experiments. In this talk, I will present related analytical results on chemotaxis-consumption systems, particularly those with Dirichlet boundary conditions for the signal. One of our findings shows that bacteria populations under certain initial conditions tend to aggregate near the boundary, where signals have been prescribed. On the other hand, for the system of chemorepulsive counterpart, it is shown that a finite time blowup can be observed whenever the diffusion effect on bacteria populations is slightly weakened. Some results on the existence of bounded solutions will also be discussed.

Boundedness of classical solutions to a chemotaxis consumption model with signal dependent motility

Khadijeh Baghaei Pasargad Institute for Advanced Innovative Solutions Iran Co-Author(s):    Ali Khelghati

Abstract: My goal in this talk is to present our recent results for a chemotaxis consumption model with signal dependent motility and logistic source. For this model, we have proved that there exists a unique global classical solution which is uniformly in time bounded. This result is obtained for small initial data without any restriction on the coefficient of the logistic term. This is a joint project with Ali Khelghati.

Asymptotic profile to the blow-up solutions of parabolic-elliptic Keller-Segel-Patlak system in $\mathbb{R}^N$ with $N\ge 3$.

Xueli Bai Northwestern Polytechnical University Peoples Rep of China Co-Author(s):    Maolin Zhou

Abstract: In this talk, we obtained the exact blow-up profiles of solutions of the Keller-Segel-Patlak system in the whole space with dimensions $N\ge 3$ which partially solves an open problem proposed by P. Souplet and M. Winkler in [Souplet, Winkler, CMP, 2019]. To establish this achievement, we develop the zero number argument for nonlinear equations with unbounded coefficients and construct a family of auxiliary backward self-similar solutions through nontrivial ODE analysis.

Boundedness in a class of Keller--Segel models with dissipative gradient terms

Alessandro Columbu Universit\`a degli Studi di Cagliari Italy Co-Author(s):    Tongxing Li; Daniel Acosta Soba; Giuseppe Viglialoro;

Abstract: We study a class of zero-flux attraction-repulsion chemotaxis models, characterized by nonlinearities laws for the diffusion of the cell density, the chemosensitivities and the production rates of the chemoattractant and the chemorepellent. Additionally, a source involving also the gradient of the biological distribution is incorporated. In this talk we will see a sketch of the proof for the existence of solutions and we will find a condition for the boundedness.

Some results on Keller-Segel(-Navier)-Stokes model with indirect signal production

Feng Dai Huazhong University of Science and Technology Peoples Rep of China Co-Author(s):    Bin Liu

Abstract: In this talk, we consider the Keller-Segel(-Navier)-Stokes model with indirect signal production in a bounded domain with smooth boundary. Under appropriate regularity assumptions on initial data, the global solvability and asymptotic stabilization to the associated initial-boundary value problem are obtained. In comparison with the existing results for the case of direct signal production, our results reveal the regularizing effect of the indirect signal production mechanism on the Keller-Segel(-Navier)-Stokes system.

Quantitative analysis and its applications for Keller-Segel type systems

Mengyao Ding Institute for Advanced Study in Mathematics of HIT Peoples Rep of China Co-Author(s):    Yuzhou Fang; Chao Zhang

Abstract: In this work, to address the asymptotic stability of chemotaxis systems incorporating various mechanisms, we employ the De Giorgi iteration method to quantitatively analyze the upper bound of solutions. The refined upper bound estimate obtained in the present paper illustrates how various factors influence the upper bound, which can then be used to determine the large-time behaviours of solutions. To show the wide applicability of our findings, we investigate the asymptotic stability of a chemotaxis model with nonlinear signal production and a chemotaxis-Navier-Stokes model with a logistic source. Additionally, within the context of $p$-Laplacian diffusion, we establish H\"{o}lder continuity for a chemotaxis-haptotaxis model and a chemotaxis-Stokes model.

On a chemotaxis model with nonlinear diffusion modelling multiple sclerosis

Simone Fagioli University of L`Aquila Italy Co-Author(s):    M. Kamath, E. Radici, L. Romagnoli

Abstract: We investigated existence of global weak solutions for a system of chemotaxis type with nonlinear degenerate diffusion, arising in modelling Multiple Sclerosis disease. The model consists of three equations describing the evolution of macrophages $(m)$, cytokine $(c)$ and apoptotic oligodendrocytes $(d)$. The main novelty in our work is the presence of a nonlinear diffusivity $D(m)$, which results to be more appropriate from the modelling point of view. We first show the existence of global bounded solutions for the model. We then investigate some properties on the stationary states and pattern formation.

Properties of given and detected unbounded solutions to a class of chemotaxis models

Silvia Frassu University of Cagliari Italy Co-Author(s):    Alessandro Columbu, Giuseppe Viglialoro

Abstract: This talk deals with unbounded solutions to a class of chemotaxis systems. In particular, for a rather general attraction-repulsion model, with nonlinear productions, diffusion, sensitivities and logistic term, we detect Lebesgue spaces where given unbounded solutions blow-up also in the corresponding norms of those spaces; subsequently, estimates for the blow-up time are established. Finally, for a simplified version of the model, some blow-up criteria are proved.

Finite-time blow-up in fully parabolic quasilinear Keller--Segel systems with supercritical exponents

Mario Fuest Leibniz University Hannover Germany Co-Author(s):    Xinru Cao

Abstract: The fully parabolic quasilinear Keller--Segel system \[ \begin{cases} u_t = \nabla \cdot ((u+1)^{m-1} \nabla u - u(u+1)^{q-1} \nabla v), \ v_t = \Delta v - v + u, \end{cases} \] which we consider in a ball $\Omega \subset \mathbb R^n$, $n \ge 2$, admits unbounded solutions whenever $m, q \in \mathbb R$ satisfy $m - q < \frac{n-2}{n}$. These are necessarily global in time if $q \leq 0$ and finite-time blow-up is known to be possible if $q > 0$ and $\max\{m, q\} \geq 1$. Utilizing certain pointwise upper estimates for $u$, we are able to give an affirmative answer to the (for nearly a decade formerly open) question whether solutions may blow up in finite time if $\max\{m, q\} < 1$. If $n = 2$, for instance, we construct solutions blowing up in finite time whenever ($m-q < 0$ and) $q < 2m$.

Analysis of a Navier-Stokes-Cahn-Hilliard system with unmatched densities and chemotaxis

Andrea Giorgini Politecnico di Milano Italy Co-Author(s):

Abstract: We consider the initial-boundary value problem for an incompressible Navier-Stokes-Cahn-Hilliard model with non-constant density and chemotaxis for soluble species proposed by Abels, Garcke and Gr\{u}n in 2012. In this talk I will present recent results concerning the global existence of weak solutions and propagation of regularity. A main feature of this contribution is the lack of the regularization effect given by the logistic source in the equation for the density of the soluble species. This is a joint work with Jingning He (The Hong Kong Polytechnic University) and Hao Wu (Fudan University).

Global boundedness and blow-up in a repulsive chemotaxis-consumption system

Dongkwang Kim Ulsan National Institute of Science and Technology, Department of Mathematical Sciences Korea Co-Author(s):    Jaewook Ahn, Kyungkeun Kang

Abstract: In this presentation, we explore the parabolic-elliptic chemotaxis-consumption system of repulsion type in higher dimensions under no-flux/Dirichlet boundary conditions. We discuss the criteria for solutions to remain bounded over time and the conditions under which blow-up occurs. Specifically, we show that the system admits globally bounded solutions when the diffusion of the organisms is enhanced, or when it is weakened but the boundary data for the signal substance is sufficiently small. Furthermore, we demonstrate that when the diffusion is further weakened and the boundary data for the signal is appropriately large, the system possesses blow-up solutions.

Global solvability and immediate regularization of measure-type population densities in a flux-limited Keller--Segel system

Shohei Kohatsu Tokyo University of Science Japan Co-Author(s):

Abstract: We consider a flux-limited Keller--Segel system in a bounded domain, for which global existence and boundedness of classical solutions with linear diffusion and regular initial data were considered by Winkler (Math.\ Nachr.; 2022; 295; 1840--1862). It is shown that under conditions on the strength of diffusion and flux limitation, for any Radon measure initial data the system admits a global solution which immediately becomes smooth and classical, and approaches the given initial data in an appropriate sense.

On a chemotaxis-May-Nowak Model for virus infection with superlinear dampening

Yan Li Nanjing University of Posts and Telecommunications Peoples Rep of China Co-Author(s):

Abstract: In this talk, we consider a three-component reaction-diffusion system, originating from the classical May-Nowak model for viral infections with superlinear dampening $-\mu u^{\alpha}$. Existence of classical solutions is verified and large time behavior of the solutions are investigated under the condition $\alpha>\frac{n+2}{2}$, while generalized solutions are constructed for arbitrary $\alpha>1$. In addition, based on an analysis of a certain eventual Lyapunov-type functional, we prove that whenever $\alpha>\max\{1,\frac{n}{2}\}$, the corresponding generalized solution asymptotically enjoys relaxation by approaching the nontrivial homogeneous steady states.

Boundedness in a two-dimensional doubly degenerate nutrient taxis system

Yuxiang Li School of Mathematics, Southeast University Peoples Rep of China Co-Author(s):    Zhiguang Zhang

Abstract: In this talk, we study the doubly degenerate nutrient taxis system \begin{align} \begin{cases}\tag{$\star$}\label{eq-0.1} u_t=\nabla \cdot(u v \nabla u)-\chi \nabla \cdot\left(u^{2} v \nabla v\right)+ u v, \ v_t=\Delta v-u v \end{cases} \end{align} in a smooth bounded domain $\Omega \subset \mathbb{R}^2$, where $\chi>0$. We prove that for all reasonably regular initial data, the corresponding homogeneous Neumann initial-boundary value problem for \eqref{eq-0.1} possesses a global bounded weak solution which is continuous in its first and essentially smooth in its second component. There exists $u_{\infty}\in L^{\infty}(\Omega)$ such that this solution possesses the convergence property that \begin{align}\label{eq-0.2} u(t) \stackrel{*}{\rightharpoonup} u_{\infty}\quad \text{ and } \quad v(t) \rightarrow 0 \quad \text { in } L^{\infty}(\Omega) \quad \text { as } t \rightarrow \infty. \tag{$ \star \star$} \end{align} Furthermore, the limit $u_{\infty}$ in \eqref{eq-0.2} exhibits spatially heterogeneous under a criterion on the initial smallness of the second component.

Global bounded weak solutions to a 3D chemotaxis-Stokes system with slow p-Laplacian diffusion and rotation

Zhongping Li China West Normal University Peoples Rep of China Co-Author(s):    Haolan He

Abstract: In this talk, we are concerned with the following chemotaxis-Stokes system with p-Laplacian diffusion and rotation $\begin{equation*} \left\{ \begin{split} &n_{t}+u\cdot \nabla n=\nabla\cdot(|\nabla n|^{p-2}\nabla n)-\nabla\cdot (nS(x,n,c)\nabla c),&&x\in \Omega ,t> 0,\ & c_{t}+u\cdot \nabla c=\Delta c-nc,&&x\in \Omega ,t> 0,\ & u_{t}+\nabla P=\Delta u+n\nabla \phi,&&x\in \Omega ,t> 0,\ &\nabla \cdot u=0, &&x\in \Omega ,t> 0 \end{split} \right. \end{equation*}$ in a smooth bounded domain $\Omega\in\mathbb{R}^3$. We show the boundedness of the weak solutions to the 3D chemotaxis-Stokes system with p-Laplacian diffusion under no-flux boundary conditions/Dirichlet signal boundary condition.

Qualitative properties of solutions to a class of chemotaxis system

Monica Marras University of Cagliari Italy Co-Author(s):    F. Ragnedda S. Vernier-Piro, V. Vespri

Abstract: We study qualitative properties as blow-up phenomena, decay in time, boundedness, global existence and H\older regularity for solutions of some classes of chemotaxis systems. In particular we consider a degenerate chemotaxis systems with porous media type diffusion and a source term satisfying the Hadamard growth condition. We prove the H\older regularity for bounded solutions to parabolic-parabolic as well as for parabolic-elliptic chemotaxis systems.

Properties of blow-up points in a parabolic-parabolic chemotaxis system with spatially heterogeneous logistic term

Masaaki Mizukami Kyoto University of Education Japan Co-Author(s):    Mario Fuest, Johannes Lankeit

Abstract: This talk discusses possible points of blow-up in a chemotaxis system with spatially heterogeneous logistic term in two-dimensional smoothly bounded domains under the Neumann boundary conditions and initial conditions. About this problem, a property of possible blow-up points is recently studied; in the parabolic-elliptic setting, it was shown that finite-time blow-up of the classical solution can only occur in points where a coefficient is zero. In this talk we give a recent development of a result about a property of blow-up points in the system in the parabolic-parabolic setting. This is a joint work with Dr. Mario Fuest (Leibniz University Hannover) and Professor Johannes Lankeit (Leibniz University Hannover).

Global solvability of predator-prey model with prey-taxis or predator-taxis

Guoqiang Ren Huazhong University of Science and Technology Peoples Rep of China Co-Author(s):    Bin Liu and Jianshe Yu

Abstract: In this paper, we study a two-species chemotaxis-Navier-Stokes system with Lotka-Volterra type competitive kinetics: $n_t+u\cdot\nabla n=\Delta n-\chi_1\nabla\cdot(n\nabla w)+n(\lambda_1-\mu_1n^{\theta-1}-a_1v)$; $v_t+u\cdot\nabla v=\Delta v-\chi_2\nabla\cdot(v\nabla w)+v(\lambda_2-\mu_2v-a_2n)$; $w_t+u\cdot\nabla w=\Delta w-w+n+v$; $u_t+\kappa(u\cdot\nabla)u=\Delta u+\nabla P+(n+v)\nabla\phi$; $\nabla\cdot u=0$, $x\in \Omega$, $t>0$ in a bounded and smooth domain $\Omega\subset \mathbb{R}^2$ with no-flux/Dirichlet boundary conditions, where $\chi_1, \chi_2$ are positive constants. We present the global existence of generalized solution to a two-species chemotaxis-Navier-Stokes system and the eventual smoothness already occurs in systems with much weaker degradation $(\theta>1)$, again under a smallness condition on $\lambda_1, \lambda_2$.

Stabilization and pattern formation in a chemotaxis model with acceleration

Weirun Tao Southeast University Peoples Rep of China Co-Author(s):    Chunlai Mu, Zhi-An Wang

Abstract: In this talk, we consider a new type of chemotaxis model with acceleration, which assumes that species` advective acceleration instead of velocity in the classical chemotaxis model is proportional to the chemical signal concentration gradient. This new model has an additional equation governing the velocity field with more delicate boundary conditions. We shall introduce some results on global existence, stabilization, and pattern formation of the model with/without the logistic source term.

Well-posedness results on an oncolytic virotherapy model

Xueyan Tao Ocean University of China Peoples Rep of China Co-Author(s):    Shulin Zhou

Abstract: One of the most promising strategies to treat cancer is attacking it with viruses. This talk begins with an introduction to a haptotactic cross-diffusion system modeling oncolytic virotherapy. After reviewing some existing results on this model, we shall report some recent results on global existence and asymptotic behavior of solutions to this system.

Existence and blow-up results to quasilinear chemotaxis-haptotaxis system

Jagmohan Tyagi Indian Institute of Technology Gandhinagar India Co-Author(s):    Poonam Rani

Abstract: We discuss the following quasilinear chemotaxis-haptotaxis system: $\begin{align*} u_{t} &= \nabla\cdot(D(u)\nabla u)- \chi \nabla\cdot(S(u)\nabla v)-\xi\nabla\cdot(u\nabla w), \; x \in \Omega\text{, }t>0, \ v_{t} &=\Delta v -v+ u, \; x \in \Omega\text{, }t>0, \ w_{t} &=-vw, \; x \in \Omega\text{, }t>0, \ \end{align*}$ under homogeneous Neumann boundary conditions in a smooth, bounded domain $\Omega \subset \mathbb{R}^{n}, n\geq 3.$ We show that for $\frac{S(s)}{D(s)} \leq A (s+1)^{\alpha}$ for $\alpha < \frac{2}{n}$ and under suitable growth conditions on $D,$ there exists a uniform-in-time bounded classical solution to the above system. Also, we establish that for radial domains, when the opposite inequality is satisfied, the corresponding solutions blow-up in finite or infinite-time.

Some discussions regarding the seminal Keller-Segel model with positive total flux

Giuseppe Viglialoro Universit \\`a degli Studi di Cagliari Italy Co-Author(s):    Silvia Frassu, Yuya Tanaka

Abstract: Since the advent of the seminal Keller-Segel models describing chemotaxis phenomena involving some cell and chemical distributions, the results obtained for related variants are innumerable. Nevertheless, the common denominator of such studies focuses on the assumption that the equation for the cells obeys a zero-flux boundary condition (impenetrable domains). The aim of this talk is to discuss preliminary results and share considerations on chemotaxis models where the total flux has a positive sign (penetrable domains). This is a joint project with Silvia Frassu and Yuya Tanaka.

Global existence and eventual smoothness of a Keller-Segel-consumption system involving local sensing and growth term

Liangchen Wang Chongqing University of Posts and Telecommunications Peoples Rep of China Co-Author(s):

Abstract: In this talk, we will consider a Keller-Segel-consumption system involving local sensing and growth term. For all suitably regular initial data, this system possesses global classical solutions in two-dimensional counterpart, whereas in the case of higher spatial dimensions ($n\geq3$), globally-defined classical solutions were also constructed under some restriction conditions. In higher-dimensional settings, it is asserted that certain weak solutions exist globally, which become smooth after some waiting time.

The global solvability and asymptotic behavior for doubly degenerate nutrient model with large initial data

Duan Wu Paderborn university Peoples Rep of China Co-Author(s):

Abstract: In this talk, we focus on the doubly degenerate nutrient model under no-flux boundary conditions in a two-dimensional smoothly bounded convex domain. In the previous related works, the results concerning the large time behavior were constrained to one dimension. In this work, we not only prove the global existence of solutions, but also show the asymptotic behavior.

On an inhomogeneous incompressible Navier-Stokes system with chemotaxis modeling vascular network

Zhaoyin Xiang University of Electronic Science and Technology of China Peoples Rep of China Co-Author(s):    Yazhi Xiao, Lu Yang

Abstract: In this talk, we focus on an inhomogeneous incompressible Navier-Stokes system with chemotaxis modeling vascular network in a bounded domain. Precisely, the system consists of an inhomogeneous incompressible Navier-Stokes equations for the density of the endothelial cells and their velocity, coupled to a reaction-diffusion equation for the concentration of the chemoattractant, which triggers the migration of the endothelial cells and the blood vessel formation. The global solvability and vanishing viscosity limit of finite energy weak solutions will be investigated under suitable initial-boundary value conditions. The solutions also satisfy a relative energy inequality, which ensures the weak-strong uniqueness property. This is a joint work with Dr Yazhi Xiao and Dr Lu Yang.

Local existence and global boundedness for a chemotaxis system with gradient dependent flux limitation

Jianlu Yan Nanjing University of Aeronautics and Astronautics Peoples Rep of China Co-Author(s):    Yuxiang Li

Abstract: In this talk, we investigate a Keller-Segel system with flux limitation under no-flux boundary conditions in a ball. It is proved that the problem possesses a unique classical solution that can be extended in time up to a maximal $T_{\max}\in (0,\infty]$. Moreover, we show that the above solution is global and bounded in certain subcritical cases. This is a joint work with Yuxiang Li (SEU).

Global existence and stabilization of weak solutions to a degenerate chemotaxis system arising from tumor invasion

Tomomi Yokota Tokyo University of Science Japan Co-Author(s):    Sachiko Ishida

Abstract: In this talk we consider a degenerate chemotaxis system arising from tumor invasion. We discuss global existence and stabilization of weak solutions to the system by a similar argument for the corresponding degenerate Keller-Segel system. This study is based on a joint paper with Professor Sachiko Ishida (J. Differential Equations, 371 (2023), 450-480).

Ratio-dependent motility in biological diffusion models

Changwook Yoon Chungnam National University Korea Co-Author(s):

Abstract: In this study, we explore the application of ratio-dependent motility to biological diffusion models, providing a framework for understanding how organisms adjust their movement based on the ratio of available resources to population density. This approach reflects a more realistic perspective on biological systems, where resource competition, rather than absolute abundance, determines motility. We apply this motility concept in both chemotaxis and prey-predator models, demonstrating its potential to enhance global solvability and provide deeper insights into the dynamics of resource-limited environments.

On a Keller-Segel chemotaxis system with flux limitation and nonlinear signal production

Wenji Zhang Hunan University of Science and Technology Peoples Rep of China Co-Author(s):

Abstract: It is well known that critical exponents distinguishing between existence and blow-up have been established for the Keller-Segel chemotaxis system with flux limitation and with nonlinear signal production respectively. On this basis, we have obtained a critical parameter differentiating global existence from finite-time blow-up when this model combined with flux limitation and nonlinear signal production. This is an extension of previous results, and the monotonicity of the first solution component is the key gradient having this result.

Some results in Keller-Segel chemotaxis systems

Pan Zheng Chongqing University of Posts and Telecommunications Peoples Rep of China Co-Author(s):

Abstract: It is well-known that chemotaxis is a common natural phenomenon describing a biochemical process through which the movement of an organism or entity is not only regulated by random diffusion, but also controlled by the concentration gradient of a chemical stimulus in the local environment. In this talk, I first mention some previous mathematical advances of chemotaxis systems. Then I introduce our recent results about Keller-Segel chemotaxis systems with indirect signal mechanisms.