Talks

Jaewook Ahn(Dongguk University)

• Title : Bounded vs. blow-up in a repulsive chemotaxis-consumption model

• Abstract : This talk considers a parabolic-elliptic repulsive chemotaxis-consumption system for (u,v) in an n-dimensional ball, where n>=3. Under a no-flux boundary condition for u and a Dirichlet condition (v=M) for v, a key feature of this model is characterized by a diffusion coefficient D(u) that extends the porous diffusion coefficient u^{m-1}. For m>1, solutions remain globally bounded for any M>0. This boundedness result also holds for the range 0<m<=1, provided that M is sufficiently small. In contrast, for the case where 0<m<2/n, we show that solutions can experience blow-up when M is large. This talk is based on joint work with K. Kang and D. Kim.

 Yutaro Chiyo(Tokyo University of Science)

• Title : Global existence, boundedness, stabilization and blow-up in several attraction-repulsion chemotaxis systems

• Abstract : In this talk we consider several attraction-repulsion chemotaxis systems. In particular, we show results on global existence, boundedness and stabilization by using energy methods. Also we present results on blow-up based on moment-type functionals. 

Mario Fuest(Leibniz Universität Hannover)

• Title : Upper estimates for the Hausdorff dimension of the temporal singular set in chemotaxis-fluid systems 

• Abstract : Due to the well-known challenges in the existence theory of the unperturbed Navier-Stokes equations in three-dimensional domains,one should not expect to obtain global classical solutions to the chemotaxis-fluid system, where these equations are coupled via buoyancy and transportation to a chemotaxis-consumption model. Nonetheless, one can aim to show partial regularity results. After briefly reviewing how to obtain a global weak solution, the talk focuses on the so-called temporal singular set, i.e., the set of times for which no neighbourhood $U$ exists such that the solution is smooth in $U \times \Omega$. We prove in particular that its Hausdorff dimension is at most $1/2$ and that it is hence in some sense smaller than the Cantor set. 

Kentaro Fujie(Tohoku University)

• Title : Entropy production estimates in chemotaxis systems and its application 

• Abstract : In this talk, we consider some quasilinear Keller--Segel system. In the study of large time behaviour of solutions to the system, its entropy functional plays an important role. On the other hand, in several cases, the entropy structure has a poor information. In stead of the entropy, we focus on the entropy production of the system and derive its estimate. Moreover, we will apply this estimate to obtain global existence of some one-dimensional quasilinear Keller--Segel system. 

Hirofumi Izuhara(University of Miyazaki)

• Title : Predator-prey interaction and cross-diffusion 

• Abstract : Cross-diffusion may be an important driving force of pattern formation in population models.Recently, a relation between cross-diffusion and reaction-diffusion systems has been revealed from the mathematical modeling point of view. In this talk, a predator-prey model with cross-diffusion and a reaction-diffusion system with two behavioral states in the predator population are discussed. We assume that the predators have identical behavioral characteristics except for their mobility and searching activity for preys: we consider two states, namely less mobile predators searching for preys more actively than mobile predators. The predator-prey model with cross-diffusion can be derived from the reaction-diffusion system with two behavioral states in predator via fast reaction limit. We also discuss the existence of traveling waves for the simplified cross-diffusion model.

Jie Jiang(Chinese Academy of Sciences)

• Title : On a Keller-Segel Chemotaxis Model with Signal-dependent Motility 

• Abstract : We report our recent work on analysis of a chemotaxis PDE model involving signal-dependent motility, which was originally proposed by Keller and Segel in their seminal work in 1971. The system features a signal-dependent motility function (diffusion rate), which may vanish or explode as the signal concentration becomes unbounded. We develop  new methods to study its global well-posedness and qualitative behavior of classical solutions. The key idea consists of an introduction of several auxiliary functions satisfying elliptic/parabolic problems, together with delicate applications of various comparison skills. The talk is based on my recent joint works with Kentaro Fujie (Tohoku University), Philippe Laurençot (University of Savoie Mont Blanc & CNRS), Yanyan Zhang (ECNU), and Yamin Xiao (HEBTU), respectively.

Hai-Yang Jin(South China University of Technology)

• Title : Global dynamics of the toxicant-taxis model with Robin boundary conditions  

• Abstract : In this talk, we are concerned with a spatiotemporal population-toxicant model with toxicant-taxis in a bounded domain with Robin boundary conditions. Based on the energy estimates, we first establish the global existence of classical solution.  Moreover, depending on the input rate function, we also show the global stabilization of constant steady state, non-constant positive steady states as well as the periodic solution. 

Kyungkeun Kang(Yonsei University)

• Title : Existence and applications of weak solutions for nonlinear diffusion equations with drift

• Abstract : We investigate weak solutions to porous medium and fast diffusion equations with drift terms, where the drift satisfies a scaling invariance condition involving its L^q norm. As applications, we revisit a variety of nonlinear diffusion systems, refining and extending existing results on existence and regularity. In a recent development, we establish the existence of nonnegative weak solutions with gradient estimates in the presence of a measure-valued external force.

DongKwang Kim(UNIST)

• Title : Global Classical Solutions for a 3D Axisymmetric Chemotaxis-Fluid System Without Swirl

• Abstract : We consider a chemotaxis-consumption model coupled with the three-dimensional axisymmetric incompressible Navier-Stokes equations without swirl. The tensor-valued chemotactic sensitivity is assumed to be axisymmetric and subject to a natural decay condition. Under suitable compatibility conditions on the initial data, we establish the global existence and boundedness of classical solutions.

Yong-Jung Kim(KAIST)

• Title : Heterogeneous Diffusion in Biological Models

• Abstract : Classical diffusion models with constant diffusion coefficients have been successfully applied in numerous contexts. In particular, various forms of reaction-diffusion equations have been widely used to describe and predict biological phenomena. In recent years, there has been growing interest in models with spatially varying diffusion coefficients. Moreover, in many applications—such as chemotaxis—pure diffusion is insufficient to explain observed behavior, and advection terms are often added. In this talk, we introduce heterogeneous diffusion models of the Fokker–Planck type and present several applications in which these models naturally arise, including chemotaxis, epi- demic spread, and thermal diffusion. Through these examples, we explore how heterogeneous diffusion can serve as an alternative to advection and provide insight into more intrinsic modeling approaches.

Philippe Laurençot(Université Savoie Mont Blanc)

• Title : A chemotaxis model with indirect signal production and phenotype switching in space dimension 2

• Abstract : The dynamics of a partially diffusive chemotaxis model with indirect signal production and phenotype switching is investigated in a two-dimensional bounded domain with smooth boundary and homogeneous Neumann boundary conditions. The initial boundary value problem is shown to be globally well-posed, the boundedness of the solutions depending on the initial total mass $M$ of the population. This phenomenon is reminiscent from the dynamics of the classical two-dimensional Keller-Segel chemotaxis model and the connection between the two models is studied as well (joint works with Christian Stinner, Darmstadt).

Johannes Lankeit(Leibniz Universität Hannover)

• Title : Energy-based blow-up in variants of the Keller-Segel system

• Abstract : One of the classical blow-up proofs for the Keller-Segel system relies on its energy functional; also how to leverage this functional for a proof of this blow-up occurring in finite time has long been known (in radial settings). In this talk, we will look at different variants of chemotaxis systems, where it can facilitate blow-up proofs, although, at a first glance, its application may seem less easily possible. In particular, we will consider blow-up in the fully parabolic chemotaxis system with a source of logistic type. (The latter topic is based on joint work with Mario Fuest and Masaaki Mizukami.) 

Jihoon Lee(Chung-Ang University)

• Title : Regularity criterion for the solutions to the oxygen consumption model.

• Abstract : The oxygen consumption dynamics model of B. subtilis has been studied mathematically in many works, often in conjunction with the Navier–Stokes equations. In this talk, we aim to consider the existence of weak solutions as well as various regularity conditions, both for the equations coupled with the Navier–Stokes system and for those studied independently.

Min-Gi Lee(Kyungpook National University)

• Title : A limit of microscopic dweller-wanderer system to a chemotaxis model

• Abstract : In this talk, we study a derivation of a chemotaxis model where species exhibits adaptive diffusion depending on the local information of population and food density. More specifically, the flux of population is the negative gradient of $\gamma(m)u$, where $m$ is the food density, $u$ is the population density, and $\gamma(\cdot)$ is a decreasing function. This type of flux law has attracted considerable attention in the reaction-diffusion community. Funaki, Mimura, and Urabe (2012) presented a derivation by hydrodynamic limit of a kinetic model where species may exists either as a fast diffusive  or a slow diffusive mode; they convert to each other quickly to meet the the quasi-equilibrium ratio between them. One small concern is that there has been an employment of asymptotic power laws $\gamma(m) = \frac{1}{1 + m^p}$ in literature while the derived $\gamma(m)$ is bounded from below by the diffusion constant of slow diffusive mode, away from 0. To fit the power laws in, one must drop the diffusion term of the slow mode, which might cause technical challenges. We show that the limit presented by Funaki, Mimura, and Urabe is still done the same with $0$ diffusive dwelling mode. (This is joint work with Kyunghan Choi and Yong-Jung Kim)

Wenbin Lyu(Shanxi University)

• Title : Logistic damping effect in the Chemotaxis model with density-independent motility 

• Abstract : In this talk, I will report some results on a parabolic-parabolic chemotaxis model with density-independent  motility and general logistic source in an n-dimensional smooth bounded domain with Neumann boundary conditions. Under the minimal conditions for the density-independent 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. 

Masaaki Mizukami(Kyoto University of Education)

• Title : Behaviour of solutions to a model for tuberculosis granuloma formation

• Abstract : This talk deals with a tuberculosis granuloma formation model, which is proposed by Feng (2024). For the ODE setting Feng showed that the reproduction number is important to determine behaviour of solutions: If the reproduction number is less than 1 then an infection-free equilibrium is locally asymptotically stable, and if the reproduction number is greater than 1 then the infection-free equilibrium is unstable. On the other hand, for the PDE setting, global existence of classical/weak solutions to the model was established in 2-/3-dimensional settings, respectively in Fuest-Lankeit–M. (2025). The purpose of this talk is, for the next step, to show that a classical solution of the problem exists globally (also in higher dimensional settings) and converges to an infection-free equilibrium if the reproduction number is less than 1 and the initial data is small in some sense. This talk is based on a joint work with Dr. Yuya Tanaka (Kwansei Gakuin University).

Zhi-An Wang(The Hong Kong Polytechnic University)

• Title : Global well-posedness and Turing–Hopf bifurcation of prey-taxis systems with hunting cooperation 

• Abstract : This talk is concerned with a predator–prey system with hunting cooperation and prey-taxis under homogeneous Neumann boundary conditions. We establish the existence of globally bounded solutions in two dimensions. In three or higher dimensions, the global boundedness of solutions is obtained for the small prey-tactic coefficient. By using hunting cooperation and prey species diffusion as bifurcation parameters, we conduct linear stability analysis and find that both hunting cooperation and prey species diffusion can drive the instability to induce Hopf, Turing and Turing–Hopf bifurcations in appropriate parameter regimes. It is also found that prey-taxis is a factor stabilizing the positive constant steady state. We use numerical simulations to illustrate various spatiotemporal patterns arising from the abovementioned bifurcations including spatially homogeneous and inhomogeneous time-periodic patterns, stationary spatial patterns and chaotic fluctuations.

Tomomi Yokoka(Tokyo University of Science)

• Title : Mathematical analysis of a quasilinear chemotaxis system for tumor invasion

• Abstract : In this talk we consider a quasilinear chemotaxis system arising from tumor invasion. Under some relation between the power of diffusion and the power of chemotaxis we show that the system possesses a global-in-time solution which stabilizes to the constant steady state. The method is based on the maximal regularity for parabolic equations used for the corresponding quasilinear Keller-Segel system. We also discuss the possibility of blow-up in the simplified system. This study is based on a joint work with Professor Sachiko Ishida (Chiba University). 

Changwook Yoon(Chungnam National University)

• Title : Diffusion under starvation in biological modeling 

• Abstract : Diffusion under resource limitation provides a coherent framework for understanding population movement in biology. In chemotaxis systems, starvation-driven nonlinear diffusion captures how organisms adjust their movement under limited resources, paralleling ratio-dependent approaches where the balance between population density and resource availability governs motility. This nonlinear formulation offers natural advantages for the well-posedness of the models. Beyond chemotaxis, the framework can be applied to a variety of ecological contexts, from spatial competition to predator–prey interactions under resource stress. In this talk, we will introduce several modeling scenarios where starvation-driven dispersal may play a role, and present numerical observations for discussion. 

Student Talk

1) Minyoo Kim(KAIST) 

• Title : Fractionation by anisotropic persistent random walk and two-coefficient diffusion law  

• Abstract : Random movement of microscopic particles in heterogeneous environments leads to fractionation phenomena, with the Soret effect being one of the most representative examples. This raises a fundamental question: what characteristics of random movement give rise to such fractionation phenomena? We investigate whether the persistence of a random-walk system has such a property and show that fractionation occurs only when the persistence is anisotropic. This is shown by investigating the convergence of a heterogeneous persistence random-walk system to a resulting anisotropic diffusion equation. Numerical simulations of the diffusion equation are compared with a Monte Carlo method and solutions to the recursive relations. 

2) Junseong Park(KAIST) 

• Title : Darcy-type Diffusion in Heterogeneous porous media with Permeable boundaries   

• Abstract : Darcy's law and the permeable boundary condition(PBC) represent two seemingly different approaches to modeling the role of permeability in fluid flow and diffusion. However, this difference arises not from distinct physical mechanisms but from whether permeability is applied to the entire domain or only to a boundary. In this talk, we unify these two perspectives using a discrete-time random walk model on a one-dimensional lattice with spatially heterogeneous permeability. We also show that the model converges to a heterogeneous diffusion equation in the diffusion limit. This result provides a unified framework to understand both bulk and boundary effects of permeability. This is a joint work with professor Yongjung Kim.