Szymon Cygan (Heidelberg University & Wroclaw University)   

Kotaro Hisa (Tohoku University)   

Abstract: We consider necessary conditions and sufficient conditions on the solvability of the Cauchy--Dirichlet problem for a fractional semilinear heat equation in open sets (possibly unbounded and disconnected) with a smooth boundary.
Our conditions enable us to identify the optimal strength of the admissible singularity of initial data for the local-in-time solvability and they differ in the interior of the set and on the boundary of the set.

Lingling Hou (Hong Kong Polytechnic University & Xinjjiang University)   

Abstract: This paper establishes the existence of traveling wave solutions to a reaction-diffusion equation coupled with a singularly perturbed first ordinary differential equation with a small parameter $\varepsilon>0$. The system is a toy model for biological pattern formation. Traveling wave solutions correspond to heteroclinic orbits of a fast-slow system. Under some conditions, the reduced problem has a heteroclinic orbit with jump discontinuity, while the layer problem has an orbit filling the gap. We thus construct a singular orbit by piecing together these two orbits. The traveling wave solutions is obtained in the neighborhood of the singular orbit. However, unlike the classical FitzHugh-Nagumo equations, the singular orbit contains a fold point where the normal hyperbolicity breakds down and the standard Fenichel theory is not applicable. To circumvent this difficulty we employ the directional blowup method for geometric desingularization around the fold point.

Yuki Kaneko (Kanto Gakuin University)   

Abstract: We discuss the asymptotic behavior of a Stefan-type free boundary problem for a kind of multi-stable reaction-diffusion equations. This problem, modeling the spreading of biological species, was proposed by Y. Du and Z.G. Lin in 2010, where the spreading-vanishing dichotomy was proved for the free boundary problem of a monostable diffusion equation with a Neumann boundary condition. When we consider a multi-stable diffusion equation with a Dirichlet boundary condition, the asymptotic behaviors are classified into four cases: big spreading, small spreading, vanishing and a borderline behavior between big spreading and small spreading. Moreover, the big spreading solution converges to a propagating terrace when the Stefan coefficient is sufficiently large. This is based on a joint work with Professor Yoshio Yamada (Waseda University).

Jo Kubokawa (Meiji University)   

Abstract: This study investigates the pulsating traveling wave of the Mitchell-Schaeffer model numerically. The Mitchell-Schaeffer model is a mathematical model that describes the dynamics of cardiac action potentials. Numerical simulation of this model under periodic boundary condition shows a pulsating traveling wave solution, which is a time-periodic deformation of a pulse-type traveling wave solution by pulse-to-pulse strong interactions. We investigate the singular limit problem of the Mitchell-Schaeffer model and present a pulsating traveling wave numerically. This is based on a joint work with H.Ninomiya.

Ryunosuke Mori (Meiji University)   

Abstract: We consider blocking and propagation phenomena of free boundary problem for curve shortening flow with a driving force in spatially undulating cylindrical domains. In this free boundary problem, Matano, Nakamura and Lou in 2006, 2013 characterize the effect of the shape of the boundary to blocking and propagation of the solutions under some slop condition about the boundary that implies time global existence of the classical solutions. In this study, we consider the effect of the shape of the boundary to blocking and propagation of this free boundary problem under more general situation that the solutions may develop singularities near the boundary. As a result, we obtain that if the spatial undulation of the boundary is steep, then shorter interval of the undulation speeds up propagation because of the singularities of the solutions.

Finn Münnich (Heidelberg University)   

Abstract: Reaction-diffusion-ODE systems are emerging in modeling of biological pattern formation based on the coupling of diffusive and non-diffusive nonlinear processes. They may exhibit patterns with singularities such as jump-discontinuities. We first establish a general framework for the stability analysis of steady states by providing nonlinear stability and instability conditions for bounded stationary solutions of reaction-diffusion-ODE systems. Then we consider Turing patterns. A necessary condition for the appearance of Turing patterns is diffusion-driven instability (DDI). We derive sufficient conditions for DDI in reaction-diffusion-ODE systems. Furthermore, we summarize when we cannot expect stable Turing patterns even if we have DDI.

Keita Nakajima (Meiji University)   

Abstract: The heart pumps blood through the body by regularly propagating the action potential. However, it is said that abnormal electrical signals are generated when the heart becomes spatially heterogeneous due to myocardial infarction.
I was interested in this phenomenon and conducted research on the mechanism of abnormal electrical signals using the Aliev-Panfilov model, which is a mathematical model of the heart. In this study, we assume that the cardiac cells damaged by myocardial infarction can be represented by small diffusion coefficients. Numerical simulations of the Aliev-Panfilov model in the case of spatial heterogeneity of diffusion show the complicated dynamics including spiral waves. To investigate the mechanism of the spontaneous generation of spiral waves, we investigate the effect of variable diffusion on propagation.
First, we go back to the derivation of the diffusion coefficient from the transition probability and generalize the repulsive/attractive transition. Then we derive the singular limit problem of the Allen-Cahn-Nagumo equation with the generalized diffusion term. We present the relationship between the diffusion coefficients and propagation. Applying this observation to the Aliev-Panfilov model, we discuss the mechanism of the generation of spiral waves. This is based on the joint work with H. Ninomiya.

Ryosuke Nishide (University of Tokyo)   

Abstract: Pattern formation and dynamics on curved surfaces are ubiquitous, especially in biological systems. The behavior of Turing patterns on curved surfaces was first considered by Turing himself in 1952 and has been of interest ever since. In this talk, we report that the propagation of Turing patterns is driven by surface curvature: Turing patterns are static on a flat surface but can propagate on an axisymmetric surface. Numerical and theoretical analyses reveal that the conditions for the onset of pattern propagation depend on the symmetry of the surface and the pattern.

Hyunjoon Park (Meiji University)   

Abstract: We consider the singular limit problem of Allen-Cahn equation when the diffusivity depends on the solution(or density) with anisotropic diffusivity. It is known that the solution dependent diffusivity affects the speed of the mean curvature flow motion, whereas the anisotropic diffusivty leads to the anisotropic mean curvauture flow of the interface. In this study, we focus on understanding the interface motion of the Allen-Cahn equation when diffusivity has both the density dependent diffusion and the anisotropic diffusion. We first give a formal asymptotic expansion which allows us to expect the interface motion equation. We also give a brief sketch that the expected interface motion does related to the solution by constructing a pair of sub- and super-solutions.

Florian Salin (Tohoku University)   

Abstract: This poster will discuss a numerical method for a fractional nonlinear diffusion equation on bounded domains. This equation arises by combining fractional (in space) diffusion with a nonlinearity of porous medium or fast diffusion type. It is well-known that, in the porous medium case, the energy of the solutions to this equation decays algebraically, and in the fast diffusion case, solutions extinct in finite time. Based on a discretization of the fractional Laplacian recently introduced by Huang and Oberman, we will present a scheme that preserves the energy decay estimates as well as the finite time extinction phenomenon. For that, we will introduce an implicit Euler scheme that satisfies the same structural energy inequalities as the continuous equation, and introduce discrete functional analysis tools to adapt continuous arguments to the discrete level.

Kotaro Sato (Tohoku University)   

Abstract: This poster is concerned with doubly-nonlinear evolution equations where the energy functional is not convex and the dissipation potential functional has singularity and degeneracy. Since the energy functional is not convex, the solutions are not necessarily continuous in time. In this poster, a parabolic regularized equation with a (strong) dissipation term is introduced and its vanishing-viscosity limit is considered via arc-length reparameterizations.

Izumi Takagi (Tohoku University)   

Abstract: There is no mathematical definition of "pattern". Even the "Turing pattern" is not defined uniquely. However, it is counterproductive to give a rigorous definition of pattern, because it contains a variety of phenomena. Instead, we discuss what kind of properties makes us recognize there is a pattern in specific examples.

Koichi Taniguchi (Tohoku University)   

Abstract: We study the problems of uniqueness for Hardy-H\'enon parabolic equations, which are semilinear heat equations with the singular potential (Hardy type) or the increasing potential (H\'enon type) in the nonlinear term. To deal with the Hardy-H\'enon type nonlinearities, we employ weighted Lorentz spaces as solution spaces. Our purpose is to prove unconditional uniqueness and non-uniqueness for Hardy-H\'enon parabolic equations in the weighted Lorentz spaces. The results extend the previous works on the Fujita equation and Hardy equations in Lebesgue spaces. This is based on the joint work with Noboru Chikami (Nagoya Institute of Technology), Masahiro Ikeda (RIKEN/Keio University) and Slim Tayachi (Universit\'e de Tunis El Manar).

Conghui Zhang (Beijing University of Civil Engineering and Architecture)   

Abstract: I will consider a receptor-based model which arises from modeling of interactions between intracellular processes and diffusible signaling factors. We prove the existence of stationary solutions with jump discontinuity by a variational method and show that under a suitable topology, they are always stable. Then we consider the asymptotic behavior of stationary solutions as the diffusion coefficient tends to infinity and zero. In addition, some numerical simulations are presented to illustrate the theoretical results.

10:00 -- 10:45   Shigeru Kondo (Osaka University)

How to transfer the 2D pattern to 3D shape

Abstract: Turing's reaction-diffusion principle revealed how to create spatial order in the body of an organism, and is now accepted by experimental biologists as a fundamental principle in embryology. However, Turing's theory is generally only a two-dimensional, equally spaced pattern, and some additional mechanism is needed to create the characteristic three-dimensional structure of an organism. We believe that one such mechanism is the wrinkling of the cuticle during molting of exoskeletons.
The horn precursors of beetles have a one-layered pouch-like structure. The pouch has a pattern of wrinkles just like those produced in Turing's simulation. During pupation, the wrinkles elongate, causing the precursor to grow into a giant, characteristic horn morphology, and the morphological changes are encoded by the shape (orientation and density) of the wrinkles. In my talk, I will report the results of my analysis of the principle of converting 2D patterns to 3D using biting insect horns as research material.
This talk is based on the joint work with Keisuke Matsuda (Osaka University).

11:00 -- 11:45   Hirokazu Ninomiya (Hokkaido University)

Abstract: In 1952, Turing proposed the mechanism of pattern formation in which a stable equilibrium of some kinetic system is destabilized by diffusion. In the case of two-component reaction--diffusion systems, however, the diffusion coefficients should be different for Turing's instability. I will give an example of a kinetic system with a asymptotically stable equilibrium, while the corresponding two-component reaction--diffusion system with equal diffusions has a family of unstable stationary solutions that is arbitrarily close to the homogeneous stationary solution.

13:00 -- 13:45   Hideo Ikeda (University of Toyama)

Abstract: Mass-conserving reaction-diffusion systems with bistable nonlinearity are considered under general assumptions, which are useful models for studying cell polarity formation, whose process is key in cell division and differentiation.
The existence of stationary solutions with a single internal transition layer is shown by using the analytical singular perturbation theory. Moreover, a stability criterion for the stationary solutions is provided by calculating the Evans function. Finally, we discuss the stability of double layer stationary solutions, which is an ongoing work. This talk is based on joint work with Masataka Kuwamura (Kobe University).

14:00 -- 14:45   Yasumasa Nishiura (Hokkaido University)

Abstract: The focused investigation on the confined assembly of block copolymer particles (BCP) within emulsion droplets has garnered significant attention. This attention is attributed to the fact that self-assembly within flexible geometries serves as an efficient and promising approach for the preparation of unconventional polymer materials, which are not attainable in either bulk states or solutions. This presentation centers on elucidating the intricate process of forming polyhedral block copolymer particles (PBCP) at the nanoscale, characterized by cubic, octahedral, and various geometric structures. These structures represent a novel class, hitherto unexplored and unprecedented in fabrication.
The emergence of these distinct cornered morphologies introduces an intriguing and counterintuitive phenomenon, intricately linked to process parameters, such as evaporation rates and initial concentration, while maintaining other variables as constants. Employing a system of coupled Cahn-Hillard (CCH) equations, we delve into uncovering the underlying mechanisms steering the formation of polyhedral particles. This exploration places emphasis on the critical role of controlling relaxation parameters for the shape variable 'u' and micro-phase separation 'v'.
While initially perceived as a macroscopic model serving as a metaphor or a qualitative representation to comprehend averaged behaviors, it is revealed that this model is not merely an abstraction. Instead, we establish a substantial correspondence between the experimental settings and the model parameters. This underscores the capability of the macroscopic model to encapsulate the microscopic details of particle shape, challenging previous perceptions and enhancing our understanding of the intricacies involved. This is a joint work with Hiroshi Yabu, Edgar Avalos, and Takashi Teramoto.