Special Session 13: Propagation Phenomena in Reaction-Diffusion Systems

Transition layer structures in reaction-diffusion models with Perona-Malik diffusion

Raffaele Folino Universidad Nacional Autonoma de Mexico Mexico Co-Author(s):

Abstract: In this talk, I consider a reaction-diffusion equation in a bounded interval of the real line with no-flux boundary conditions. In particular, the linear diffusion (typical of the classical reaction-diffusion models) is replaced by the (nonlinear) Perona-Malik diffusion and the reaction term is the derivative of a double well potential with wells of equal depth. After investigating the associated stationary problem and highlighting the differences with the standard results (linear diffusion), we focus the attention on the long time dynamics of solutions, proving either exponentially or algebraic slow motion of profiles with a transition layer structure. This is a joint work with Alessandra De Luca (University of Turin) and Marta Strani (University of Venice).

Traveling Wave Analysis in Receptor-Mediated Models Incorporating Hysteresis Effects

Lingling Hou College of Mathematics and System Science Xinjiang University Peoples Rep of China Co-Author(s):    Izumi Takagi

Abstract: In this talk, we investigate the existence of traveling wave solutions in a reaction-diffusion ordinary differential system, which is coupled by a set of three ordinary differential equations and a reaction-diffusion equation. We employ the geometric singular perturbation theory to demonstrate the existence of the traveling wave solutions. Subsequently, we utilize the contraction mapping principle to prove the uniqueness of the wave speed. At the end of the paper, we conduct numerical simulations for a specific model that meets the hypothetical conditions, validating the obtained results.

Surface curvature drives propagation and chaos of Turing pattern

Shuji Ishihara The University of Tokyo Japan Co-Author(s):    Ryosuke Nishide

Abstract: We study the Turing pattern on curved surfaces. Since the seminal work by A. Turing many researchers have investigated the pattern formation on curved surfaces such as spheres and tori, in which it has been presumed that the Turing pattern is static on curved surfaces, as it is on a flat plane. We show that the Turing pattern on curved surfaces actually moves on curved surfaces. We mainly study reaction-diffusion systems on an axisymmetric surface with periodic boundary conditions, with parameters showing Turing pattern on a flat plane. Our numerical and theoretical analyses reveal that there exist propagation solutions along the azimuth direction, and the symmetries of the surface as well as pattern are involved for the initiation of the pattern propagation. By applying weakly non-linear analysis, we derive the amplitude equations and show that the intricate interactions between modes rise on curved surface and results in the initiation of pattern propagation and even more complex behaviors such as oscillation and chaos. This study provides a novel and generic mechanism of pattern propagation that is caused by surface curvature (which is not possible in 1D systems), as well as new insights into the potential role of surface geometry in pattern dynamics.

Propagating front solutions to Fisher-KPP equation with time-fractional derivative

Hiroshi Ishii Hokkaido University Japan Co-Author(s):

Abstract: In this talk, we address the Fisher-KPP equation with a Caputo derivative as the time derivative and discuss the long-time behavior of the front solution. After briefly explaining the background of the model, we will introduce numerical results and discuss the expected properties of the solution. Additionally, to characterize the long-time behavior of the solution, we assume that the solution asymptotically behaves like a traveling wave solution and present the results of our analysis of potential traveling wave solutions to which the solution may converge. We will also explain the usefulness of these results.

Unbounded traveling wave solutions for reaction-diffusion equations

Ryo Ito Kanagawa University Japan Co-Author(s):    Hirokazu Ninomiya

Abstract: We consider unbounded traveling wave solutions for one dimensional reaction-diffusion equations. Main interest of this talk is existence of unbounded traveling wave solutions and the relation between unbounded and bounded traveling wave solutions. We prove that there exists a threshold speed, called the minimal speed, that separates the existence and non-existence of unbounded wave solutions under few technical assumptions for nonlinearity, especially it includes bistable type nonlinearity, and we reconsider min-max type characterization of the threshold speeds.

Large time behavior of solutions of a cooperative system with population flux by attractive transition

Kousuke Kuto Waseda University Japan Co-Author(s):    Ryuichi Kato

Abstract: In this talk, we consider a cooperative model with cross-diffusion terms of attractive transition type. In the weak cooperative 3D case, the time global well-posedness of classical non-stationary solutions is shown. Especially in the case of large random diffusion coefficients, we show that any nonstationary positive solution asymptotically approaches the coexistence constant steady state at time infinity by constructing a Lyapunov function. We also discuss the relationship between the bifurcation diagram of the steady states obtained by Adachi-Kuto (2024) and the long-time behavior of the non-stationary solutions.

Asymptotic behavior of spreading fronts in an anisotropic multi-stable equation on $\mathbb{R}^N$

Hiroshi Matsuzawa Kanagawa University Japan Co-Author(s):    Mitsunori Nara

Abstract: In this talk, we consider the Cauchy problem for an anisotropic reaction-diffusion equation with a multi-stable nonlinearity on $\mathbb{R}^N$ and study the asymptotic behavior of its solutions. Matano, Mori, and Nara (2019) previously investigated this problem with a bistable nonlinearity, demonstrating that, under suitable conditions on the initial function, the solution develops a spreading front closely approximated by the expanding Wulff shape for sufficiently large times. In this talk, we extend their results to cases involving multi-stable nonlinearities, where the nonlinearity can be decomposed into multiple bistable components.

Compact traveling wave solutions to a mean-curvature flow with driving force

Harunori Monobe Osaka Metropolitan University Japan Co-Author(s):

Abstract: Mean-curvature flow with a driving force appears in various mathematical problems such as motion of soap films, grain boundaries and singular limit problems of various reaction diffusion systems, e.g., Allen-Cahn-Nagumo equation. In this talk, we show the existence and uniqueness of traveling waves, composed of a Jordan curve (or closed surface), for an anisotropic curvature flow with a driving force. We call such a solution ``compact traveling wave in this talk. Our aim is to investigate the condition of external driving force when the curvature flow has traveling waves. This is a joint work with H. Ninomiya.

Blocking and propagation in two-dimensional undulating cylinders with spatial periodicity

Ryunosuke Mori Meiji University Japan Co-Author(s):    Hiroshi Matano

Abstract: We consider blocking and propagation phenomena of mean curvature flow with a driving force in two-dimensional undulating cylinders with spatial periodicity. In this problem, Matano, Nakamura and Lou in 2006 prove the tmie global existence of the classical solutions under some boundary-slope condition that means the bumps of the boundary is not steep. Moreover they characterize the effect of the shape of the boundary to blocking and propagation by introducing a notion called maximal opening angle. However if we do not assume the boundary-slope condition, then the classical solutions do not always exist in time globally and their characterization by the maximal opening angle is not always applicable. In this talk, we consider the effect of the shape of the boundary to blocking and propagation of this problem under more general situation that the solutions may develop singularities near the boundary.

Front propagation for the bistable reaction-diffusion equation on unbounded metric graphs

Yoshihisa Morita Ryukoku Joint Research Center Sci & Tech Japan Co-Author(s):

Abstract: We are concerned with the front propagation for the reaction-diffusion equation with bistable nonlinearity on metric graphs composed of half-lines and branching points. We demonstrate the asymptotic behavior of front-like entire solutions that either propagate beyond the branching points or are blocked by them on certain specific metric graphs. This talk is based on the works Jimbo-M (2019, 2021, 2024), Iwasaki-Jimbo-M (2022), and M (2023).

Reaction-diffusion systems of topological signals coupled by the Dirac operator: a new framework for the emergence of stationary and dynamical Turing patterns.

Riccardo Muolo Tokyo Institute of Technology Japan Co-Author(s):    Ginestra Bianconi (Queen Mary U London and The Alan Turing Institute, UK) & Timoteo Carletti (U Namur and naXys, Belgium)

Abstract: Pattern formation is a key feature of many natural and engineered systems, ranging from ecosystems to neural dynamics. Turing instability provides one of the most famous theories for pattern formation in a continuous domain, which was later extended to networked systems, where the dynamical variables interact in the nodes and flow among nodes by using network links. However, in a number of real systems, including the brain and the climate, dynamical variables are not only defined on nodes but also on links, triangles and higher-dimensional simplexes, leading to topological signals. The discrete topological Dirac operator is emerging as the key operator that allows cross-talk between signals defined on simplexes of different dimensions, for instance among nodes and links signals of a network. Here, we propose a mathematical framework able to generate stationary and dynamical Turing patterns of topological signals defined on nodes and links of networks. This framework accounts for a rich dynamical behavior even without the (Hodge-Laplacian) diffusion term, i.e., occurring solely due to the Dirac operator. This work opens a new framework displaying a rich interplay between topology and dynamics with possible applications to brain and climate networks.

Propagation and Blocking of Bistable Waves by Variable Diffusion

Hirokazu Ninomiya Meiji University Japan Co-Author(s):    Keita Nakajima

Abstract: Biological diffusion processes are often influenced by environmental factors. In this talk, we investigate the effects of variable diffusion, which depends on a point between the departure and arrival points, on the propagation of bistable waves. This process includes neutral, repulsive, and attractive transitions. By using singular limit analysis, we derive the equation for the interface between two stable states and examine the relationship between wave propagation and variable diffusion. More precisely, when the transition probability depends on the environment at the dividing point between the departure and arrival points, we derived an expression for the wave propagation speed that includes this dividing point ratio. This shows that, asymptotically, the boundary between wave propagation and blocking in a one-dimensional space corresponds to the case where the transition probability is determined by a dividing point ratio of 3:1 between the departure and arrival points. Furthermore, as an application of this concept, we consider the Aliev-Panfilov model to explore the mechanism for generating target patterns.

Long time dynamics of a reaction-diffusion model of obesity-induced Alzheimers disease and its control strategies

rana parshad iowa state university USA Co-Author(s):    Ranjit Upadhyay, Debashish Pradhan, Parimita Roy

Abstract: Evidence suggests that obesity, diabetes, and aging notably increase susceptibility to dementia related conditions such as Alzheimers disease. This work explores the correlations between obesity, diabetes, and this disease. It introduces a reaction diffusion model encompassing variables like glucose dynamics, insulin levels, microglia, cytokines, plaques, neurofibrillary tangles, and cognitive decline. The system proposed is an example of a partly dissipative system, as there is a lack of smoothing in several of the state equations. Despite this challenge, we explore the long time behavior of this system, via showing the existence of a global attractor. Several control strategies for the disease are also proposed.

Convergence to forced waves of the Fisher-KPP equation in a shifting environment by utilizing a relative entropy

Masahiko Shimojo Tokyo Metropolitan University Japan Co-Author(s):    Jong-Shenq Guo, Karen Guo

Abstract: This talk aims to investigate the stability of forced waves for the Fisher-KPP equation in a shifting environment, without imposing the monotonicity condition on the shifting intrinsic growth term. A new method is introduced to derive the stability of forced waves under certain perturbation of a class of initial data.

Entire solutions with and without radial symmetry in balanced bistable reaction-diffusion equations

Masaharu Taniguchi Research Institute for Interdisciplinary Science, Okayama University Japan Co-Author(s):

Abstract: Let $n\geq 2$ be a given integer. In this paper, we assert that an $n$-dimensional traveling front converges to an $(n-1)$-dimensional entire solution as the speed goes to infinity in a balanced bistable reaction-diffusion equation. As the speed of an $n$-dimensional axially symmetric or asymmetric traveling front goes to infinity, it converges to an $(n-1)$-dimensional radially symmetric or asymm\ etric entire solution in a balanced bistable reaction-diffusion equation, respectively. We conjecture that the radially asymmetric entire solutions obtained in this paper are associated with the ancient solutions called the Angenent ovals in the mean curvature flows.