Special Session 11: Eigenvalue problems in reaction-diffusion equations and applications

Asymptotic behavior of the principal eigenvalue for cooperative periodic-parabolic systems and applications

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

Abstract: The effects of spatial heterogeneity on population dynamics have been studied extensively. However, the effects of temporal periodicity on the dynamics of general periodic-parabolic reaction-diffusion systems remain largely unexplored. As a first attempt to understand such effects, we analyze the asymptotic behavior of the principal eigenvalue for linear cooperative periodic-parabolic systems with small diffusion rates. As an application, we show that if a cooperative system of periodic ordinary differential equations has a unique positive periodic solution which is globally asymptotically stable, then the corresponding reactiondiffusion system with either the Neumann or regular oblique derivative boundary condition also has a unique positive periodic solution which is globally asymptotically stable, provided that the diffusion coefficients are sufficiently small. The role of temporal periodicity, spatial heterogeneity and their combined effects with diffusion will be studied in subsequent papers for further understanding and illustration.

A scattering theory on hyperbolic spaces

Lu Chen Beijing Institute of Technology Peoples Rep of China Co-Author(s):

Abstract: In this talk, we introduce a theoretical framework for scattering theory on hyperbolic spaces. We give the accurate characterization of the asymptotic behavior of the Green functions and use them to establish the ingoing and outgoing radiation conditions. Finally, we also discuss a hyperbolic Rellich lemma.

Investigating receptor-based models with hysteresis

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

Abstract: In this presentation, we delve into the existence of traveling wave solutions within a coupled system comprising a reaction-diffusion ordinary differential equation and a trio of additional ordinary differential equations. We apply geometric singular perturbation theory to establish the presence of these traveling wave solutions. Following this, we employ the contraction mapping principle to affirm the uniqueness of the wave speed. To substantiate our theoretical findings, we conclude with numerical simulations for a particular model that aligns with our theoretical assumptions, thereby validating our results.

The global dynamics of an age-structured model with spatial structure

Hao Kang Tianjin University Peoples Rep of China Co-Author(s):    Hao Kang, Rui Peng and Maolin Zhou

Abstract: In this paper we investigate the global dynamics of an age-structured model with spatial structure including random diffusion and advection, and with a monotone nonlinearity in the birth rate. The existence, uniqueness and global stability of a positive equilibrium are given briefly via the theory of monotone dynamical systems. More interesting, we obtain the asymptotic behavior of principal eigenvalue and asymptotic profiles of the equilibrium under the large advection, small diffusion and large diffusion, respectively, which are new compared with the previous work on the diffusive age-structured models. Our tool is the principal spectral theory of linear age-structured operators with diffusion and advection. The proofs are based on the construction of new super-/sub-solutions to solve the issue of the nonlocal birth term which is specific in age-structured models.

Systems of parabolic equations with delays: Continuous dependence on parameters

Marek Kryspin Wroclaw University of Science and Technology Poland Co-Author(s):    Janusz Mierczy\`{n}ski

Abstract: The presentation will cover results concerning linear non-autonomous systems of parabolic partial differential equations with delay. More specifically, I will present theorems regarding the existence and uniqueness of such initial-boundary value problems. Additionally, I will discuss theorems on the regularization of solutions over time and the continuous dependence of solutions on parameters (not just the initial condition) among other functional coefficients of the equation. These types of equations are important for many reasons; however, it should be noted that they generate dynamical systems, and their random versions generate measurable skew-product semi-flows for which the theory of Lyapunov exponents, exponential separation, and Oseledets decomposition is still being developed. Moreover, models based on this type of differential equations are important in mathematical ecology, particularly when modeling interactions between species. The presentation will be based on joint work with Janusz Mierczy\`{n}ski.

On principal eigenvalue for time-periodic parabolic operators

Shuang Liu Beijing Institute of Technology Peoples Rep of China Co-Author(s):    Yuan Lou

Abstract: In this talk, we shall discuss some qualitative properties of the principal eigenvalue for a linear time-periodic parabolic operator under zero Neumann boundary conditions. Various asymptotic behaviors of the principal eigenvalue and its monotonocity, as a function of the diffusion rate and frequency, will be discussed. This analysis will lead to the classification of the topological structures of the level sets for the principal eigenvalue, in the plane of frequency and diffusion rate. This will help us better understand the joint effects of various parameters on the principal eigenvalue.

VANISHING PROPERTY OF PRINCIPAL EIGENFUNCTION FOR COOPERATIVE ELLIPTIC SYSTEMS

Suying Liu Northwestern Polytechnical University Peoples Rep of China Co-Author(s):    Xueli Bai, Ziyang Chen, Yumeng Yin

Abstract: It is well-known that the normalized principal eigenfunction of single elliptic eigenvalue problem with Neumann boundary condition converges to 0 as d goes to 0 for those points satisfies $c(x)

Principal eigenvalues for elliptic operators with large drift

Yuan Lou Shanghai Jiao Tong University Peoples Rep of China Co-Author(s):    Shuang Liu

Abstract: The study on the qualitative properties of principal eigenvalues for second order elliptic operators with drift has a long history. In recent years there are some growing interest in investigating the asymptotic behaviors of such principal eigenvalues with large drift rates. In this talk we will give a brief review and discuss some recent works. This talk is mainly based on joint work with Shuang Liu (Beijing Institute of Technology) .

Nonradial boundary spiky steady states of chemotaxis systems in a symmetric convex planar domain.

Hongze Wang Chinese university of Hong Kong (Shenzhen) Peoples Rep of China Co-Author(s):    Hongze Wang, Xuefeng Wang

Abstract: We investigate the existence of the non-radial steady states of the Keller-Segel model in a bounded convex planar domain that is symmetric with respect to two orthogonal directions via global bifurcation. It is shown that non-radial steady states exist if the chemotactic coefficient exceeds a critical threshold. To model the cell aggregation, one of the most important phenomena in chemotaxis, we also show that boundary spiky solutions exist if the chemotactic coefficient tends to infinity. Our results provide a new insight on the mechanism of the pattern formation and cell aggregation in a bounded convex planar domain with two orthogonal directions.

Linear viscous instability of boundary layer flow

Di Wu South China University of Technology Peoples Rep of China Co-Author(s):    Q, Chen, N. Masmoudi, Y. Wang, Z. Zhang

Abstract: The Tollmien-Schlichting (T-S) waves play a key role during the early stage of the boundary layer transition. In a breakthrough work (Duke Math Jour, 165(2016)), Grenier, Guo and Nguyen gave a first rigorous construction of the T-S waves of temporal mode for the incompressible fluid. In this talk, we show two results about the Tollmien-Schlichting waves: 1. For the incompressible case, we confirm the existence of neutral curve by constructing stable and neutral stable Tollmien-Schlichting waves. 2. We construct the unstable Tollmien-Schlichting waves of both temporal and spatial mode to the linearized compressible Navier-Stokes system around the boundary layer flow in the whole subsonic regime.

Basic reproduction ratios for time-periodic homogeneous evolution systems

Lei Zhang Shaanxi Normal University Peoples Rep of China Co-Author(s):    Fengbin Wang, Xiaoqiang Zhao

Abstract: This report will focus on the basic reproduction ratio in homogeneous evolutionary systems, particularly investigating the equivalence between the basic reproduction ratio and the stability of the corresponding homogeneous system, as well as methods for numerically computing this ratio. The main difficulties will be addressed in the following two aspects: the invariance of homogeneous mappings does not guarantee order-preserving; the cone spectral radius of a compact, positively order-preserving homogeneous mapping may not be continuous.

The nonexistence on the limit of elliptic operators with large drift

Maolin Zhou Nankai University Peoples Rep of China Co-Author(s):

Abstract: The asymptotic behavior of the principal eigenvalue of elliptic operators has been widely investigated for last decades. All previous studies are focusing on how to give an better estimate on the limit for different kinds of operators. In this talk, we will show the first counterexample such that the limit does not exist for some operators with degeneracy.

Linear stability/instability and nonlinear dynamics of the 3-jet zonal flow

Hao Zhu Nanjing University/University of Vienna Peoples Rep of China Co-Author(s):

Abstract: The classical Rayleigh`s criterion gives a necessary condition for the linear instability of a zonal flow on a sphere. In this talk, we will see that it is far from sufficient for the 3-jet flow, and will give the sharp criteria. Mechanisms that induce unstable eigenvalues are provided. Furthermore, we will discuss nonlinear dynamics near the $3$-jet flow, as well as near some general steady flows.