Special Session 6: Propagation dynamics of PDEs: recent progress and trends

Bistable pulsating fronts for reaction-diffusion equations with advection in periodic media

Weiwei Ding South China Normal University Peoples Rep of China Co-Author(s):    Xing Liang, Linfeng Xu

Abstract: This talk focuses on bistable pulsating fronts for multidimensional reaction-diffusion equations with advection in periodic media. We will establish the existence of pulsating fronts in rapidly oscillating media and characterize the homogenized wave limit. In addition, we will present estimates for the convergence rate of the wave speed as the period approaches zero, and discuss the influence of advection on the propagation dynamics.

Spreading in a shifting environment: linear and nonlinear features

Thomas Giletti University Clermont-Auvergne France Co-Author(s):

Abstract: In this talk, we will discuss the large time behavior of solutions of reaction-diffusion equations and systems with shifting heterogeneities. Such a situation arises in the modeling of population dynamics under an environmental change, due e.g. to global warming. We will be concerned with the extinction or survival of species, and in the latter case in how it spreads ahead of the heterogeneity. While the key issue is whether the species is able to outpace the climate, which may sound relatively straightforward, we will find the picture may quickly become more intricate than expected. In particular, we will see how nonlinear features may strongly impact the outcome due for instance to Allee effects or inter-species competition. In particular, we will see that the size of the initial population may play a crucial part into its survival at large times.

Spreading dynamics for the Lotka-Volterra system with general initial supports: the strong competition

Hongjun Guo Tongji University Peoples Rep of China Co-Author(s):

Abstract: This talk concerns the spreading dynamics of a high-dimensional strong competition Lotka-Volterra system where two species initially occupy disjoint measurable (possibly unbounded) subsets in R^N, which are called initial support. Recently, Hamel and Rossi introduced some new geometric notions, such as bounded or unbounded directions and positive-distance interior, for single-species equations with general initial supports. Under these notions and appropriate assumptions, we characterize directional spreading behavior for the two-species system: precise spreading speeds and sets for both species are derived.

Reaction-diffusion equations in R^N: convergence to fronts

Francois Hamel Aix-Marseille University France Co-Author(s):    Hongjun Guo and Luca Rossi

Abstract: This talk is concerned with reaction-diffusion-advection equations in homogeneous or spatially periodic media. I will discuss the asymptotic properties of the solutions of the Cauchy problem, under an assumption of weak stability of the constant steady states 0 and 1, or for Fisher-KPP reactions. I will especially show that front profiles appear, along sequences of times and points, in the large-time dynamics of the solutions, whether their initial supports be bounded or unbounded. I will also discuss further geometrical properties of the asymptotic invasion shapes of invading solutions. The talk is based on some joint works with Hongjun Guo and Luca Rossi.

Propagation dynamics of a nonautonomous epidemic system with nonlocal diffusion

Zhucheng Jin University of Science and Technology of China Peoples Rep of China Co-Author(s):

Abstract: In this talk, we investigates the spreading speeds and generalized traveling wave solutions for a nonautonomous epidemic system with nonlocal dispersal. The main challenge arises from the lack of a comparison principle for such nonmonotone systems. To characterize the spreading speed of the system, we first establish the spreading speed of a scalar nonautonomous KPP equation. Then, using a derived key pointwise estimate, we compare the solution of the system with that of the scalar KPP equation and thus obtain the spreading speed of the system. Combining with the spreading speed results, we demonstrate the nonexistence of generalized traveling wave solutions with small wave speeds in average sense. By constructing proper super and sub solutions, we establish the existence of generalized traveling wave solutions and study the limiting behavior of the wave profile using the uniform persistence theory. This talk is based on a joint work with Prof. Xiongxiong Bao.

Spreading speeds of diffusive age-structured equations in slowly and rapidly varying media

Hao Kang Tianjin University Peoples Rep of China Co-Author(s):    Shuang Liu

Abstract: We investigate the effects of the spatial period in determining the spreading speed of age-structured reaction-diffusion equations in spatially periodic media. We establish the asymptotic behavior of the spreading speed when the spatial period tends to zero and infinity, respectively. By comparing the spreading properties between the age-structured models and the classic Fisher-KPP reaction-diffusion equations, we show that the introduction of the age structure leads to more complicated spreading dynamics. Specifically, in rapidly oscillating media, there does not exist a definitive relationship between their spreading speeds. In contrast, in slowly oscillating media, the two spreading speeds coincide and can be characterized by a family of periodic Hamilton-Jacobi equations, with the zero order term determined by an age-structured equation parameterized by the spatial variable. Finally, we present an explicit formula for the limiting spreading speed in patchy environments, via constructing the viscosity solutions of the associated Hamilton-Jacobi equation. This is a joint work with Shuang Liu in BIT.

Free boundary problems for the spread of ecosystem engineers

Frithjof Lutscher University of Ottawa Canada Co-Author(s):

Abstract: Propagation dynamics describing the spread of invasive species are historically studies on static landscapes. Many such species, however, alter the landscape through which they spread in order to improve conditions for their own survival. Such species are referred to as ``ecosystem engineers``. To describe the propagation dynamics for such species, we have to include the interactions between the species and the landscape, which makes the landscape dynamic. We formulate a two-sided free boundary problem for the spread of ecosystem engineers. We study the existence and uniqueness of traveling waves for this system under different assumptions of the population dynamics (monostable or bistable) and different scenarios for the movement condition of the free boundary. We discover a large range of qualitatively different profiles of traveling waves under different parameter regimes.

Dynamics of Curves under the Anisotropic Area-Preserving Curvature Flow

Hirokazu Ninomiya Meiji University Japan Co-Author(s):    Harunori Monobe

Abstract: The area-preserving curvature flow was first studied by Gage in 1986. He proved that any smooth closed planar curve evolving under this flow exists for all time and converges to a circle enclosing the same area. A natural question is how the dynamics change when the coefficients depend on the anisotropy. In this talk, we study the anisotropic area-preserving curvature flow for planar curves, where both the external forcing term and the coefficient of the curvature may exhibit anisotropy depending on the orientation of the curve. Under suitable assumptions on the anisotropic coefficients, we prove the uniqueness of traveling wave solutions. Moreover, we show that any closed convex curve evolving under the flow converges to this traveling wave solution.

Infinite cascade of traveling waves in a nonlocal model of gravitational fingering

Iuliia Petrova University of Sao Paulo (USP) Brazil Co-Author(s):    Sergey Tikhomirov

Abstract: We introduce a new nonlocal model for gravitational fingering and study its wave propagation dynamics. Gravitational fingering is an instability in the displacement of miscible fluids in porous media, governed by Darcy's law, which occurs when a lighter fluid lies below a heavier one and gravity drives an exchange of positions. Our main result establishes the existence of a propagating terrace consisting of infinitely many traveling waves in a semi-discrete Transverse Flow Equilibrium (TFE) model. The talk is based on ongoing joint work with Sergey Tikhomirov (PUC-Rio, Brazil) and extends earlier results on two-tubes models of gravitational fingering (SIMA, arxiv:2401.05981)

Front propagation dynamics in Fisher-KPP equations on unbounded metric graphs

Wenxian Shen Auburn University USA Co-Author(s):    Hewan Shemtaga, Selim Sukhtaiev

Abstract: This talk is concerned with front propagation dynamics in Fisher-KPP equations on unbounded metric graphs. Such equations can be used to model the evolution of populations living in environments with network structure. There are several studies on front propagation phenomenon in bistable equations on unbounded metric graphs. It is known that, in such equations, the network structure of the underlying environment may block the propagation of the fronts. It will be shown in this talk that the network structure of the environments does not block the propagation of the fronts in Fisher-KPP equations. In particular, it will be shown that the Fisher-KPP equation on an unbounded graph with finite many edges has the same spreading speed $c^*$ as the Fisher KPP equation on the real line $\mathbb{R}$ and has a generalized traveling wave connecting the stable positive constant solution and the trivial solution with averaged speed $c$ for any $c>c^*$.

Polyhedral entire solutions in reaction-diffusion equations

Masaharu Taniguchi Okayama University Japan Co-Author(s):

Abstract: We study polyhedral entire solutions to a bistable reaction-diffusion equation in $\mathbb{R}^{n}$. We consider a pyramidal traveling front solution to the same equation in $\mathbb{R}^{n+1}$. As the speed goes to infinity, its projection converges to an $n$-dimensional polyhedral entire solution. Conversely, as the time goes to $-\infty$, an $n$-dimensional polyhedral entire solution gives $n$-dimensional pyramidal traveling front solutions. The result in this paper suggests a correlation between traveling front solutions and entire solutions in general reaction-diffusion equations or systems.

Propagation for heterogeneous reaction-diffusion equations on adjacent domains

Andrea Tellini Universidad Politécnica de Madrid Spain Co-Author(s):    Henri Berestycki, Luca Rossi

Abstract: We consider a system of two Fisher-KPP-type reaction-diffusion equations formulated on adjacent domains, with different diffusion coefficients and reaction parameters. The two equations are coupled via a transmission condition at their common boundary. We will show how these heterogeneities influence the global behavior of the system, focusing in particular on the asymptotic speed of propagation. These results are based on a joint work with Henri Berestycki (EHESS - Paris) and Luca Rossi (Sapienza University - Rome).

Stability of traveling wave solutions for the singular Keller-Segel model with logistic source

Zhian Wang The Hong Kong Polytechnic University Hong Kong Co-Author(s):    Jingyu Li

Abstract: The classical singular Keller-Segel system with logistic growth has been shown to admit a unique traveling wave solution (up to a translation) with a minimal (critical) wave speed, yet the stability of this solution remains unproven. In contrast, in the absence of logistic growth, the system has a unique traveling wave solution with a single and well-defined wave speed whose stability has been well established in the literature. This study establishes the nonlinear stability of traveling wave solutions for the singular Keller-Segel system with logistic growth, covering both critical and supercritical wave speed scenarios. To our knowledge, this work marks the first stability result concerning traveling wave solutions of chemotaxis-growth systems. We prove our stability result by following three steps. First, we derive several novel auxiliary results regarding the traveling wave solutions that are indispensable for subsequent stability analysis. Next, by leveraging the anti-weighted function technique, we identify appropriate weight functions needed in our stability analysis. Finally, by employing the weighted energy method and conducting sophisticated coupling estimates, we show that the unique traveling wave solution is nonlinearly asymptotically stable under small perturbations.

Propagation dynamics of reaction and diffusion equations in a time-heterogeneous shifting environment

Xiaoqiang Zhao Memorial University of Newfoundland Canada Co-Author(s):    Lei Zhang and Xiao-Qiang Zhao

Abstract: In this talk, I will first give a brief review of the propagation dynamics of evolution equations in a shifting environment. Then I will report our recent research on a large class of time and space heterogeneous reaction-diffusion equations. We obtain the leftward and rightward spreading speeds in terms of the shifting speed and the spreading speed of an associated limiting equation. We also establish the existence, uniqueness and nonexistence of the forced traveling wave, and further apply this result to obtain the existence of almost periodic traveling waves for nonautonomous SIS epidemic models.