Schedule for: 22w5165 - Interfacial Phenomena in Reaction-Diffusion Systems

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

In this talk I will report on a spectral optimization result obtained in collaboration with Idriss Mazari (Univ. Paris-Dauphine). More precisely, our goal is to optimize the principal eigenvalue of a space-time periodic cooperative operator acting on vector-valued functions. In this problem, the principal eigenvalue is understood as a function of the off-diagonal elements of the coupling matrix. It turns out that this is not a convex optimization problem and that the construction of optimizers, both minimizers and maximizers, requires a new method. We devise such a method by taking inspiration in a matrix-theory paper of 2007 by Neumann and Sze.

The symmetry of solutions of elliptic equations is a classical and challenging problem in PDEs, strictly linked with stability. We consider in this talk reaction-diffusion equations and we ask whether the 1-dimensional symmetry eventually emerges in the long time, for solutions which are initially non-symmetric. We will present a satisfactory answer in the case of the Fisher-KPP equation, together with some counter-examples and open questions. This topic is the object of a joint work with F. Hamel.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

In this work, we consider the speed selection problem of the scalar reaction-diffusion equations and the Lotka-Volterra competition systems. Compared with the classical result on single equations by Muratov et. al., we propose a sufficient and necessary criterion to this long-standing problem of monostable dynamical systems for the first time. Moreover, our results can further reveal the essence of the linear determined problem from a new viewpoint on the decay rate of travelling wave solutions.

In this talk I will consider mean curvature flows in a cylinder with Robin boundary conditions. The boundary slopes are unbounded when the flow goes to infinity, and so there is no uniform-in-time gradient estimates. Nevertheless, by using the zero number argument we can present interior uniform-in-time gradient estimates. Then we show that, in $1D$ case, the flow converges as $t\to \infty$ to a Grim Reaper whose whole profile lies in the cylinder; in $nD$ case, the flow propagates to infinity with exponential speed: $u\sim e^{(N-1)t}e^{\frac{|x|^2}{2}}$. (Joint work with X. Wang and L. Yuan) .

The talk is about the large time dynamics of bounded solutions of reaction-diffusion equations with unbounded initial support in $\mathbb{R}^N$. I will present a Freidlin-Gartner type variational formula for the spreading speeds in any direction. This provides a description of the asymptotic shape of the level sets of the solutions at large time. The formula involves notions of bounded and unbounded directions of the initial support. The results hold for a large class of reaction terms and for solutions emanating from initial conditions with general unbounded support. I will also discuss the sharpness of the results and I will list some counterxamples when the assumptions are not all fulfilled. The talk is based on some joint works with Luca Rossi.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

We revisit the classical question of pushed versus pulled fronts in reaction-diffusion equations. Due to the shape of nonlinear terms, the traveling front can be of pushed, pulled, or pushmi-pullyu type. We explore the identification criteria and introduce a new quantity named as the “shape defect function”, which is crucial in our analysis. We prove the nonperturbative asymptotic stability of traveling wave solutions in precise moving frames for both pulled and pushmi-pullyu types using two key tools: a weighted Hopf-Cole transform and a relative entropy approach to weighted $L^1 \to L^\infty$ decay. This is joint work with Chris Henderson and Lenya Ryzhik.

In this talk, I will report our recent research on monotone evolution systems with asymptotic translation invariance. Under an abstract setting we establish the existence of spatially inhomogeneous steady states and the asymptotic propagation properties for a large class of such systems. Then we apply the developed theory to study traveling waves and spatio-temporal propagation patterns for a reaction-diffusion equation in a cylinder under shifting environment and an asymptotically homogeneous KPP-type equation.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

In this talk I will report some recent results on the rate of spreading of the nonlocal Fisher-KPP equation with free boundary and radial symmetry. It is well known that the local diffusion version of this problem always has a finite spreading speed, determined by the associated traveling wave solutions. This is no longer the case for the nonlocal diffusion problem, whose spreading speed may or may not be finite, depending on whether a certain threshold condition is satisfied by the kernel function in the diffusion term. I will present some sharp estimates on the rate of spreading for the nonlocal problem and reveal some fundamental changes of behavior between dimension one and high dimensions. This is joint work with Dr Wenjie Ni.

For a balanced bistable reaction-diffusion equation, an axisymmetric traveling front has been well studied. We prove that an axially asymmetric traveling front with any positive speed does exist in a balanced bistable reaction-diffusion equation. Our method is as follows. We use a pyramidal traveling front for an imbalanced reaction-diffusion equation whose cross section has a major axis and a minor axis. Preserving the major axis and the minor axis to be given constants and taking the balanced limit, we obtain an axially asymmetric traveling front in a balanced bistable reaction-diffusion equation. This traveling front is monotone decreasing with respect to the traveling axis, and its cross section is a compact set with a major axis and a minor axis.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

In this talk, we investigate the accelerating propagation dynamics of a nonlocal model with periodic time delay, which arises from the study of stage-structured invasive species subject to seasonal successions.

In this talk we discuss the existence of pulsating travelling waves for the so-called Kermack-Mckendrick epidemic system with diffusion and posed in a multi-dimensional periodic medium. Specifically, we prove that there exists a semi-inifite interval of admissible wave speeds such that for each wave speed in this interval and in each direction of propagation, there exists a pulsating travelling wave solution of the epidemic system which is globally bounded. Our proof is based on the reformulation of the pulsating wave profile equation as a non-degenerate space and time periodic problem for some particular directions a propagation. The case of a general direction of propagation is obtained by using a limiting argument on the directions of propagation.

Area-preserving curvature flows in a two-dimensional homogeneous medium have been studied for several decades. In 1986, Gage showed that an initially convex closed curve remains convex and converges to a circle as time goes to infinity. However, in many applications, the medium is not homogeneous. In this talk, we consider the area preserving flow in an inhomogeneous medium when the area enclosed by the interface is small. I will explain the reduced equation of its center. The talk is based on some joint works with R. Lui.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

In this talk, I will present some recent progress on reaction-diffusion fronts in spatially periodic bistable media. The results include: existence of pulsating fronts with large periods, existence of and an explicit formula for the limit of front speeds as the spatial period goes to infinity, convergence of pulsating front profiles to a family of front profiles associated with spatially homogeneous equations. This talk is mainly based on a joint work with Francois Hamel and Xing Liang.

Consider the following SKT competition model $$\left\{ \begin{array}{ll} u_t=\epsilon^2 u_{xx}+(a_1-b_1u-c_1 v)u,\\ v_t=[(1+\gamma u)v]_{xx}+(a_2-b_2u-c_2v)v, \end{array} x\in \mathbb{R},\; t>0; \right.$$ with small diffusion rate $\epsilon >0$ and non-small cross diffusion rate $\gamma>0$, $a_i,b_i,c_i>0$( $i=1,2$) and $d_2>0$. By applying geometric singular perturbation argument and topological index method, Yaping Wu and Ye Zhao proved that the system in strong competition case has a family of stable travelling waves connecting $(0,a_2/c_2)$ and$(a_2/b_2,0)$ with locally unique slow speed $c=\epsilon c_\epsilon$ and both components of the waves have transition layers. In this talk we shall mainly talk about our recent work on the higher order two-scale expansion of the waves with transition layer for small $\epsilon >0$ and the long time behavior and the asymptotic speed of the shifts of two interacting waves. It is a joint work with Dr. Hao Zhang (Capital Normal University) and Prof. Shin Ichiro Ei (Hokkaido University).

In this talk, we study an SIR epidemic model with nonlocal dispersal. We characterize the asymptotic spreading speed of the disease, and describe the asymptotic behavior of solutions, by applying a Liouville-type theorem for entire solutions of a class of reaction-diffusion systems with nonlocal dispersal. This is joint work with Jong-Shenq Guo and Amy Ai-Ling Poh.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

In this talk, we first present a Liouville type theorem for entire solutions of a class of reaction-diffusion systems with fractional diffusion. Then an application of this theorem to the spreading dynamics in some ecological systems is to be addressed. This talk is based on a joint work with Masahiko Shimojo.

For compact convex surfaces Huisken proved in 1985 that Mean Curvature Flow (MCF) deforms them into "round points," i.e. after appropriate rescaling they converge to a sphere. On the other hand, solutions with noncompact convex initial data can converge after rescaling to cylinders. In a rescaled version of MCF these cylinders appear as fixed points. In this talk I will report on ongoing work with Daskalopoulos, Sesum, Bourni, Langford, and Nguyen on the solutions of rescaled MCF that connect these fixed points, and on what they mean for MCF

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Homogenization is a general principle that the dynamics of physical processes occurring in periodic or random environments often become effectively homogeneous in the long-time-large-scale limit. I will presents results showing that homogenization occurs for reaction-diffusion equations with both time-periodic-spatially-random and space-time-random KPP reactions and coefficients. These results rely on two crucial new tools: virtual linearity of KPP reaction-diffusion dynamics and a non-autonomous versions of Kingman’s subadditive ergodic theorem.

In this talk, we are concerned with traveling wave solutions of the classical two-species Lotka–Volterra competitive system with diffusion. There are several types of waves which connect different equilibria of the system. In particular, we will focus on the existence of monostable waves with pulse-front profiles.

5-day workshop participants are welcome to use BIRS facilities (TCPL ) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 11AM.