Schedule for: 20w5205 - Interfacial Phenomena in Reaction-Diffusion Systems (Online)

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

Consider the following Keller-Segel chemotactic model, which was first proposed by Keller and Segel in 1971 to model the bacteria population chemotaxis in a capillary tube \begin{equation}\label{KS} (1)\hspace{0.5cm}\left\{\hspace{-0.5cm} \begin{array}{lll} &\frac{\partial b}{\partial t}=\frac{\partial}{\partial x} \Big({\frac{\partial b}{\partial x}} -\beta \frac{b}{s}\frac{\partial s}{\partial x} \Big), &x\in R,t>0\\[3mm] &\frac{\partial s}{\partial t}= -bs^{-\alpha}, &x\in R,t>0. \end{array} \right. \end{equation} where $b(x,t)$ is the density of bacteria and $s(x,t)$ is the concentration of chemo-attractant. For the case $1>\alpha>1-\beta$, and for any $c>0$ and $s_\infty>0$ Keller and Segel found explicit presentation of positive wave solutions $(B(x-ct),S(x-ct))$ of system (1) satisfying $$ S(-\infty)=0, S(\infty)=s_\infty>0, B(-\infty)=B(+\infty)=0, $$ which can explain the wave phenomena of the bacteria pulses observed in the experiment. In this talk we shall talk about our recent work on the spectral stability/instability of the whole family of explicit traveling waves $(B(x-ct),S(x-ct))$ in some weighted spaces, by applying detailed spectral analysis, Evan's function method and numerical simulation. We shall also talk about our work on the local well-posedness of solution for the original Keller-Segel model (1). It's a joint work with Yi Li, Yong Li and Hao Zhang.

This talk concerns a hyperbolic model of cell-cell repulsion with a dynamics in the population of cells. More precisely, we consider a population of cells producing a field (the “pressure”) which induces a motion of the cells following the opposite of the gradient. The field indicates the local density of population and we assume that cells try to avoid crowded areas and prefer locally empty spaces which are far away from the carrying capacity. We analyze the well-posedness property of the associated Cauchy problem on the real line. We start from bounded initial conditions and we consider some invariant properties of the initial conditions such as the continuity, smoothness and monotony. We also describe in detail the behavior of the level sets near the propagating boundary of the solution and we find that an asymptotic jump is formed on the solution for a natural class of initial conditions. Finally, we prove the existence of sharp traveling waves for this model, which are particular solutions traveling at a constant speed, and argue that sharp traveling waves are necessarily discontinuous. This analysis is confirmed by numerical simulations of the PDE problem. This is a joint work with Xiaoming Fu and Pierre Magal.

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!

This talk will be devoted to the existence of pulsating travelling front solutions for spatially periodic heterogeneous reaction-diffusion equations in arbitrary dimension, in the multistable case. In general, the notion of a single front is not sufficient to understand the dynamics of solutions, and we instead observe the appearance of a so-called propagating terrace. This roughly refers to a finite family of stacked fronts connecting intermediate stable steady states and whose speeds are ordered. Surprisingly, for a given equation, the shape of this terrace (i.e., the involved intermediate steady states or even their number) may depend on the direction of the propagation. This in turn raises some difficulties in the spreading shape of solutions of the evolution problem. The presented results come from a series of collaborations with W. Ding, A. Ducrot, H. Matano and L. Rossi.

Motivated by the two-dimensional nucleation of crystal growth, we consider the initial-value problem of the level-set mean curvature flow equation with discontinuous source terms. We discuss uniqueness and existence of viscosity solutions and study the asymptotic shape of solutions. Applying the game-theoretic interpretation for this problem, we also study the asymptotic speed of solutions. This talk is based on a joint work with K. Misu (Hokkaido University).

Consider a general semilinear elliptic equation with Neumann boundary conditions. A seminal result of Casten, Holland (1978) and Matano (1979) states that, if the domain is convex and bounded, any stable solution is constant. In this talk, we will investigate whether this classification result extends to convex unbounded domains, or to some non-convex domains. These questions involve the geometry of the domain in a rather intricate way. In particular, our results recover and extend some classical results on De Giorgi's conjecture about the classification of stable solutions of the Allen-Cahn equation in $R^n$.

The interplay between reaction-diffusion evolution and spatial boundary has received a great deal of recent attention. In this talk, we consider an essential example: reaction-diffusion equations in the half-space. Using the maximum principle and the sliding method, we handle a host of reactions (monostable, ignition, and bistable) under a wide class of boundary conditions (Dirichlet and Robin). We consider the existence and uniqueness of steady states, the asymptotic speed of propagation, and the existence of traveling waves. This is joint work with Henri Berestycki.

In 1996, Aoki, Shida and Shigesada proposed a three-component reaction-diffusion model describing the spread of the early farming during the New Stone Age. By numerical simulations and some formal linearization arguments, they concluded that there are four different types of spreading behaviors depending on the parameter values. In this talk, we give theoretical justification to all of the four types of spreading behaviors observed by Aoki et al. We also investigate the case where the motility of the hunter-gatherers is not equal to that of the farmers, which is not discussed in the paper of Aoki et al.

Wave propagation for the two-species Lotka-Volterra competition models has been studied widely. In this talk, we shall focus on the bistable waves and discuss some recent progress.

This talk is concerned with the spatial spreading speed of the following Keller-Segel chemoattraction system, \begin{equation*}\label{abstract-eq1} \begin{cases} u_t=u_{xx}-\chi(uv_x)_x +u(a-bu),\quad x\in{\mathbb {R}},\\ 0=v_{xx}- \lambda v+\mu u,\quad x\in{\mathbb {R}}, \end{cases} \end{equation*} where $\chi$, $a$, $b$, $\lambda$, and $\mu$ are positive constants, and $u(t,x)$ and $v(t,x)$ represent the population densities of a mobile species and a chemo-attractant, respectively. It is well known that, in the absence of chemotaxis (i.e. $\chi=0$), the population of the mobile species spreads at the asymptotic speed $c_0^*=2\sqrt a$. It will be shown in this talk that the chemotaxis neither speeds up nor slows down the spatial spreading of the mobile species provided that the logistic damping constant $b$ is large relative to the chemotaxis sensitivity coefficient $\chi$.

In this talk, we will show some recent results about the limit problem of the principal eigenvalue for some second elliptic and parabolic operators in one dimensional space when the advection coefficient converges to infinity. It has some applications to the existence and stability of solutions of single equations and systems. This is a joint work with Shuang Liu, Yuan Lou and Rui Peng.

In this talk, I will present an ongoing work in collaboration with Danielle Hilhorst about the singular limit of a large class of strongly coupled, strongly competitive two-species reaction--diffusion systems. Particular cases are the standard Lotka--Volterra system, the Potts--Petrovskii cross-taxis system and the SKT cross-diffusion system. We focus on the singular limit of traveling waves and use the sign of the wave speed as a criterion to compare dispersal--growth strategies.

The situation is the following: a line, having a strong diffusion on its own, exchanges mass with the half plane below, supposed to be a reactive medium. A front propagates both on the line and below, and one wishes to describe its shape. This setting was proposed (collaboration with H. Berestycki and L. Rossi) as a model of how biological invasions can be enhanced by transportation networks. Numerical simulations, due to A.-C. Coulon, reveal an a priori surprising phenomenon: the solution is not monotone in the direction orthogonal to the line. We will try to understand this feature in the particular case of a free boundary problem that can be obtained as a limiting case of the original reaction-diffusion system, amd discuss further features of the free boundary, such as its shape at infinity, or what happens when the diffusion on the line becomes infinite. Joint work with L. Caffarelli.

In this talk, I will present recent results on the propagation speed in a reaction diffusion system with an anomalous Levy process diffusion, modeled by a nonlocal equation with a fractional Laplacian and a generalized KPP type monostable nonlinearity. Given a typical Heavy side initial datum, we show that the speed of interface propagation displays an algebraic rate behavior in time, in contrast to the known linear rate in the classical model of Brownian motion and the exponential rate in the KPP model with the anomalous diffusion, and depends on the sensitive balance between the anomaly of the diffusion process and the strength of monostable reaction. In particular, for the combustion model with a fractional Laplacian $(-\Delta)^{s}$, we show that the speed of propagation transits continuously from being linear in time, when a traveling wave solution exists for $s \in (1/2, 1)$, to being algebraic in time with a power reciprocal to $2s$, when no traveling wave solution exists for $s \in (0, 1/2)$. The talk is based on a joint work with Jerome Coville and Mingfeng Zhao.

Mean Curvature Flow defines a gradient-like dynamical system on the space of convex hypersurfaces. I will discuss what is known about the fixed points and connecting orbits of this flow.