Schedule for: 24w5273 - Propagation and Stability in Evolution Equations

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I will consider nonnegative bounded solutions of a class of reaction-diffusion equations set in the whole space R^N. It follows from a celebrated result of C. Jones that the solutions that are initially compactly supported and converge to the maximal steady state at large time, have their level sets which become asymptotically locally planar at large time. I will discuss some recent results of that type for solutions with unbounded initial support and for various classes of reactions. Some counter-examples will also be listed. The talk is based on some joint works with Luca Rossi.

The model under study is a linear transport equation in the upper half plane, together with a nonlinear and nonlocal Dirichlet condition that couples the values of the unknown function at the boundary to those inside. The objective (joint work with G. Faye and M. Zhang) is to understand its precise asymptotics in time. Its structure is reminiscent of that of the "Road-field model" introduced by Berestycki, Rossi and the author. Its primary motivation is the study of the nonlocal Kermack-McKendrick model for the spread of epidemics, and it presents specific issues calling for a mathematical study of its own. This first part is devoted to the derivation of the model and to a discussion of some of its general properties.

In this talk, I shall provide a classification of steady solutions to two-dimensional incompressible Euler equations in terms of the set of flow angles. The first main result asserts that the set of flow angles of any bounded steady flow in the whole plane must be the whole circle unless the flow is a parallel shear flow. In an infinitely long horizontal strip or the upper half-plane supplemented with slip boundary conditions, besides the two types of flows appeared in the whole space case, there exists an additional class of steady flows for which the set of flow angles is either the upper or lower closed semicircles. This type of flows is proved to be the class of non-shear flows that have the least total curvature. As consequences, we obtain Liouville-type theorems for two-dimensional semilinear elliptic equations with only bounded and measurable nonlinearity, and the structural stability of shear flows whose all stagnation points are not inflection points, including Poiseuille flow as a special case. Our proof relies on the analysis of some quantities related to the curvature of the streamlines. This talk is based on a joint work with Huan Xu and Chunjing Xie.

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 present results showing that homogenization occurs for reaction-diffusion equations with both 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, I will introduce some results about bistable pulsating fronts in slowly oscillating one-dimensional environments. First we will show the uniform boundedness of the fronts for any positive period. Then we will show the asymptotical behavior of the speed and profile when the period increasing to infinity. This is a joint work with Francois Hamel and Weiwei Ding.

In this talk, we first prove a stability theorem for traveling waves in a class of non-cooperative reaction-diffusion systems with nonlocal dispersal of equal diffusivities. Our stability criterion is in the sense that the initial perturbation is such that a suitable weighted relative entropy function is bounded and integrable. Then we apply our main theorem to derive the stability of traveling waves for some specific examples of non-cooperative systems arising in ecology and epidemiology.

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We describe some results and open problems concerning the occurrence of sharp interfaces in planar traveling waves. We also comment on the stability of such waves and discuss their parameter dependence. The problem arises in the context of a contact inhibition problem.

In this talk we will present a recent work completed and in progress with Ariela Briani and Hitoshi Ishii where we extend to fully nonlinear operators the well known result on thin domains of Hale and Raugel. The test function approach seems to be very revealing for these problems where the equations converge to lower dimensional domains. Even in the case of the Laplacian the conditions are more general.

Domain geometry can strongly influence steady-state solutions to reaction-diffusion equations. For example, the whole space hosts a multitude of positive bistable steady states, while the Dirichlet half-space admits only one. In this talk, we explore this contrast in epigraphs. The multiplicity of bistable steady states in an epigraph depends on its large-scale structure. Epigraphs that are bounded from below support only one positive solution, while those containing a non-convex cone sport infinitely many. The arguments include moving planes and a connection with constant mean curvature surfaces.

Compartmental models are reaction-diffusion systems, that appear in the modeling of chemical reactions, population dynamics, or epidemiology (the most famous example being in this last case the SIR model of Kermack-McKendrick). In this talk, I will present some results concerning such compartmental models with several interacting population. More specifically, we shall show how the interactions between all the populations lead to a partial propagation of the traits. This talk is based on some results obtained in collaboration with Samuel Tréton (Rouen University)

In this talk I will report some recent progress on a free boundary model, where the free boundary conditions are deduced from the biological assumption that the species expands or shrinks its population range through its members at the range boundary keeping the population density there at a "preferred level". It turns out that this strategy would make the species a super invader: no matter whether the growth function $f(u)$ is monostable, bistable (strong Allee) or of combustion type (weak Allee), the species always spreads successfully.

We will discuss ancient solutions to the mean curvature flow and the Ricci flow and mention some classification results of those. These solutions are important in singularity analysis of geometric flows since they model the singularities.

Logarithmic diffusion is observed in several fields of science, such as the central limit approximation of Carleman’s model based on the Boltzmann equation, a model for long Van-der-Waals interactions in thin fluid films, and the evolution of conformal metric under the Ricci flow on the plane. We focus on the spreading and extinction phenomena of the solution to the logarithmic diffusion equation on a line, in the presence of a logistic reaction term. A Liouville-type theorem will be introduced to understand the extinction and interfacial phenomena from the point of entire solutions. This is a joint work with Hiroshi Matsuzawa (Kanagawa University), Harunori Monobe (Osaka Metropolitan University), and Eiji Yanagida (University of Tokyo).

In this talk, we consider the singular limit of the Allen-Cahn equation with porous medium diffusion. Unlike the linear diffusion, the solution has asymmetrical diffusivity near two stable steady states 0 and 1, which may cause inconsistent propagation movement of the interface and the free boundary. In order to describe the propagation of the interface, we give a sketch on how to construct a pair of sub- and super-solutions.

Incorporating time delay and Stefan type free boundary into reaction-diffusion equation yields a nonlocal problem. Under a KPP type setting we establish a dichotomy on propagation or vanishing. When propagation happens, the spreading speed is shown to exist and it is determined nonlinearly by a delay-induced nonlocal elliptic problem in half line. This talk is based on a joint work with Yihong Du and Ningkui Sun.

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I will cover the following topics (3 per lecture): 1. Local existence 2. Evolution of curvature and total curvature 3. Blow-up results 4. 1st variation and $L^1$ contractivity of Curve Shortening 5. Ancient solutions: Daskalopoulos-Hamilton-Sesum theorem 6. Constructing ancient solutions: Ancient Trombones, Ancient Spirals, and Ancient space Trombones

In this talk, we will be interested in the large-time spreading properties of solutions of reaction-diffusion systems from population dynamics, e.g. of competition or prey-predator type. While some situations are well-understood, in particular when a comparison principle is available or when there are only two species, in the general case this remains a mostly open problem. We will get some insight from the special case of a triangular system, where the problem reduces to a scalar equation with a so-called moving heterogeneity. Some recent results in collaboration with Léo Girardin and Hiroshi Matano will highlight the intricacy of even such a simplified situation.

In this talk, we will discuss our recent progress on the asymptotic behaviour of the principal eigenvalue of some second order operators. Through analyzing the closed orbits of the advection vector field, we obtain a complete result on how the principal eigenvalue converges for large advection. We also find a unified viewpoint to understand some previous results on this topic, like [Berestycki-Hamel-Nadirashvilli, 2005] and [Chen-Lou, 2008]. This is a joint work with Shuang Liu and Yuan Lou.

We consider the long-term dynamics of two epidemic models with free boundaries: one with local diffusion and the other with nonlocal diffusion. Both models are well-posed, and their long-term dynamical behaviors are characterized by a spreading-vanishing dichotomy. When spreading persists, we determine the spreading speed. For the local diffusion model, the spreading speed is finite and determined by an associated semi-wave problem. For the nonlocal diffusion model, a threshold condition is found in terms of the kernel functions appearing in the nonlocal diffusion terms, such that the spreading speed is finite precisely when this condition is satisfied; otherwise, it is infinite, namely accelerated spreading. Finally, for some typical classes of kernel functions, we determine the precise spreading rate of the accelerated spreading in the epidemic region. This talk is based on joint work with Professor Yihong Du and Dr. Wenjie Ni.

Let $n\geq 2$ be a given integer. 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 asymmetric entire solution in a balanced bistable reaction-diffusion equation, respectively. We conjecture that our radially asymmetric entire solutions are associated with the ancient solutions called the Angenent Ovals in the mean curvature flows.

Excitable media are nonlinear dynamical systems that can support abundant spatiotemporal patterns such as traveling fronts, pulses, spiral waves, etc. In this talk, we will study a singular limiting problem of a FitzHugh–Nagumo-type reaction-diffusion system and explore spatiotemporal patterns, which may help us understand the mechanism behind the self-organized patterns in living systems. This talk is based on a series of joint works with Yan-Yu Chen (National Taiwan University) and Hirokazu Ninomiya (Meiji University).

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We consider a nonlocal reaction-diffusion equation that models a population structured in space and in phenotype. We assume that the population lives in a heterogeneous environment, so that the same individual may be more or less fit according to its spatial position. We give a criterion for the persistence of the population, based on the principal eigenvalue of an elliptic operator. We also give, in a particular case involving the Fisher Geometric Model, some optimization results about the shape that the environment should have to make the persistence as hard or as easy as possible. This talk is based on a work in collaboration with Luca Rossi.

In this second part, we prove that any initially localized solution will lag behind the minimal traveling waves and the delay grows logarithmically in time.

We investigate the Keller-Segel system in the plane, focusing on initial conditions with sufficient decay and either critical or supercritical mass. For the critical mass case, we identify a function $u_0$ such that any small perturbation of $u_0$ with critical mass leads to a solution that is globally defined and blows up over an infinite time. This demonstrates the non-radial stability of the infinite-time blow-up. In the supercritical mass case, we provide a new proof for the existence of blow-up solutions, including a precise expansion for the mass. This is joint work with Federico Buseghin (U. of Bath) Manuel del Pino (U. of Bath), Jean Dolbeault (U. Paris Dauphine), Monica Musso (U. of Bath) and Juncheng Wei (UBC).

We are interested in controlling the asymptotic behaviour of two competing species in an interval by controlling the size of the populations at the boundary. In particular, we want to know if it is possible to eradicate one of the species. Since the boundary controls have to satisfy positivity and boundedness constraints, classic techniques in control theory cannot be applied. In this talk, we will discuss non-controllability phenomena due to the presence of barrier solutions depending on the competition coefficients of the systems and on the length of the interval.

Please meet up at the parking lot.

Please use the meal card and get the breakfast items from Comma(or any food merchants)

I will cover the following topics (3 per lecture): 1. Local existence 2. Evolution of curvature and total curvature 3. Blow-up results 4. 1st variation and $L^1$ contractivity of Curve Shortening 5. Ancient solutions: Daskalopoulos-Hamilton-Sesum theorem 6. Constructing ancient solutions: Ancient Trombones, Ancient Spirals, and Ancient space Trombones