为交流近年来在偏微分方程及其应用领域所取得的最新研究成果，研讨相关的前沿课题，同时促进偏微分方程相关领域的专家，特别是青年学者之间的交流与合作，华南理工大学数学学院和中国人民大学数学科学研究院将于2020年10月24日-25日举办《2020年秋季华工-人大PDE网上青年研讨会》。 会议将邀请多名专家进行专题学术报告。

我们诚邀您出席此次学术活动；此次研讨会将使用腾讯会议进行线上报告，无任何费用，我们热情期待您的参与支持！

学术委员会：刘正荣   楼元   Izumi Takagi  朱长江

会议组委会：金海洋    向  田

会议主办单位：

华南理工大学数学学院

中国人民大学数学科学研究院

会议资助：

国家自然科学基金

广州市科技计划项目

中国人民大学“双一流”基金

日程安排：

24日会议日程： 2020, 10.24,  8:30—17:00，

腾讯会议ID：  724 432 866

会议链接： https://meeting.tencent.com/s/XpyhT70Dwd7U

25 日会议日程： 2020, 10.25,  8:30—17:00，

腾讯会议ID：  254 945 553

会议链接：https://meeting.tencent.com/s/G1oUlUiUAMR9

《2020年秋季华工-人大PDE网上青年研讨会》

日程安排

2020年10月24日，周六，下午
腾讯会议ID： 724 432 866
时间	会议内容
	Chair:  Izumi Takagi
14:00-14:40	聂华（陕西师范大学）
14:40-15:20	孙建文（兰州大学）
15:20-15:30	休息
	Chair:  刘正荣
15:30-16:10	何小清（华东师范大学）
16:10-16:50	林可（西南财经大学）

2020年10月25日，周日，下午
腾讯会议ID： 254 945 553
时间	会议内容
	Chair:   陶有山
14:00-14:40	李芳（中山大学）
14:40-15:20	彭红云（广东工业大学）
15:20-15:30	休息
	Chair:  向昭银
15:30-16:10	周茂林（南开大学）
16:10-16:50	季善明（华南理工大学）

《2020年秋季华工-人大PDE网上青年研讨会》

报告题目与摘要

On the effects of carrying capacity and intrinsic growth rate on single and multiple species

何小清

华东师范大学

报告摘要：In this talk, we report some recent progress on the impacts of spatial heterogeneity and temporal periodicity on population dynamics and evolution. We will focus on exploring and comparing the qualitative properties and long time dynamical behaviors of these population models under various hypotheses on the correlations between the two quantities: the carrying capacity and the intrinsic growth rate.

Variational approach of critical sharp front speeds in degenerate diffusion model with time delay

季善明

华南理工大学

报告摘要： For the classical reaction diffusion equation, the a priori speed of fronts is determined exactly in the pioneering paper (Benguria and Depassier 1996 Commun. Math. Phys. 175 221–227) by variational characterization method. In this talk, we study the age-structured population dynamics using a degenerate diffusion equation with time delay.We show the existence and uniqueness of sharp critical fronts, where the sharp critical front is C1-smooth when the diffusion degeneracy is weaker with 1 < m < 2, and the sharp critical front is non-C1-smooth (piecewise smooth) when the diffusion degeneracy is stronger with m > 2, and the non-critical waves are C2-smooth. We give a new variational approach for the critical wave speed and investigate how the time delay affects the propagation mechanism of fronts. It is shown that the time delay slows down the critical wave speed.

The dynamics of a Fisher-KPP nonlocal diffusion model with free boundaries

李芳

中山大学

报告摘要： We introduce and study a class of free boundary models with ``nonlocal diffusion", which are natural extensions of the free boundary models in [1] and elsewhere, where ``local diffusion" is used to describe the population dispersal, with the free boundary representing the spreading front of the species. We show that this nonlocal problem has a unique solution defined for all time, and then examine its long-time dynamical behavior when the growth function is of Fisher-KPP type. We prove that a spreading-vanishing dichotomy holds, though for the spreading-vanishing criteria significant differences arise from the well known local diffusion model in [1]. Furthermore, we establish a threshold condition on the kernel function such that spreading grows linearly in time exactly when this condition holds, while when the kernel function violates this condition, accelerating spreading happens.

[1] Y. Du, Z. Lin, Spreading-Vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal., 42 (2010) 377-405.

Concentration profiles of endemic equilibrium for a reaction-diffusion-advection epidemic model

李慧聪

中山大学

报告摘要： We study a reaction-diffusion-advection SIS epidemic model with mass action infection mechanism in a one dimensional bounded domain. We mainly focus on the asymptotic behavior of EE in three cases: large advection; small diffusion of the susceptible population; small diffusion of the infected population. Our main results show that the asymptotic profiles of the susceptible and infected populations obtained here are very different from that of the corresponding system without advection and that of the system with standard incidence infection mechanism.

Global boundedness, long time behavior, and pattern formation driven by the parametrization of a nonlocal Fisher-KPP problem

李静

中央民族大学

报告摘要：  In this talk, we will discuss the global boundedness and the long time behavior of solutions for Cauchy problem of a class of nonlinear nonlocal reaction-diffusion equations which arise in modeling emergence and evolution of a biological species. By introducing a localization technique to generalized the methods for local problem on bounded domain to Cauchy problem of nonlocal problem, the “Hair trigger effect” and the “Ellee effect” are detected respectively for different type of sources and initial data.

Sharp conditions on global existence and blow-up in a degenerate two-species and two-chemical system

林可

西南财经大学

报告摘要： We consider a degenerate chemotaxis model with two-species and two-stimuli in dimension $dgeq 3$ and find two critical lines intersecting at one same point which separate the global existence and blow up of weak solutions to the problem. More precisely, above lines (i.e. subcritical case), the problem admits a global weak solution obtained by the limits of strong solutions to an approximated system. Based on the second moment of solutions, initial datas are constructed to make sure blow up occurs in finite time both on and below lines (i.e. critical and supercritical cases). In addition, the existence or non-existence of minimizers of free energy functional is discussed on critical lines and the solutions exist globally in time if the size of initial data is small. We also investigate a simultaneous blow-up phenomenon. Join work with Professor J.A. Carrillo.

Hopf bifurcation from spike solutions for some Turing systems

梅林峰

浙江师范大学

报告摘要： We develop a way to study the existence and linear stability of Hopf bifurcation from spike solutions for some classical Turing systems. The core of our argument are the two nonlocal eigenvalue problems: one comes from the linearization around the spike solution, leading to the existence of Hopf bifurcation; the other comes from the linearization around the periodic spike solution, leading to the linear stability of the Hopf bifurcation. Application of the theory to the classic Gierer-Meinhardt, Gray-Scott system, and Schnakenberg system leads to the existence of stable, or unstable Hopf bifurcations, respectively. Some numerical computation is needed in the process.

Invasion analysis on a predator-prey system in open advective environments

聂华

陕西师范大学

报告摘要： We investigate a reaction-diffusion-advection system which characterizes the interactions between the predator and prey in advective environments, such as streams or rivers. In contrast with non-advective environments, the dynamics of this system is more complicated. It turns out that there exists a critical mortality rate of the predator and two critical advection rates, which classify the dynamic behavior of this system into two or three scenarios, that is, (i) both populations go extinct; (ii) the predator can not invade and the prey survives in the long run;(iii) the predator can invade successfully when rare and it will coexist permanently with the prey. Specially, the predator can invade successfully when rare if both the mortality rate of the predator and the advection rate are suitably small. Furthermore, by the global bifurcation theory and some auxiliary techniques, the existence and uniqueness of coexistence steady states of this system are established. Finally, by means of numerical simulations, the effects of diffusion on the dynamics of this system are investigated. The numerical results show that the random dispersals of both populations favor the invasion of the predator.

On a parabolic-hyperbolic chemotaxis system with discontinuous data: Well-posedness, stability and regularity

彭红云

广东工业大学

报告摘要： We shall present a system of conservation laws derived from the chemotaxis models describing the initiation of tumor angiogenesis. This system has been well-studied with smooth initial data. However, many interesting questions are still open for the discontinuous initial data. In this talk, we shall present some recent results on the well-posedness of the solutions to the chemotaxis system with the discontinuous initial data.

Global existence and finite time blow-up of solutions of the Gierer-Meinhardt system

彭锐

江苏师范大学

报告摘要： In this talk, we are concerned with a Gierer-Meinhardt system with zero Neumann boundary condition in a bounded smooth domain. We obtain new sufficient conditions for global existence and finite time blow-up of solutions, especially in the critical exponent cases: p-1=r and qr=(p-1)(s+1). This is joint work with Fang Li (Sun Yat-sen University) and Xianfa Song (Tianjin University).

The periodic nonlocal dispersal problem and applications

孙建文

兰州大学

报告摘要：In this talk, we give the recent studies on the periodic nonlocal dispersal equations. We study the basic properties of eigenvalue and limiting behavior of positive solutions. The effect of degeneracy and heterogeneity is also investigated.

The singularity arising in the area-preserving curvature flow

王小六

东南大学

报告摘要： In the area-preserving curvature flow, an embedded convex closed curve evolves into a circle as time goes to infinity, while an immersed curve may have singularity during its evolution. In this talk, we investigate the property of such singularity. It is shown that the singularity must be type-II. The asymptotic shape of evolving curve near the singularity point will also be discussed.

Some results on Keller-Segel(-fluid) systems

王玉兰

西华大学

报告摘要： We will talk about convergence of solutions to a class of parabolic Keller-Segel systems, possibly coupled to the Navier-Stokes equations, to solutions of its parabolic-elliptic counterpart. We shall first establish a general result. This general result will thereafter be concretized in the context of two examples.

Global existence and boundedness in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion and general sensitivity

郑甲山

鲁东大学

报告摘要：In this paper, we consider the chemotaxis-Navier-Stokes system with nonlinear diffusion and rotational flux in a bounded domain with smooth boundary in three dimensions, which describes the motion of oxygen-driven swimming bacteria in an incompressible fluid. With developing some new methods, it is proved that under the condition $m>frac{10}{9}$ and proper regularity hypotheses on the initial data, the corresponding initial-boundary problem possesses at least one global bounded weak solution, which is uniformly bounded. In view of $S$ is a tensor-valued chemotactic sensitivity, it is easy to see that the restriction on $m$ here is optimal, and thus solve the problem left in Bellomo-Belloquid-Tao-Winkler ([1]) and Tao-Winkler ([24]). This result significantly improves or extends previous results of several authors.

A new type of nonlocal Stefan problem

周茂林

南开大学

报告摘要： In this talk, we will discuss a new type of nonlocal Stefan problem, especially its similarity and difference with the classical Stefan problem. It is a joint work with Xinfu Chen and Fang Li.

Dynamics of Competition-Diffusion-Advection Systems

周鹏

上海师范大学

报告摘要： In this talk, I will firstly review some classical results on competitive ODE and reaction diffusion models. Then focusing on the non-self-adjoint systems involving advection terms, I will report some recent development in this direction with an emphasis on how to deal with the advection terms which causes some difficulties in the qualitative analysis of both semi-trivial and coexistence steady states.