报告题目1：Effects of some new free boundary conditions on the nonlocal KPP equation with free boundary

报告人：杜一宏教授，澳大利亚新英格兰大学

报告时间：2024年9月21日

报告地点：20-200

报告摘要：I will report some recent results on the nonlocal reaction diffusion equation $u_t-dL[u]=f(u)$ with a KPP type reaction term $f(u)$ over a changing interval $[g(t), h(t)]$, viewed as a 
model for the spreading of a species with population range $[g(t), h(t)]$ and density $u(t,x)$. The nonlocal diffusion operator $L[u]$ has the form $L[u](t,x)=\int_{g(t)}^{h(t)}J(x-y)u(t,y)dy-u(t,x)$ 
while the free boundaries are governed by $ h’(t)=\mu\int_{g(t)}^{h(t)}K(h(t)-x)u(t,x)dx$,$ g’(t)=-\mu\int_{g(t)}^{h(t)}K(x-g(t))u(t,x)dx$, as well as $u(t, g(t))=u(t, h(t))=0,$ where $K(z)$ is 
nonnegative and continuous for $z\geq 0$ with $K(0)>0$.
Depending on the relationships between $K$ and $J$, new behavior may appear. The basic model of Cao-Du-Li-Li(JFA2019) corresponds to the case that $K(z)=\int_z^\infty J(x)dx$. Some new 
relations between $J$ and $K$ will be examined.
The talk  is based on joint works with Xin Long, Wenjie Ni, Fernando Quiros and Tanshan Yi.

报告题目2：The existence and boundary asymptotic behavior of boundary blow-up solutions to Monge-Ampere equations

报告人：张学梅教授，华北电力大学

报告时间：2024年9月21日

报告地点：20-200

报告摘要：In this talk we consider the existence and the asymptotic behavior of boundary blow-up solutions for a class of Monge-Ampere equations. This is a series of joint work with Professor Yihong Du and Professor Meiqiang Feng.

报告题目3：Asymptotic behavior for a viscous Hamilton-Jacobi equation with degenerate gradient nonlinearity

报告人：张正策教授，西安交通大学

报告时间：2024年9月21日

报告地点：20-200

报告摘要：In this talk, we consider the asymptotic behavior of gradient blowup solutions for a viscous Hamilton-Jacobi equation with degenerate gradient nonlinearity. In the one-dimensional case, the gradient grow-up rate is established by the method of matched asymptotic expansions for infinite time gradient blowup solutions. In the higher dimensional case, the gradient blowup rate is established in suitable ranges of exponents for the finite time gradient blowup and time-increasing solutions. As a by-product which is of independent interests itself, the gradient estimate near boundary for the corresponding elliptic equation is derived under weaker assumptions on the inhomogeneous term. They are joint works with Caihong Chang, Bei Hu and Qiangchang Ju.

报告题目4：Starting-point Entropy and Shigesada–Kawasaki–Teramoto Model

报告人：陈秀卿教授，中山大学

报告时间：2024年9月21日

报告地点：20-200

报告摘要：The existence of global weak solutions to the cross-diffusion model of Shigesada, Kawasaki, and Teramoto for an arbitrary number of species is proved. The model consists of strongly coupled parabolic equations for the population densities in a bounded domain with no-flux boundary conditions, and it describes the dynamics of the segregation of the population species. The diffusion matrix is neither symmetric nor positive semidefinite. A new logarithmic entropy allows for an improved condition on the coefficients of heavily nonsymmetric diffusion matrices, without imposing the detailed-balance condition that is often assumed in the literature. Furthermore, the large-time convergence of the solutions to the constant steady state is proved by using the relative entropy associated to the logarithmic entropy.

报告题目5：Dynamics of consumer-resource reaction-diffusion models: single and multiple consumer species

报告人：何小清教授，华东师范大学

报告时间：2024年9月21日

报告地点：20-200

报告摘要：A consumer-resource reaction-diffusion model with a single consumer species was proposed and experimentally studied by Zhang et al. in 2017. Analytical study on its dynamics was further performed by He et al. in 2019. In this work, we completely settle the conjecture proposed by He et al. in 2019 about the global dynamics of the consumer-resource model for small yield rate. We then study a multi-species consumer-resource model in both homogeneous and heterogeneous environments under various circumstances.

报告题目6：Speed of time almost periodic traveling waves for rapidly/ slowly oscillating reaction-diffusion equations

报告人：丁维维教授，华南师范大学

报告时间：2024年9月21日

报告地点：20-200

报告摘要：This talk is concerned with the wave propagation dynamics of time almost periodic reaction-diffusion equations. Assuming the existence of a time almost periodic traveling wave connecting two stable steady states, we focus on the asymptotic behavior of wave speeds in both rapidly and slowly oscillating environments. We prove that, in the rapidly oscillating case, the average speed of the time almost periodic wave converges to the constant wave speed of the homogenized equation. On the other hand, in the slowly oscillating case, the average speed converges to the arithmetic mean of the wave speeds for a family of equations with frozen coefficients. These explicit formulas for the limits of speeds also show the significant influences of temporal variations on wave propagation phenomena. Even in the periodic environment, it can alter the sign of bistable wave speeds.

报告题目7：Boundedness in a two-dimensional doubly degenerate nutrient taxis system

报告人：李玉祥教授，东南大学

报告时间：2024年9月21日

报告地点：20-200

报告摘要：In this talk, we consider a no-flux initial-boundary value problem for the doubly degenerate nutrient taxis system in a smoothly bounded convex domain. We prove that for all reasonably regular initial data, the problem possesses a global bounded weak solution. This work improves the previous works.

报告题目8：移动环境中一类时滞非合作系统的传播性质

报告人：林国教授，兰州大学

报告时间：2024年9月21日

报告地点：20-200

报告摘要：本报告介绍我们最近关于移动环境中一类时滞非合作系统的传播性质。这类模型部分源于一类时滞传染病模型，该模型描述未成年易感者、感染者以及成年个体之间的相互作用。已有工作研究了成年个体一致持久时感染者的空间传播问题。我们关注成年个体具有常数渐近传播速度时候未成年感染者的空间扩张行为。主要是获得了不同情形下感染者的不同渐近传播速度。

报告题目9：Bistable dynamics in a single species model with seasonality

报告人：王金凤教授，哈尔滨师范大学

报告时间：2024年9月21日

报告地点：20-200

报告摘要：We will discuss some basic dynamics for a single species ODE model, with strong Allee effect growth in good season.

报告题目10：几类k-Hessian型方程（组）解的相关问题研究

报告人：冯美强教授，北京信息科技大学

报告时间：2024年9月21日

报告地点：20-200

报告摘要：在这个报告中，我将介绍近些年在k-Hessian 型方程（组） Dirichlet 和边界爆破问题解的研究方面所取得的一些新结果。这些结果主要是和华北电力大学张学梅教授合作完成。

报告题目11：Uniqueness of critical points of the second Neumann eigenfunctions ontriangles

报告人：姚若飞教授，华南理工大学

报告时间：2024年9月21日

报告地点：20-200

报告摘要： The hot spots conjecture, proposed in 1974 by Jeff Rauch, states that the second Neumann eigenfunction of the Laplacian attains its global maximum (hottest point) exclusively on the boundary of the domain. Notably, for triangular domains, the nonexistence of interior critical points was recently established by Judge and Mondal in [Ann. Math., 2022]. Nevertheless, several pertinent questions surrounding the second Neumann eigenfunction in triangles lingered unanswered. In this talk, we delve into these unresolved issues such as (1) the uniqueness of non-vertex critical point, (2) the sufficient and necessary conditions for the existence of non-vertex critical point, (3) the exact location of the global extrema, (4) the location of endpoints of the nodal line, and so on. Our findings not only confirm both the original theorem and Conjecture 13.6 proposed by Judge and Mondal in [Ann. Math., 2020], but also achieve a key objective outlined in the Polymath 7 research thread 1 led by Terence Tao. Our approach employs the continuity method via domain deformation. This is a joint work with Prof. Hongbin Chen and Prof. Changfeng Gui.

报告题目12：Does the limit of the principal eigenvalue for the second order elliptic operator with large advection always exist?

报告人：白学利教授，西北工业大学

报告时间：2024年9月21日

报告地点：20-200

报告摘要：In this talk, we consider the following elliptic eigenvalue problem$ -\Delta\varphi(x)-2s\mathbf{v}\cdot\nabla\varphi(x)+c(x)\varphi(x)=\lambda(s)\varphi(x)$on a bounded domain $\Omega$ with the Neumann boundary condition. There are numerous studies focusing on the convergence of the limit of the principal eigenvalue $\lambda(s)$ as $s\to+\infty$, under some conditions. However, the non-existence of the limit of the principal eigenvalue remains an open problem. In view of this, we construct the first counterexample such that $\lambda(s)$ does not converge as $s\to+\infty$.

报告题目13：Weak Diffusive Stability Induced by High-order Spectral Degeneracies

报告人：吴启亮教授，美国俄亥俄大学

报告时间：2024年9月21日

报告地点：20-200

报告摘要：The Lyapunov stability of equilibria in dynamical systems is determined by the interplay between the linearization and nonlinear terms. In this talk, we present our recent results on the case when the spectrum of the linearization is diffusively stable with high-order spectral degeneracy at the origin. Roll solutions at the zigzag boundary of the Swift-Hohenberg equation are shown to be nonlinearly stable, serving as examples that linear decays weaker than the classical diffusive decay, together with quadratic nonlinearity, still give nonlinear stability of spatially periodic patterns. The study is conducted on two physical domains: the 2D plane and the infinite 2D torus. Linear analysis reveals that, instead of the classical $t^{-1}$ diffusive decay rate, small perturbations of zigzag stable roll solutions decay with slower algebraic rates ($t^{-3/4}$ for the 2D plane; $t^{-1/4}$ for the infinite 2D torus) due to the highorder degeneracy of the translational mode at the origin in the Bloch-Fourier spaces. The nonlinear stability proofs are based on decompositions of the neutral translational mode and the faster decaying modes, and fixed-point arguments, demonstrating the irrelevancy of the nonlinear terms.

邀请人：非线性分析与PDE团队