报告题目1：Global classical solutions to a two-dimensional chemotaxis-fluid system involving signal-dependent degenerate diffusion

报告人：王一夫教授，北京理工大学

报告时间：2024年10月28日（周一）

报告地点：20-202

报告摘要：This talk is concerned with the two-dimensional chemotaxis- fluid model which accounts for signal-dependent motilities of microbial populations interacting with an incompressible liquid through transport and buoyancy. Here smooth motility functionsatisfies  on but , . For all reasonably regular initial data,  the corresponding initial boundary value problem possesses global classical solutions under the suitable smallness mass of microbial populations; when   the logistic kinetics is taken into account in the microbial populations, this problem possesses global bounded classical solutions, which can converge toward (1,0,0) as time tends to infinity when a certain small mass is imposed on  signal,  and thereby extends the recent results for the fluid-free analogue to the system coupled to a Navier-Stokes equations.

报告题目2：On autoignition of co-flow laminar jets with nonlinear diffusion

报告人：黄锐教授，华南师范大学

报告时间：2024年10月28日（周一）

报告地点：20-202

报告摘要：This talk is concerned with mathematical analysis of a model for autoignition of laminar co-flow jets with nonlinear diffusion. We provide detailed analysis of the model that reveals dependency of the autoignition position on principal physical and geometric parameters involved.

报告题目3：On a Lotka-Volterra competitive patch model

报告人：周鹏教授，上海师范大学

报告时间：2024年10月28日（周一）

报告地点：20-202

报告摘要：A Lotka-Volterra patch model describing the competition between two aquatic species is investigated, where organisms are supposed to take both random and advective movements. The global dynamics is determined in different settings of parameters, where sufficient analysis on the principal eigenvalue is performed to determine the local dynamics of the boundary equilibria and a technical approach is developed to establish the nonexistence of any positive equilibrium for any number of patches. This work can be seen as a further development of a series of works on the corresponding space-continuous PDE model.

报告题目4：Spatio-temporal Dynamics in a Reaction-Diffusion Equation with Nonlocal Spatial Memory

报告人：宋永利教授，杭州师范大学

报告时间：2024年10月28日（周一）

报告地点：20-202

报告摘要：To model a single-species cognitive movement, we formulate a reaction-diffusion equation with nonlocal spatial memory and investigate its dynamics. We explore the influence of the perceptual scale on the stability and Turing bifurcation. When the random diffusion is dominant, the perceptual scale does not affect the stability, but when the memory-based diffusion is dominant, there exist Turing bifurcations induced by the perceptual scale. Then the joint effect of the perceptual scale and the memory delay on the stability and spatio-temporal dynamics is investigated to show rich spatiotemporal dynamics via Turing--Hopf bifurcation and double Hopf bifurcation. Finally, we apply our analysis to an application and illustrate our theoretical results with numerical simulations. This is a joint work with Shuyang Xue and Hao Wang.

报告题目5：Mixed local and nonlocal problems with nonstandard growth

报告人：张超教授，哈尔滨工业大学

报告时间：2024年10月28日（周一）

报告地点：20-202

报告摘要：In recent years, the mixed local and nonlocal problems have been widely studied owing to its essential applications in plasma physics and biology. In this talk, we consider the mixed local and nonlocal functionals with nonstandard growth. By utilizing the expansion of positivity, we study the local behaviour of the minimizers of such problems, involving the local boundedness, the local Holder continuity, and Harnack type inequality.

报告题目6：On several reaction-diffusion SIS epidemic models

报告人：崔仁浩教授，哈尔滨师范大学

报告时间：2024年10月28日（周一）

报告地点：20-202

报告摘要：We are concerned with several reaction-diffusion SIS epidemic models in advective heterogeneous environments, and investigate the combined effect of spatial heterogeneity and individual dispersal on dynamics of infected diseases. Threshold- type results on the global dynamics in terms of the basic reproduction number are established. We further investigate the effects of diffusion, advection and mechanism on asymptotic profiles of the endemic equilibrium. The individuals concentrate at the downstream end when the advection rate tends to infinity. As the diffusion rate of the infected individuals tends to zero, the density of the infected may vanish on the habitat. The results may provide some implications on disease control and prediction.

报告题目7：Several population-toxicant PDE models

报告人：黄启华教授，西南大学

报告时间：2024年10月28日（周一）

报告地点：20-202

报告摘要：In this presentation, we will introduce several PDE models that describe the interactions between populations and toxicants in polluted aquatic environments. These models include size-structured population models affected by pollution, reaction-diffusion-advection models applicable to river ecosystems, and toxicant-taxis models relevant to lake ecosystems. Theoretical and numerical findings will be utilized to identify pivotal factors influencing population persistence and extinction, along with the spatial distributions of populations and toxicants.

报告题目8：Prevention of Infinite-time Blowup by Slightly Super-linear Degradation in a Keller--Segel System with Density-suppressed Motility

报告人：江杰研究员，中国科学院精密测量科学与技术创新研究院

报告时间：2024年10月28日（周一）

报告地点：20-202

报告摘要：In this talk, we consider an initial-Neumann boundary value problem for a Keller--Segel system with non-local Fokker-Planck type diffusion and source terms. Infinite-time blowup of the classical solution was previously observed for its source-free version when dimension $N\geq2$. In this talk, we will report our recent result that with any source term involving a slightly super-linear degradation effect on the density, of a growth order of $s\log s$ at most, the classical solution is uniformly-in-time bounded when $N\leq3$, thus preventing the infinite-time explosion detected in the source-free counter-part. By contrast, we recall that there are finite-time blowups in Keller-Segel system with Fick type diffusion even when slightly super-linear degradation gets involved. Thus, our result reveals an important difference between Fokker-Planck type diffusion and Fick type diffusion in Keller--Segel models. We will first outline the comparison method developed by the speaker to study the homogeneous problem and we review some previous results concerning global boundedness as well as infinite blowups. Then, we show that an improved comparison argument by introducing a new auxiliary variable, together with a construction of an entropy-like inequality will yield to the desired blowup-prevention result.

报告题目9：Asymptotic sets of spreading for spatially periodic reaction-diffusion equations

报告人：郭宏骏教授，同济大学

报告时间：2024年10月28日（周一）

报告地点：20-202

报告摘要：In this talk, we discuss the Cauchy problem for the reaction-diffusion equation in spatially periodic media with unbounded initial support. We prove a variational formula for the spreading speed and equivalent formula for the spreading sets, which could illustrate the asymptotic shape of the level sets of solutions at large time.

报告题目10：A generalization of Darboux-Froda theorem and its applications

报告人：易泰山教授，中山大学

报告时间：2024年10月28日（周一）

报告地点：20-202

报告摘要：In real analysis, the Darboux-Froda theorem states that all discontinuities of a real-valued monotone functions of a real variable are at most countable.  In this paper, we extend this theorem to a family of  monotone real vector-valued functions  of a real variable arising from dynamical systems. To this end, we explore some essential characteristics  of countable and uncountable sets by the notions of strong cluster points, upper and lower strong cluster points, and establish the existence of strong cluster point sets, upper and lower strong cluster point sets for an uncountable set. With the help of these cluster point sets, we establish a jump lemma that help characterize the discontinuities of the family of monotone vector-functions.  Then we  introduce the notion of distinction set and prove the existence of a distinction set.  Making use of  the upper and lower strong cluster points of the distinction set and the jump lemma, we prove the Darboux-Froda extension theorem. Moreover, we also present some applications of the generalized Darboux-Froda theorem.

报告题目11：具奇性敏感度趋化模型的一些新结果

报告人：李彬教授，宁波工程学院

报告时间：2024年10月28日（周一）

报告地点：20-202

报告摘要：本次报告首先介绍几类趋化模型以及其定性分析方面的研究进展，其次探讨奇性敏感、混合阻尼、非线性信号产生等机制对相关模型解性质的影响，最后简要说明相关证明的主要思路。部分结果与重庆师范大学谢莉教授合作完成。

报告题目12：A Hamilton-Jacobi approach for asymptotic spreading of competitive species

报告人：刘爽教授，北京理工大学

报告时间：2024年10月28日（周一）

报告地点：20-202

报告摘要：In this talk, we shall discuss some spreading properties of the Lotka-Volterra competition-diffusion system. When the initial data vanish on a right half-line, we derive the exact spreading speeds and prove the convergence to homogeneous equilibrium states between successive invasion fronts. Our method is based on the Hamilton-Jacobi approach for Fisher-KPP equation due to Freidlin, Evans and Souganidis. Our main result settles an open question raised by Shigesada et al. in 1997, and shows that one of the species spreads to the right with a nonlocally pulled front. This is a joint work with King-Yeung Lam and Qian Liu.

报告题目13：On a (cross-)diffusive SIS epidemic model with power-type incidence

报告人：李慧聪教授，中山大学

报告时间：2024年10月28日（周一）

报告地点：20-202

报告摘要：We study global existence, boundedness, and convergence of non-negative classical solutions of a Neumann initial-boundary value problem for a cross-diffusive SIS (susceptible-infected-susceptible) epidemic model with power-type infection mechanism generalizing the mass action type. The cross-diffusive term describes the effect that the susceptible individuals tend to move away from higher density of the infected population. Global existence and boundedness of classical solutions are obtained in certain parameter ranges, and threshold/non-threshold long-time behaviors of global and bounded solutions

邀请人：非线性分析与PDE团队