偏微分方程及其应用青年学者研讨会会议通知偏微分方程及其应用青年学者研讨会将于 2018 年 12 月 7 日至 12 月 9 日在华东师范大学数学科学学院召开本次研讨会旨在加强国内偏微分方程及其相关领域的青年学者之间的联系, 交流最新研究成果, 促进彼此间的合作, 研讨源于物理几何生

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1 偏微分方程及其应用青年学者研讨会华东师范大学数学科学学院中国上海 2018 年 12 月 7 日 12 月 9 日

2 偏微分方程及其应用青年学者研讨会会议通知偏微分方程及其应用青年学者研讨会将于 2018 年 12 月 7 日至 12 月 9 日在华东师范大学数学科学学院召开本次研讨会旨在加强国内偏微分方程及其相关领域的青年学者之间的联系, 交流最新研究成果, 促进彼此间的合作, 研讨源于物理几何生物以及非线性分析中的前沿课题会议期间统一安排食宿, 费用自理会议组织委员会 : 王丽萍赵纯奕黄侠会议报到 : 会议将于 2018 年 12 月 7 日 14:00 到 21:00 在沪华国际大酒店大厅办理报到手续酒店地址 : 上海市闵行区剑川路 368 号 ( ) 会议地点 : 华东师范大学闵行校区数学楼 401 报告厅联系信息 : 王丽萍 lpwang@math.ecnu.edu.cn 赵纯奕 cyzhao@math.ecnu.edu.cn 黄侠 xhuang1209@gmail.com

3 会议日程 12 月 8 日地点 : 数学科学学院 401 报告厅主持人时间报告信息叶东 08:55 09:00 开幕式徐露 09:00 09:40 09:45 10:25 Mean field equations on tori and the associated pre-modular forms 陈志杰清华大学 Bifurcation results of two semilinear elliptic equations 杜卓然湖南大学 10:25 10:55 照相 & 茶歇王阳 10:55 11:35 午餐 Bound state solutions for the supercritical fractional Schrödinger equation 敖微微武汉大学秋实阁二楼李敬宇 13:30 14:10 14:15 14:55 Nonlocal power-type curvature flows of immersed locally curves 王小六东南大学 Blow-up phenomena for the constant scalar curvature and constant boundary mean curvature equation 陈学长南京大学 14:55 15:20 茶歇向田 15:20 16:00 16:05 16:45 一种带有非局部吸收项目的退化抛物方程解的性态分析白学利西北工业大学 Analysis on cross-diffusive SIS epidemic models 李慧聪中山大学 17:45 晚宴沪华国际大酒店

4 12 月 9 日地点 : 数学科学学院 401 报告厅主持人时间报告信息郭千桥 09:00 09:40 09:45 10:25 Clustering Phase Transition Layers with Boundary Intersection for an Inhomogeneous Allen-Cahn Equation 杨军华中师范大学 On the local uniqueness of the blow up solutions 杨文中科院武汉物理与数学所 10:25 10:45 茶歇杨敏波 10:45 11:25 11:30 12:10 Study of some shallow water wave equations with peakons 魏龙杭州电子科技大学 New solutions for Neumann problems in 邓圣兵西南大学午餐秋实阁二楼下午自由讨论午餐地点 : 秋实阁二楼 ( 数学科学学院对面 ) 8 日晚宴地点 : 沪华国际大酒店

5 报告摘要 ( 报告人字母序 ) Bound state solutions for the supercritical fractional Schrödinger equation 敖微微武汉大学 We prove the existence of positive solutions to the supercritical nonlinear fractional Schrodinger equation, with as, where for. We prove that if as, then for, this problem admits a continuum of solutions. More generally, for, conditions for solvability are also provided. This result is the extension of (Davila- Del Pino-Musso-Wei JDE 2007) to the fractional case. The main contributions for the fractional case are the existence of a smooth, radially symmetric, entire solution of and the analysis of its properties. The difficulty here is the lack of phase plane analysis for a nonlocal ODE; instead we use conformal geometry methods together with Schaaf s argument. 一种带有非局部吸收项目的退化抛物方程解的性态分析白学利西北工业大学我们考虑一种描述二维简单闭合曲线的退化抛物方程, 其中非局部项的出现, 导致这一问题不满足比较原理我们利用能量方法,LS 不等式等技巧, 克服了这一困难在适当范围内, 给出了解的爆破条件和整体存在条件进一步, 对于任意整体有界的解, 我们证明了它一定具有轨道收敛性以上工作跟中山大学李芳副教授和东南大学王小六副教授共同合作完成

6 Blow-up phenomena for the constant scalar curvature and constant boundary mean curvature equation 陈学长南京大学 Inspired by the non-compact examples for the Yamabe equation due to Brendle, Brendle- Marques, we construct an example to show that the full set of the solutions to the constant scalar curvature and constant boundary mean curvature equation is non-compact. More precisely, for each fixed positive constant, there exists an integer, when, the aforementioned set for all conformal metrics with scalar curvature 1 and boundary mean curvature is non-compact in -topology. This is joint with Nan Wu. Mean field equations on tori and the associated pre-modular forms 陈志杰清华大学 Let be a flat torus and consider the mean field equation with four singular sources: (1) where is the Dirac measure at. The solvability of (1) essentially depends on the moduli in a sophisticated manner. In this talk, I will introduce a precise characterization of those s in terms of zeros of pre-modular forms such that (1) has solutions on such. This is based on joint work with Prof. Chang-Shou Lin and Ting-Jung Kuo. New solutions for Neumann problems in 邓圣兵西南大学 In this talk, I will present some results about the existence of bubbling solutions for Neumann problems in, which is related to the critical Trudinger-Moser trace inequality. In particular, we construct positive solutions for critical Neumann problem that concentrates around some points of the boundary. This is a joint work with M. Musso.

7 Bifurcation results of two semilinear elliptic equations 杜卓然湖南大学 We consider the mean field equation on the unit sphere where is a real parameter. We obtain the existence of non-axially symmetric solutions bifurcating from at the values for any odd integer, by using bifurcation theory. For Allen-Cahn equation involving fractional Laplacian, its all periodic solutions admit lower bound. By using bifurcation theory we prove that the lower bound of all periods is not larger than some exact positive constant. This is a joint work with Changfeng Gui, Jiaming Jin and Yuan Li. Analysis on cross-diffusive SIS epidemic models 李慧聪中山大学 We are concerned with two frequency-dependent SIS epidemic reaction-diffusion models in heterogeneous environment, with a cross-diffusion term modeling the effect that susceptible individuals tend to move away from higher concentration of infected individuals. It is first shown that the corresponding Neumann initial-boundary value problem possesses a unique global classical solution which is uniformly-in-time bounded in any spatial dimension. It is further shown that the models admit threshold-type dynamics in terms of the basic reproduction number. Our results on the asymptotic profiles of endemic equilibrium illustrate that restricting the motility of susceptible population may eliminate the infectious disease entirely for the model with constant total population but fails for the model with varying total population. In particular, this implies that such cross-diffusion does not contribute to the elimination of the infectious disease modelled by the second one.

8 Nonlocal power-type curvature flows of immersed locally curves 王小六东南大学 We provide sufficient conditions on an initial curve for the area preserving and the length preserving curvature flows of curves in a plane, to develop a singularity at some finite time or converge to an m-fold circle as time goes to infinity. For the area-preserving flow, the positivity of the enclosed algebraic area determines whether the curvature blows up in finite time or not, while for the length-preserving flow, it is the positivity of an energy associated with initial curve that plays such a role. Study of some shallow water wave equations with peakons 魏龙杭州电子科技大学 In this talk,we consider the Cauchy problems of some shallow water wave equations with peakons. We focus on the local well-posedness, wave-breaking and persistence properties of these equations. Meanwhile, from the viewpoint of Lie symmetry analysis, we obtain some conserved quantities for these equations and present a general procedure of how these conserved quantities come about. Based on these conserved quantities, some analytical results are presented. Clustering Phase Transition Layers with Boundary Intersection for an Inhomogeneous Allen-Cahn Equation 杨军华中师范大学 We consider the nonlinear problem of inhomogeneous Allen Cahn equation where is a bounded domain in with smooth boundary, is a small positive parameter, denotes the unit outward normal of, is a positive smooth function on. Let be a curve intersecting orthogonally with at exactly two points and dividing into two parts, which satisfies stationary and non degeneracy conditions with respect to the functional

9 . We will show the existence of a solution with clustering transition layers near with mutual distance, provided that is small and away from a discrete set of values at which resonance occurs. On the local uniqueness of the blow up solutions 杨文中科院武汉物理与数学所 In this talk, we shall discuss the local uniqueness and non degeneracy results on the blow up solutions of the Mean Field equation. Based on these properties, we derive some conclusion about the Mean Field entropy and some symmetry result on the solution of the Mean Field equation on torus.

10 与会人员信息 ( 姓氏拼音排序 ) 序号姓名单位邮箱 1 敖微微武汉大学 wwao@whu.edu.cn 2 白学利西北工业大学 mybxl110@163.com 3 陈学长南京大学 xuezhangchen@nju.edu.cn 4 陈志杰清华大学 zjchen@math.tsinghua.edu.cn 5 崔仁浩哈尔滨师范大学 renhaocui@163.com 6 邓圣兵西南大学 shbdeng@swu.edu.cn 7 杜卓然湖南大学 duzr@hnu.edu.cn 8 高俊磊华东师范大学 gaojunlei123math@sina. com 9 郭千桥西北工业大学 gqianqiao@nwpu.edu.cn 10 郭倩华东师范大学 @qq.com 11 韩敬文华东师范大学 jwhan950106@126.com 12 胡良根宁波大学 hulianggen@tom.com 13 黄侠华东师范大学 xhuang1209@gmail.com 14 姜佳欣华东师范大学 @qq.com 15 金伟华东师范大学 jin_wei5@163.com 16 金云娟华东师范大学 jyj0926@126.com 17 李芳上海师范大学 lifwx@shnu.edu.cn 18 李芳中山大学 lifang55@mail.sysu.edu.cn 19 李慧聪中山大学 lihuicong@mail.sysu.edu.cn 20 李敬宇东北师范大学 lijy645@nenu.edu.cn

11 序号姓名单位邮箱 21 李震昊华东师范大学 22 刘萍哈尔滨师范大学 23 刘豫宁上海纽约大学 24 陆秋平扬州大学 25 马宁哈尔滨师范大学 26 钮维生安徽大学 27 潘兴斌华东师范大学 28 皮慧荣广西大学 29 秦增芸华东师范大学 30 孙中原哈尔滨师范大学 31 汤厚志华东师范大学 32 王国爽哈尔滨师范大学 33 王金凤哈尔滨师范大学 34 王琦上海理工大学 35 王丽萍华东师范大学 36 王伟宇华东师范大学 37 王小六东南大学 38 王阳杭州电子科技大学 39 王轶明华东师范大学 40 王中亮第二军医大学 41 魏龙杭州电子科技大学

12 序号姓名单位邮箱 42 吴伟华东师范大学 43 向田中国人民大学 44 项杏飞同济大学 45 谢宇华东师范大学 46 徐露湖南大学 47 严煜浩华东师范大学 48 杨军华中师范大学 49 杨敏波浙江师范大学 50 杨文武汉物理数学所 51 姚张锋华东师范大学 52 叶东华东师范大学 53 袁海荣华东师范大学 54 张璐华东师范大学 55 张亚萍华东师范大学 56 张一平华东师范大学 57 章志兵安徽工业大学 58 曾小雨武汉理工大学 59 赵纯奕华东师范大学 60 张艳艳华东师范大学 61 郑有泉天津大学 62 周风华东师范大学

Introduction to Hamilton-Jacobi Equations and Periodic Homogenization

Introduction to Hamilton-Jacobi Equations and Periodic Yu-Yu Liu NCKU Math August 22, 2012 Yu-Yu Liu (NCKU Math) H-J equation and August 22, 2012 1 / 15 H-J equations H-J equations A Hamilton-Jacobi equation