报 告 人：杜一宏 教授
报告题目：Propagation dynamics of reaction-diffusion equations with a new free boundary condition
报告时间：2024年04月30日（周二）上午10：00-11：00
报告地点：静远楼1508学术报告厅
主办单位：数学与统计学院、数学研究院、科学技术研究院
报告人简介：
杜一宏, 澳大利亚科学院院士, 博士生导师，1962年出生。于1978年至1988年在山东大学获得学士、硕士和博士学位，导师为郭大钧教授。1988年至1991年赴英国Heriot-Watt University大学做Research Fellow，1991年至1992年在澳大利亚新英格兰大学做Research Fellow，合作导师为国际著名数学家、澳大利亚科学院院士E.N. Dancer教授。随后历任澳大利亚新英格兰大学讲师、高级讲师、副教授、教授。2021年当选澳大利亚科学院院士。目前主要研究兴趣包括非线性椭圆型和抛物型偏微分方程、自由边界问题、非线性泛函分析及其应用。已在国际一流数学杂志包括JEMS、ARMA、PLMS、JFA、JMPA、TAMS、AIHP、SIAM、IUMJ、CVPDE、Nonlinearity、JDE等发表学术论文170余篇。已发表论文完全他引次数超过5000次，多次入选Web of Science高被引学者。出版个人研究专著1部。多次组织国际性学术会议，多次担任国际大会执行和学术委员会委员，多次被邀请参加国际学术会议并做全会报告或者邀请报告。获校长杰出研究奖(Vice-Chancellor's Award for Excellence in Research)。已连续多次主持澳大利亚国家研究基金（ARC）。
报告摘要：
I will report some recent results on the reaction diffusion equation $u_t-du_{xx}=f(u)$ with standard nonlinear functions $f(u)$ 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 free boundaries $x=g(t)$ and $x=h(t)$ are not governed by the same Stefan condition as in Du and Lin (2010) and other previous works; instead, they satisfy a related but different set of equations obtained from a “preferred population density” assumption at the range boundary, which allows the population range to shrink as well as to expand. I will demonstrate that the longtime dynamics of the model exhibits persistent propagation with a finite asymptotic propagation speed determined by a certain semi-wave solution, and the density function converges to the semi-wave profile as time goes to infinity. The asymptotic propagation speed is always smaller than that of the corresponding classical Cauchy problem where the reaction-diffusion equation is satisfied for $x$ over the entire real line with no free boundary. Moreover, when the preferred population density used in the free boundary condition converges to 0, the solution $u$ of our free boundary problem converges to the solution of the corresponding classical Cauchy problem, and the propagation speed also converges to that of the Cauchy problem.
