Speaker: Yihong Du, doctoral supervisor, Australian Academy of Science

Date: May 9, 2024

Time: 9:00-9:45

Location: B1032, Zhixin Building, Shandong University

Sponsor: School of Mathematics, Shandong University

Abstract:

I will report some recent results on the reaction diffusion equation$u_t-du_{xx}=f(u)$ 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.

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https://www.view.sdu.edu.cn/info/1020/190641.htm