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​Mini-course on Free Boundary Models with Local and Nonlocal Diffusion

日期:2020-07-27点击数:

Yihong Du (University of New England)

Lanzhou University

https://meeting.tencent.com

July 27: Diffusion, propagation and free boundaries: An overview

Conference ID: 763367921; Conference password: 202007

This part gives a brief review of the mathematical theory devoted to the modeling of populationpropagation, involving reaction diffusion equations with local as well as nonlocal diffusion, withor without free boundary.

July 28: Free boundary models with local diffusion in 1-D

Conference ID: 624947572; Conference password: 202007

We will look at the Fisher-KPP equation with diffusion and free boundaries in one spacedimension, and start with some basic techniques to treat questions including existence,comparison principle, spreading-vanishing dichotomy. It will end with a brief review of currentresearch on more general situations, including heterogeneous environment, and systems ofequations.

Main references:

Y. Du and Zhigui Lin, Spreading-vanishing dichotomy in the diffusive logistic model with a freeboundary, SIAM J. Math. Anal., 42(2010), 377-405.Y. Du and Bendong Lou, Spreading and vanishing in nonlinear diffusion problems with freeboundaries, J. Eur. Math. Soc., 17(2015), 2673-2724.Y. Du, Mingxin Wang and Maolin Zhou, Semi-wave and spreading speed for the diffusivecompetition model with a free boundary, J. Math. Pure Appl., 107(2017), 253-287.Conference ID: 763367921; Conference password: 202007

July 29: Free boundary models with local diffusion in n-D

Conference ID: 311783805; Conference password: 202007

The high dimension version of the free boundary problem (without radial symmetry) is muchmore challenging mathematically. This part will examine the notions of weak solution and theirregularity, and also discuss some current work in that direction.

Main references:

Y. Du and Z.M. Guo, The Stefan problem for the Fisher-KPP equation, J. Diff. Eqns., 253(2012),996-1035.1Y. Du, Hiroshi Matano and Kelei Wang, Regularity and asymptotic behavior of nonlinear Stefanproblems, Arch. Rational Mech. Anal., 212(2014), 957-1010.Y. Du, Hiroshi Matsuzawa and Maolin Zhou, Spreading speed and profile for nonlinear Stefanproblems in high space dimensions, J. Math. Pures Appl., 103(2015), 741-787.

July 30: Free boundary models with nonlocal diffusion:Part 1

Conference ID: 447921656;Conference password: 202007

We will consider the nonlocal version of the 1-D Fisher-KPP equation with free boundary, andexamine several basic questions including existence and uniqueness, comparison principles, andspreading-vanishing dichotomy. Some recent results on related models will also be discussed.

Main references:

Jiafeng Cao, Y. Du, Fang Li and Wan-Tong Li, The dynamics of a Fisher-KPP nonlocal diffusionmodel with free boundaries, J. Functional Anal., 277 (2019), 2772-2814.Y. Du and Wenjie Ni, Approximation of random diffusion equation by nonlocal diffusion equationin free boundary problems of one space dimension, preprint, 2020.(arXiv:2003.05560)

July 31: Free boundary models with nonlocal diffusion:Part 2

Conference ID: 742430861; Conference password: 202007

We will continue the discussion on the 1-D Fisher-KPP model with nonlocal diffusion and freeboundaries, and show how the spreading speed can be determined. Some related recent workswill also be discussed.

Main references:

Y. Du, Fang Li and Maolin Zhou, Semi-wave and spreading speed of the nonlocal Fisher-KPPequation with free boundaries, preprint, 2019.(arXiv: 1909.03711)Y. Du, Wan-Tong Li, Wenjie Ni and Meng Zhao, Finite or infinite spreading speed of an epidemicmodel with free boundary and double nonlocal effects, preprint, 2020.

Aug. 1: Questions and open discussions

Conference ID: 228281128; Conference password: 202007

We will have discussions about problems and questions in the field and possible future directions.Any questions from the audience are very welcome.2