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Interfaces in the Fisher equation and a Hamilton-Jacobi equation

Banff International Research Station for Mathematical Innovation and Discovery

We consider the dynamics of interfaces in the Fisher-KPP equation. It is known that solutions of this equation exhibit interfaces that correspond to transition layers from the trivial steady state to a positive steady state. If an initial value decays rapidly in space, then the interface moves with a constant speed that is equal to the minimal speed of traveling fronts in one-dimensional space. On the other hand, it is known that if an initial value decays slowly, the interface may move in a rather irregular way. In this talk, we show that the dynamics of interfaces for slowly decaying initial data can be described as a level set of a Hamilton-Jacobi equation. We also discuss properties of solutions of the Hamilton-Jacobi equation. This is a joint work with Hirokazu Ninomiya (Meiji University).

Mathematics; Partial differential equations; Calculus of variations and optimal control; optimization

video/mp4

Author affiliation: Tokyo Institute of Technology

2017-01-28

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/59981

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/