American Institute of Mathematical Sciences

A global well-posedness and asymptotic dynamics of the kinetic Winfree equation

 1 Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul, 08826, Korea 2 Korea Institute for Advanced Study, Hoegiro 85, Seoul, 02455, Korea 3 Department of Mathematics and Research Institute of Natural Sciences, Hanyang University, 222 Wangsimni-ro, Seongdong-gu, Seoul 04763, Korea 4 Center for Mathematical sciences, Huazhong University of Science and Technology, Luoyu road 1037, Wuhan 430074, China

* Corresponding author: Xiongtao Zhang

Received  April 2018 Published  November 2019

Fund Project: The work of S.-Y. Ha is supported by National Research Foundation of Korea(NRF-2017R1A2B2001864), and the work of J. Park has been supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (NRF-2018R1C1B5043861). The work of X. Zhang is supported by the National Natural Science Foundation of China (Grant No. 11801194)

We study a global well-posedness and asymptotic dynamics of measure-valued solutions to the kinetic Winfree equation. For this, we introduce a second-order extension of the first-order Winfree model on an extended phase-frequency space. We present the uniform(-in-time) $\ell_p$-stability estimate with respect to initial data and the equivalence relation between the original Winfree model and its second-order extension. For this extended model, we present uniform-in-time mean-field limit and large-time behavior of measure-valued solution for the second-order Winfree model. Using stability and asymptotic estimates for the extended model and the equivalence relation, we recover the uniform mean-field limit and large-time asymptotics for the Winfree model. 200 words.

Citation: Seung-Yeal Ha, Jinyeong Park, Xiongtao Zhang. A global well-posedness and asymptotic dynamics of the kinetic Winfree equation. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2019229
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