Bulletin of the
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Bull. Korean Math. Soc. 2024; 61(3): 837-848

Online first article May 17, 2024      Printed May 31, 2024

https://doi.org/10.4134/BKMS.b230489

Copyright © The Korean Mathematical Society.

Non-uniform dependence on initial data for the Fornberg--Whitham equation in $C^1(\mathbb{R})$

Yanghai Yu

Anhui Normal University

Abstract

It is shown in [1] that the Cauchy problem for the Fornberg--Whitham equation is well-posed in $C^1(\mathbb{R})$ and the data-to-solution map is H\"{o}lder continuous from $C^\alpha$ to $\mathcal{C}([0,T];C^\alpha)$ with $\alpha\in[0,1)$. In this short paper, we further show that the data-to-solution map of the Fornberg--Whitham equation is not uniformly continuous on the initial data in $C^1(\mathbb{R})$.

Keywords: Fornberg--Whitham equation, non-uniform dependence

MSC numbers: Primary 35Q53, 35B35

Supported by: Y. Yu is supported by National Natural Science Foundation of China (12101011).

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