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

Yanghai Yu
대한수학회 (The Korean Mathematical Society)|2024.05|대한수학회보 (Bulletin of the KMS)|제61권 제3호|pp. 837-848|DOI 10.4134/BKMS.b230489

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키워드

Fornberg--Whitham equationnon-uniform dependence
주제분류수학

초록

It is shown in \cite{btw} that the Cauchy problem for the Fornberg--Whitham equation is well-posed in $C^1(\R)$ 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 $C^1(\mathbb{R})$.

Abstract

It is shown in \cite{btw} that the Cauchy problem for the Fornberg--Whitham equation is well-posed in $C^1(\R)$ 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 $C^1(\mathbb{R})$.

참고문헌 (29)

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