Well-posedness and non-uniform dependence on initial data for the Fornberg–Whitham-type equation in Besov spaces

Xueyuan Qi
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Abstract

In this paper, we first establish the local well-posedness for the Fornberg–Whitham-type equation in the Besov spaces \(B^{s}_{p,r}({\mathbb {R}})\) with \( 1\le p,r\le \infty \) and \(s> max\{1+\frac{1}{p},\frac{3}{2}\}\), which improve the previous work in Sobolev spaces \( H^{s}({\mathbb {R}})= B^{s}_{2,2}({\mathbb {R}})\) with \( s>\frac{3}{2}\) (Lai and Luo in J Differ Equ 344:509–521, 2023). Furthermore, we prove the solution is not uniformly continuous dependence on the initial data in the Besov spaces \(B^{s}_{p,r}({\mathbb {R}})\) with \( 1\le p\le \infty \),\( 1\le r< \infty \) and \(s> max\{1+\frac{1}{p},\frac{3}{2}\}\).

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贝索夫空间中福恩贝格-惠瑟姆型方程的良好拟合性和对初始数据的非均匀依赖性
本文首先在贝索夫空间 \(B^{s}_{p,r}({\mathbb {R}})\) with \( 1\le p,r\le \infty \) and\(s>;max{1+\frac{1}{p},\frac{3}{2}\}), which improve the previous work in Sobolev spaces \( H^{s}({\mathbb {R}})= B^{s}_{2,2}({\mathbb {R}})\) with\( s>\frac{3}{2}\) (Lai and Luo in J Differ Equ 344:509-521, 2023).此外,我们还证明了在贝索夫空间 \(B^{s}_{p,r}({\mathbb {R}})\) with \( 1\le p\le \infty \),\( 1\le r< \infty \) and\(s> max\{1+\frac{1}{p},\frac{3}{2}\}) 中,解并非均匀连续地依赖于初始数据。
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