The well-posedness, ill-posedness and non-uniform dependence on initial data for the Fornberg–Whitham equation in Besov spaces. (April 2023)

• Record Type: Journal Article
• Authors: Guo, Yingying
• Is Part Of: Nonlinear analysis. Volume 70(2023)
• Publication Date: 2023-04
• Subjects: The Fornberg–Whitham equation -- Besov spaces -- Local well-posedness -- Ill-posedness -- Non-uniform dependence; Nonlinear functional analysis -- Periodicals; Analyse fonctionnelle non linéaire -- Périodiques; Nonlinear functional analysis; Periodicals; 515.7248
• DOI: 10.1016/j.nonrwa.2022.103791
• Languages: English
• ISSNs: 1468-1218

Abstract

Abstract: In this paper, we first establish the local well-posedness (existence, uniqueness and continuous dependence) for the Fornberg–Whitham equation in both supercritical Besov spaces B p, r s, s > 1 + 1 p, 1 ≤ p, r ≤ + ∞ and critical Besov spaces B p, 1 1 + 1 p, 1 ≤ p < + ∞, which improves the previous work (Holmes and Thompson, 2017; Holmes, 2016; Yin, 2007). Then, we prove the solution is not uniformly continuous dependence on the initial data in the Besov spaces B p, r s, s > 1 + 1 p, 1 ≤ p ≤ + ∞, 1 ≤ r < + ∞ or s = 1 + 1 p, 1 ≤ p < + ∞, r = 1 . At last, we show that the Cauchy problem for the Fornberg–Whitham equation is ill-posed in B p, ∞ σ with σ > 3 + 1 p, 1 ≤ p ≤ + ∞ .