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