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In this paper we study the harmonic map heat flow on the euclidean space R d and we show an unconditional uniqueness result for maps with small initial data in the homogeneous Besov space B ˙ p , ∞ d p (R d) where d < p < ∞. As a consequence we obtain decay rates for solutions of the harmonic map flow of the form ‖ ∇ u (t) ‖ L ∞ (R d) ≤ C t − 1 2 . Additionally, under the assumption of a stronger spatial localization of the initial conditions, we show that the temporal decay happens in a self-similar way. We also explain that similar results hold for the biharmonic map heat flow and the semilinear heat equation with a power-type nonlinearity. [ABSTRACT FROM AUTHOR]

In the paper, we revisit the uniform continuity properties of the data-to-solution map of the Camassa–Holm equation on the real-line case. We show that the data-to-solution map of the Camassa–Holm equation is not uniformly continuous on the initial data in Besov spaces B p , r s (R) with s > 1 2 and 1 ≤ p , r < ∞ , which improves the previous works Himonas et al. (2007) [23] , Li et al. (2020) [32] and Li et al. (2021) [31]. Furthermore, we present a strengthening of our previous work in Li et al. (2020) [32] and prove that the data-to-solution map for the Camassa–Holm equation is nowhere uniformly continuous in B p , r s (R) with s > max ⁡ { 1 + 1 / p , 3 / 2 } and (p , r) ∈ [ 1 , ∞ ] × [ 1 , ∞). The method applies also to the b-family of equations which contain the Camassa–Holm and Degasperis–Procesi equations. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the finite time blow-up for the high-order Camassa-Holm-Fokas-Olver-Rosenau-Qiao equations, which is a generalization of the Camassa-Holm equation and the Fokas-Olver-Rosenau-Qiao equation. We explore how high-order nonlinearities affect the dispersive dynamics and breakdown mechanism of solutions. Firstly, we established the local well-posedness for the Cauchy problem in the framework of Besov spaces. Then, we derive the precise blow-up mechanism for strong solutions by means of the transport equation theory. Finally, a sufficient condition on initial data that leads to the finite time blow-up of the second-order derivative of the solutions is described in detail. [ABSTRACT FROM AUTHOR]