On the Dynamics of Zero-Speed Solutions for Camassa–Holm-Type Equations

Abstract

Abstract In this paper, we consider globally defined solutions of Camassa–Holm (CH)-type equations outside the well-known nonzero-speed, peakon region. These equations include the standard CH and Degasperis–Procesi (DP) equations, as well as nonintegrable generalizations such as the $b$-family, elastic rod, and Benjamin-Bona-Mahony (BBM) equations. Having globally defined solutions for these models, we introduce the notion of zero-speed and breather solutions, i.e., solutions that do not decay to zero as $t\to +\infty $ on compact intervals of space. We prove that, under suitable decay assumptions, such solutions do not exist because the identically zero solution is the global attractor of the dynamics, at least in a spatial interval of size $|x|\lesssim t^{1/2-}$ as $t\to +\infty $. As a consequence, we also show scattering and decay in CH-type equations with long-range nonlinearities. Our proof relies in the introduction of suitable virial functionals à la Martel–Merle in the spirit of the works of [74, 75] and [50] adapted to CH-, DP-, and BBM-type dynamics, one of them placed in $L^1_x$ and the 2nd one in the energy space $H^1_x$. Both functionals combined lead to local-in-space decay to zero in $|x|\lesssim t^{1/2-}$ as $t\to +\infty $. Our methods do not rely on the integrable character of the equation, applying to other nonintegrable families of CH-type equations as well.