Persistence Properties and Unique Continuation of Solutions of the Camassa-Holm Equation

Abstract

Abstract. It is shown that a strong solution of the Camassa-Holm equation, initially decaying exponentially together with its spacial derivative, must be identically equal to zero if it also decays exponentially at a later time. In particular, a strong solution of the Cauchy problem with compact initial profile can not be compactly supported at any later time unless it is the zero solution.

On the behavior of the solution of the dissipative Camassa–Holm equation with the arbitrary dispersion coefficient

“…It is also worth mentioning that from the point of view of theory of water waves the fact that solutions that originate from smooth localized initial data can develop singularities only in the form of breaking waves, as proved in the paper [10], is especially interesting. In [16], among others, authors showed (for k = 0) the infinite propagation speed for the Camassa-Holm equation in the sense that a strong solution of the Cauchy problem with compact initial profile cannot be compactly supported at any later time unless it is the zero solution, which is an improvement of a first result in this direction obtained in [13] (also, see [15]). The wide range of problems for CH equation with non-zero dispersion coefficient was considered in [18,22,25].…”

“…Precisely, some arguments (see [13,15,16]) used to investigate IPS (the infinite propagation speed) property, help shed additional light on the blow-up mechanism. Also, we will see that there exist some differences between the cases k, λ = 0 and k, λ = 0 if we speak about the behavior of the solution on the above-mentioned curves.…”

Some properties of solutions to the weakly dissipative Degasperis–Procesi equation

“…(2.1) in L ∞ -space. The main idea comes from a recent work of Himonas, Misiolek, Ponce and Zhou [7].…”

“…In [7], the infinite propagation speed for the Camassa-Holm equation was established. Later, Zhou and Zhu [21] considered the similar problem on the Degasperis-Procesi equation.…”

Unique continuation property and control for the Benjamin–Bona–Mahony equation on a periodic domain

“…Unfortunately, we cannot modify the above argument to deal with the UCP for the full BBM equation, as the nonlinear term has no reason to be perturbative at the "small" frequencies ξ = ±i. We point out that a moment approach, inspired by the paper [7] that was concerned with the UCP for the Camassa-Holm equation (see also [14]), was nevertheless applied in [29] to prove the UCP for the Kadomtsev-Petviashvili (KP)-BBM-II equation.…”