The local criteria for blowup of the Dullin-Gottwald-Holm equation and the two-component Dullin-Gottwald-Holm system

Abstract

We investigate wave breaking criteria for the Dullin-Gottwald-Holm equation and the two-component Dullin-Gottwald-Holm system. We establish a new blowup criterion for the general case γ + c 0 α 2 ≥ 0 involving local-in-space conditions on the initial data.

Liouville theorems for periodic two-component shallow water systems

“…Remark 3.1. The "local-in-space" blowup results investigated in [28] look more general and more precise than Theorem 1.1. On the other hand, our proof of blowup criteria is shorter and sufficient for our purpose.…”

“…These local-in-space blow-up criteria attract our interests, as they have the specific feature of being purely local in the space variable: indeed the blow-up conditions only involve the values of u 0 (ξ) and u 0 (ξ) in a single point ξ of the real line. In [28], Duc Truc Hoang investigated wave breaking criteria for the Dullin-Gottwald-Holm equation and the two-component Dullin-Gottwald-Holm system. The author established a new blow-up criterion for the general case γ + c 0 α 2 ≥ 0 involving local-in-space conditions on the initial data.…”

“…Motivated by the works of the above authors (see [1,2,3,28]), we plan to investigate the new local-in-space blow-up criteria for some two-component shallow water systems, including Eqs. (1.2) and (4.2).…”