Blow-up phenomena and peakons for the gFQXL/gCH-mCH equation

Abstract

This paper is devoted to studying the generalized Fokas–Qiao–Xia–Li/generalized Camassa–Holm-modified Camassa–Holm (gFQXL/gCH-mCH) equation, which includes the Camassa–Holm equation, the generalized Camassa–Holm equation, the Novikov equation, the Fokas–Olver–Rosenau–Qiao/modified Camassa–Holm equation, and the Fokas–Qiao–Xia–Li/Camassa–Holm-modified Camassa–Holm equation as its special case. We first show that the gFQXL/gCH-mCH equation is locally well-posed in Besov spaces. Then, we present a blow-up criterion and a precise blow-up scenario, and utilizing the fine structure and the H1-norm conservation, we establish a finite time blow-up result with respect to the initial data. Finally, peakons are derived in an explicit formula.