Blow-up criteria for some two-component shallow water wave system

Abstract In this paper we investigate the two-component nonlinear dispersive wave equation on the real line. We establish a blow-up criterion for the system of coupled equations under certain natural initial profiles. The obtained criterion is only involved in the initial values $$u_{0}\left( x_{0}\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>u</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mfenced> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mfenced> </mml:mrow> </mml:math> and $$u_{0}^{\prime }\left( x_{0}\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msubsup> <mml:mi>u</mml:mi> <mml:mrow> <mml:mn>0</mml:mn> </mml:mrow> <mml:mo>′</mml:mo> </mml:msubsup> <mml:mfenced> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mfenced> </mml:mrow> </mml:math> at a single point $$x_{0}\in \mathbb {R}.$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>∈</mml:mo> <mml:mi>R</mml:mi> <mml:mo>.</mml:mo> </mml:mrow> </mml:math>