In this paper, we investigate the existence of a weak solution and blow-up of strong solutions to a two-component Fornberg–Whitham system. Due to the absence of some useful conservation laws, we establish the existence of a weak solution to the system in lower order Sobolev spaces \(H^{s}\times H^{s-1}\) (\(s\in (1,3/2]\)) via a modified pseudo-parabolic regularization method. And then, a blow-up scenario for strong solutions to this system is shown. By the analysis of Riccati-type inequalities recently, we present some sufficient conditions on the initial data that lead to the blow-up for corresponding strong solutions to the system.

在本文中，我们研究了二元 Fornberg-Whitham 系统弱解的存在性和强解的爆炸。由于缺乏一些有用的守恒定律，我们在低阶 Sobolev 空间中建立了系统弱解的存在\(H^{s}\times H^{s-1}\) ( \(s\in (1,3/2]\) ) 通过改进的伪抛物线正则化方法。然后，显示了该系统强解的爆炸场景。通过最近对 Riccati 型不等式的分析，我们提出了一些充分的导致系统相应强解爆炸的初始数据条件。