Existence of a weak solution and blow-up of strong solutions for a two-component Fornberg–Whitham system

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.