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 Hs×Hs-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^{s}\times H^{s-1}$$\end{document} (s∈(1,3/2]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s\in (1,3/2]$$\end{document}) 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.