Well-posedness and blow-up for a non-local elliptic–hyperbolic system related to short-pulse equation

In this paper, we investigate the non-local elliptic–hyperbolic system related to a short-pulse equation on the line. We first establish the local well-posedness of the problem in Hr(R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^r(\mathbb {R})$$\end{document}, r>32\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r>\frac{3}{2}$$\end{document}, by taking advantage of the Kato’s method. Then, we obtain a blow-up criterion for strong solutions and a result of blow-up solution. Finally, the existence of global weak solutions is proved.