Global existence and blow up of solutions for the Cauchy problem of some nonlinear wave equations

In this paper, we study the global existence and blow up for the Cauchy problem for some hyperbolic system uktt+δukt-ϕΔuk+fk(u1,u2)=λ|uk|β-1uk.k=1,2.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} u_{ktt}+\delta u_{kt}-\phi \Delta u_{k}+f_k(u_1,u_2)=\lambda |u_k|^{\beta -1}u_k. \quad k=1,2. \end{aligned}$$\end{document}Under certain conditions we prove the global existence of solutions by adapting the method of modified potential well in a functional setting of generalized Sobolev spaces, and we prove that the solution decays exponentially by introducing an appropriate Lyapunov function. By the concave method, we discuss the blow-up behavior of weak solution with certain conditions and give some estimates for the lifespan of solutions.