Profile of Blow-up Solution to Hyperbolic System with Nonlocal Term

This paper is concerned with a nonlocal hyperbolic system as follows: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \begin{array}{*{20}l} {{\left| {u_{{tt}} = \Delta u + {\left( {{\int_\Omega {vdx} }} \right)}^{p} \;{\text{for}}\;x \in \mathbb{R}^{N} ,t > 0,} \right.} \hfill} \\ {{v_{{tt}} = \Delta v + {\left( {{\int_\Omega {udx} }} \right)}^{q} \;{\text{for}}\;x \in \mathbb{R}^{N} ,t > 0,} \hfill} \\ {{u{\left( {x,0} \right)} = u_{0} {\left( x \right)},u_{t} {\left( {x,0} \right)} = u_{{01}} {\left( x \right)}\;{\text{for}}\;x \in \mathbb{R}^{N} ,} \hfill} \\ {{v{\left( {x,0} \right)} = v_{0} {\left( x \right)},v_{t} {\left( {x,0} \right)} = v_{{01}} {\left( x \right)}\;{\text{for}}\;x \in \mathbb{R}^{N} ,} \hfill} \\ \end{array} $$\end{document}where 1 ≤ N ≤ 3, p ≥ 1, q ≥ 1 and pq > 1. Here the initial values are compactly supported and Ω ⊂ ℝN is a bounded open region. The blow-up curve, blow-up rate and profile of the solution are discussed.