非局部波动方程组解的全局存在性与爆破问题

The Questions of the global existence and blow-up solutions for a system of nonlocal wave equations

本文主要处理非局部波动方程组解的全局存在与爆破问题,考虑如下非局部波动方程组的初值问题:3。所有这些初值函数都为连续的且fi(x)+gi(x)0,i=1,2,3。根据对称性,本文假定p1≤p2≤p3。

In this paper, we mainly deal with global existence and blow-up solutions for systems of nonlocal wave equations with, we consider the initial value problem for a system of nonlocal wave equations {δ^2u1/δt^2=δ^2u1/δx^2＋‖u2（·,t）‖p1,δ^2u2/δt^2＋‖u3（·,t）‖p2,δ^2u3/δt^2=δ^2u3/δx^2＋‖u1（·,t）‖p3,-∞〈x〈∞,t〉0 ui（x,0）=fi（x）,δui/δt（x,0）=gi（x）,i=1,2,3,-∞〈x〈∞ where 0〈p1,p2,p3〈＋∞,‖ui（·,t）‖=∫-∞^＋∞ φi（x）｜u（x,t）｜dx,i=1,2,3.where φi（x）≥0 and ∫-∞^＋∞ φi（x）dx=1,i=1,2,3.All of the initial values are continuous and ｜fi（x）｜＋｜gi（x）｜absolotely unequalto 0,i=1,2,3.By symmetry, we assume p1≤p2≤p3.In this paper we prove the solutions of （1） is global existence with the condition 0 〈 p2p3 ≤ 1 ;While the solutions of （1） is blowing-up with the condition, p1≥ 1 ,p1p2p3 ≤ 1 ;While the solutions of （1）is blowing-up with the condition p1≥1 ,p1p2p3〉 1.