带势的非线性Klein-Gordon方程柯西问题的稳定和不稳定集(英文)

Stable and Unstable Sets for the Cauchy Problem for Nonlinear Klein-Gordon Equation with Potential

对带势的非线性Klein-Gordon方程柯西问题,我们定义了新的对于初值的稳定和不稳定集.我们证明了如果发展进入了不稳定集,解在有限时间内爆破;如果发展进入了稳定集,解整体存在.运用势并讨论,我们回答了当初值为多少时,柯西问题的整体解存在.

For the Cauchy problem for the nonlinear Klein-Gordon equation with potential,we define new stable and unstable sets for the initial data. We prove that if during the evolution enters into the unstable set, the solution blows up in finite time. If during the evolution enters into the stable set,the solution is global. By using scaling argument,we also answer the question of how small the initial data are the global solution of the Cauchy problem exists.