带线性坍塌项和竞争势的非线性波动方程柯西问题的稳定集和不稳定集(英文)

Stable and Unstable Sets for the Cauchy Problem of the Nonlinear Wave Equation With Linear Damping and Potential Terms

本文考虑带线性坍塌项和竞争势的非线性波动方程柯西问题,定义了新的稳定集和不稳定集,证明了如果初值进入不稳定集,则解在有限时间爆破;如果初值进入稳定集,则整体解存在.运用势井讨论,回答了当初值在多么小的时候,该柯西问题的整体解存在.

For the Cauchy problem of the nonlinear wave equation with linear damping and potential terms, 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 oroblem exists.