一类非线性波动方程混合问题解的爆破

Blow up of solution for a kind of mixed problem of nonlinear wave equations

基于Sattinger在1968提出的势井理论,通过构造不稳定集,应用凸形分析方法,简单地证明了一类非线性波动方程混合问题解的爆破性.即对于半线性波动方程utt-Δu=|u|r-1u的初边值问题,当初值属于不稳定集V时,解在L2(Ω)意义下,在有限时刻发生blow up现象.

According to potential well theory advanced by Sattinger in 1968 and,by constructing a unstable set,the blow up behavior of the solution for a kind of mixed problem of nonlinear wave equations is verified readily by using the convexity method.That is,to the initial-boundary-value problem of semi-linear wave equation u_(tt)-Δu=|u|^(r-1)u,with an initial value belonging to an unstable set V,the solution will blow up under L^2(Ω) norm and at a finite moment.