球形脉冲在焦点的散射研究(Ι)

A Study of Scattering for Sphereical Pulses at Focus(I)

讨论了非线性波动方程(2t-Δx)uε+F(εα|tuε|p-1tuε)=0,(t,x)∈[0,T]×R3,uε|t=0=εU0r,r-r0ε,tuε|t=0=U1r,r-r0ε。当p>2,α=p-2时解在穿过焦点(r0,0)后的性态,其中F1在上是一致Lipschitz的。通过变量变换,将问题转化为讨论无穷远处的解,引入一个关键函数讨论脉冲波穿过焦点后(t→+∞)的性态。

We discuss the behavior of the solution to the wave equation{（δt^2-△x）u^∈＋F（∈^a｜δtu^∈｜^p-1δtu^∈）=0,（t,x）∈[0,T]×R^3,u^∈｜t=0=εU0（r,r-r0/ε）,δtu∈｜t=0=U1（r,r-r0/ε） after the focus （r0,0） ,where p 〉 2,α = p -2 and F is uniformly Lipschitiz on R. By introducing some changes of variables, the problem becomes one that we discuss the solution of a system at infinity, and study the behavior of pulses after the focus by introducing a key function.