【摘要】 本文讨论了半线性波动方程(2t-Δx)uε+F(εα|tuε|p-1tuε)=0(t,x)∈[0,∞[×R3uε|t=0=εU0(r,r-εr0),tuε|t=0=Ul(r,r-εr0)。当p>2,α=p-2时解在到达焦点(r0,0)前无穷远处的性态,其中F在R上是一致Lipschitz的。通过变量变换,将问题转化为负无穷远处的初、边值问题,证明解的存在唯一性,引入线性解讨论脉冲波在t→-∞的传播性态,并引入散射算子说明了脉冲波越过焦点的过程。更多还原

【Abstract】 We discuss the behavior of the solution to the wave equation (2t-Δx)uε+F(εα｜tuε｜p-1tuε)=0(t,x)∈[0,∞[×R3uε｜t=0=εU0(r,r-r0ε),tuε｜t=0=Ul(r,r-r0ε)at infinity before the focus,where p>2,α=p-2 and F is uniformly Lipschitiz on R.By changing the problem into a mixed problem with initial value at negative infinity and proving the existence and uniqueness of the solution to the producing problem,it discusses the propagation of pulses at t=-∞ and introduces a scattering operator to describe the process that the pulses across the focus.更多还原