R^(1+3)中球形非线性脉冲的局部存在性

Local Existence of Spherical Nonlinear Pulses in R^(1+3)

讨论非线性波动方程( 2t-Δx)uε+F(εα| tuε|p-1 tuε)=0, (t,x)∈[0,T]×R3,uε|t=0=εU0r,r-r0)ε, tuε|t=0=U1r,r-r0)ε .解的局部存在性.在建立一些必要的估计的情况下,给出小初值条件下脉冲的局部存在性,为讨论全局存在性和散射问题提供了必要的准备.

This paper discusses the local existence of spherical nonlinear pulses of wave equation(~2_t-Δ_x)u~ε+F(ε~α｜_tu~ε｜^(p-1)_tu~ε)=0,\ (t,x)∈[0,T]×R^3,u~εJB(｜)_(t=0)=εU_0(r,r-r_0ε,_tu~ε｜]_(t=0)=U_1B(()r,r-r_0ε.By giving some useful estimates,we prove the local existence with small initial data.The results lay the foundation for discussing the global existence and scattering problems.