半线性波动方程的一个注记

A Note on Semilinear Wave Equations

非线性波动方程是一类重要而典型的发展方程。本文研究非线性项依赖于未知函数本身的三维半线性波动方程的Cauchy问题,通过构造适当的解空间,利用Banach压缩映像原理,我们给出了最大模框架下局部解存在唯一性的一个简单而直接的新证明。

Nonlinear wave equation is a kind of important and typical evolution equations. In thispaper, we study the Cauchy problem of three-dimension semilinear wave equations,in which the nonlinear terms depend the unknown function itself, by constructing suitable solution space, and using the Banach contraction mapping theorem, we givea simple, direct and new proof for the existence and uniqueness of local solution inthe framework of maximum norm.