Sobolev空间中非线性波动方程的局部解的探讨

The discussion on the local solutions of nonlinear wave equations in sobolev room

对非线性偏微分方程的研究吸引着许多数学家,物理学家及工程学家.对于线性的波动方程,只要初值适当光滑,其Cauchy问题的解必具有适当的光滑性,同时在t≥0上是整体存在的,然而对于非线性波动方程,其Cauchy问题的整体经典解通常只能在时间t的一个局部范围内存在.目前对于在Sobolev空间中非线性波动方程解的渐近理论的研究,还是一个空白.现以非线性波动方程utt-Δu=f(t,x,u,Du)(t∈R+,x∈Rn)为研究对象,其在Sobolev空间中局部解存在的一个充分条件是S>n2+1,通过引入该Cauchy问题的等价积分算子,运用Fourier变换,利用Banach不动点定理。

The research on nonlinear partial differential equations attracts many mathematicians, physicians and engineers. To linear wave equations, if the initial value is appropriate smooth, its solution of Cauchy problem must have appropriate smooth characteristics, and exists integrally at t≥0. But to nonlinear wave equations, its integral solution of Cauchy problem can exist at local extension of t. At present, the research on gradual approximate theory of nonlinear wave equations in Sobolev room is still blank. The paper aims on nonlinear wave equation un-△u=f （t,x, u ,Du） （t ∈ R^ ＋ , x ∈ R^n ）, and one of its requirements for the occurrence of the local solutions n is S〉n/2-1. By inducting the equivalent integral operator of Cauchy problem, handling Fourier transform, using Banach immobility dot theorem, the paper demonstrates that the exponent of Cauchy problem of the nonlinear wave equation in Sobolev room is n / 2- 1/k-1.