一阶双曲型方程组解的衰减率

Decay Rate of Solution to Hyperbolic System of First Order

本文在研究波动方程时引入的整体Sobolev不等式推广到双曲组的情形．得到了一阶双曲组Cauchy问题解的几个衰减估计．特别是当初始资料给在指定的带权Sobolev空间中时，定理1．5的结果提供了最佳的衰减率．在定理的证明中我们将双曲组化到相应的一阶拟微分方程的情形，进而利用微局部分析建立所需要的估计．

In this paper we generalize global Sobolev inequality introduced by Klainer man in studying wave equation to the hyperbolic system case. We obtain severaJ decay estimate of solutions to hyperbolic system of first order by different norms of initial data. Particularly, the result mentioned in Theorem 1.5 offers an optimal decay rate of solutions, if the initial data belongs to the asigned weighted Sobolev space. In the proof of the theorem we reduce the estimate of solutions to hyperbolic system to the corresponding case for a scalar pseudodifferential equation of the first order, then we establish the required estimate by using microlocaJ analysis.