<正> Treves研究过Cauchy问题的唯一性中离散现象,且证明了Cauchy问题具有非平凡解的充要条件为p=1,3,5……王光寅等对Treves方程的Cauchy问题和Gousat问题的存在性的离散现象作了进一步研究,给出问题解的C~∞存在的充要条件是p≠1,3,5……文献[2]中关于Cauchy问题的定理2中充分性的证明是不充分的,但经过修正,原文中的充要条件仍能成立.本文将沿用文献[2]中的方法给出Treves方程的Cauchy问题当p=1,3,5……时有C~∞解的充分且必要条件是初始数据具有一定的相客性,同时给出其解的一些具体的表达式.

With regard to discrete phenomena in the uniqueness of the Cauchy problem, F. Treves [1] pointed out that the Canchy problemu(x, o) = ut (x, o) = o has non-trivial solutions if and only if p=l, 3, 5. ...... Wang Guangying andothers I21 studied the Cauchy problem and Gousat problem and proved that both have a unique solution if and only if p%i, 3, 5. ...... Continuing to use the methodreferred to in Note C2]> this paper studies the Cauchy problem of Treves equation where p = l, 3, 5 ...... , puts forward the proof of the theorem that with ample and necessaryconditions, there exists C" solution, and at the same time, gives a number of its concrete expressions.