Abstract
F. Treves(1) had studiied an interesting example on the discrete phenomena in the uniqueness of the initial value problem. He showed the Cauchy problem {Lpu=ux x - x2ut t+put = 0, t ≥ 0,u(x,0) = ut(x,0) = 0possessing sufficient and necessary condition of nontrivial solution is p = 1,3,5, …… Wang Guangy in et al(2) proved that both the Cauchy problem and the Goursat problem have a unique solutcon if and only if p ≠ 1,3,5, …. It is due to the equation Lpu = 0, which is of bichara cteristics, that happens such discrete phenomena and it is x = 0 that chonges the equation into a bicharacteristic one. Therefore, if we again give a certain boundary condition on x = 0, it will chingeinto a mixed problem. And what kind of new phenomena will appear? This is an interesting problem.
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