【摘 要】
：
The authors consider the Cauchy problem for the following nonlinear waveequationsutt - △u = | vl2,vtt - △v = (△u)2,u(0, x) = εuo(x), ut(0, x) = ε(x),v(0, x
【机 构】
：
DepartmentofAppliedMathematics,NorthChinaInsituteofWaterConservancyandHydroeletricPower
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The authors consider the Cauchy problem for the following nonlinear waveequationsutt - △u = | vl2,vtt - △v = (△u)2,u(0, x) = εuo(x), ut(0, x) = ε(x),v(0, x) = εvo(x), vt(0, x) = εv1(x),where x ∈ R3, t ≥ 0, ε＞ 0 is a small parameter, and obtain the sharp b
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