【摘要】 考虑非线性波方程utt-2kuxxt=g(ux)x的Cauchy问题,其中,k>0为实数,g(s)是给定非线性函数.当g(s)=sn时(n 2为整数),由Fourier变换方法和绝对值估计,证明了对任意T>0,如果初始数据u0∈W3,1(R)∩H2(R),u1∈W1,1(R)∩L2(R),则Cauchy问题存在惟一的整体光滑解u∈C∞((0,T];H∞(R))∩C([0,T];H2(R))∩C1([0,T];L2(R)).利用凸性方法,证明了相应的Cauchy问题在空间C∞((0,T];H∞(R))∩C([0,T];H2(R))∩C1([0,T];L2(R))中不存在整体广义解.更多还原

【Abstract】 This paper concerns with the Cauchy problem for the nonlinear wave equationutt-2kuxxt=g(ux)x,wherek >0 is a real number,g(s)is a given nonlinear function. Wheng(s) =sn(wheren 2 is an integer), by the Fouriertransform method and absolute value estimates we prove that for anyT >0 , the Cauchy problem admits a unique globalsmooth solutionu∈ C∞((0,T];H∞(R))∩ C([0,T];H2(R))∩ C1([0,T];L2(R)), provided that the initial datau0∈ W3,1(R)∩ H2(R), u1∈ W1,1(R)∩ L2(R).And by the convexity method, it is shown that the Cauchy problemhas no global generalized solution in the spaceC∞((0,T];H∞(R))∩ C([0,T];H2(R))∩ C1([0,T];L2(R)).更多还原