河南大学学报（自然科学版）
河南大學學報（自然科學版）
하남대학학보（자연과학판）
JOURNAL OF HENAN UNIVERSITY(NATURAL SCIENCE)
2005年
3期
6-9
,共4页
非线性波方程%Cauchy问题%整体光滑解%整体广义解
非線性波方程%Cauchy問題%整體光滑解%整體廣義解
비선성파방정%Cauchy문제%정체광활해%정체엄의해
Nonlinear wave equation%Cauchy problem%Global smooth solution%Global generalized solution
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.
攷慮非線性波方程utt-2kuxxt=g(ux)x的Cauchy問題,其中,k>0為實數,g(s)是給定非線性函數.噹g(s)=sn時(n 2為整數),由Fourier變換方法和絕對值估計,證明瞭對任意T>0,如果初始數據u0 ∈ W3,1(R)∩H2(R), u1 ∈ W1,1(R)∩ L2(R),則Cauchy問題存在惟一的整體光滑解u ∈C∞((0,T];H∞(R))∩C([0,T];H2(R))∩ C1([0,T];L2(R)).利用凸性方法,證明瞭相應的Cauchy問題在空間 C∞((0,T];H∞(R))∩ C([0,T];H2(R))∩ C1([0,T];L2(R))中不存在整體廣義解.
고필비선성파방정utt-2kuxxt=g(ux)x적Cauchy문제,기중,k>0위실수,g(s)시급정비선성함수.당g(s)=sn시(n 2위정수),유Fourier변환방법화절대치고계,증명료대임의T>0,여과초시수거u0 ∈ W3,1(R)∩H2(R), u1 ∈ W1,1(R)∩ L2(R),칙Cauchy문제존재유일적정체광활해u ∈C∞((0,T];H∞(R))∩C([0,T];H2(R))∩ C1([0,T];L2(R)).이용철성방법,증명료상응적Cauchy문제재공간 C∞((0,T];H∞(R))∩ C([0,T];H2(R))∩ C1([0,T];L2(R))중불존재정체엄의해.
This paper concerns with the Cauchy problem for the nonlinear wave equation utt-2kuxxt=g(ux)x, where k>0 is a real number, g(s) is a given nonlinear function. When g(s)=sn (where n 2 is an integer), by the Fourier transform method and absolute value estimates we prove that for any T>0, the Cauchy problem admits a unique global smooth solution u ∈C∞((0,T];H∞(R))∩C([0,T];H2(R))∩ C1([0,T];L2(R)), provided that the initial data u0 ∈ W3,1(R)∩H2(R), u1 ∈ W1,1(R)∩ L2(R). And by the convexity method, it is shown that the Cauchy problem has no global generalized solution in the space C∞((0,T];H∞(R))∩ C([0,T];H2(R))∩ C1([0,T];L2(R)).
