| 一类N维非线性波动方程的Cauchy问题 |
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| 引用本文： | 陈国旺.一类N维非线性波动方程的Cauchy问题[J].数学学报,2012(5):797-810. |
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| 作者姓名： | 陈国旺 |
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| 作者单位： | 郑州大学数学系 |
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| 基金项目： | 国家自然科学基金资助项目(10971199,11171311);河南省教育厅基金(2009C110006) |
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| 摘 要： | 证明下列非线性波动方程的Cauchy问题v_(tt)-α△v_(tt)-Δv=g(v)-αΔg(v),x∈R~N,t>0,(1)v(x,0)=v_0(x),v_t(x,0)=v_1(x),x∈R~N(2)在空间C~2(0,∞);H~s(R~N))(s>N/2)中存在唯一整体广义解v和在空间C~2(0,∞);H~s(R~N))(s>2+N/2N)中存在唯一整体古典解v,即u∈C~2(0,∞);C_B~2(R~N)).还证明Cauchy问题(1),(2)在C~3(0,∞);W~(m,p)(R~N)∩L~∞(R~N))(m≥0,1≤p≤∞)中有唯一整体广义解v和在C~3(0,∞);W~(m,p)(R~N)∩L~∞(R~N))(m>2+N/P)中有唯一整体古典解v,即v∈C~3(0,∞);C~2(R~N)∩L~∞(R~N)).
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| 关 键 词： | N维非线性波动方程 Cauchy问题 解的整体存在性 |
Cauchy Problem for an N-Dimensional Nonlinear Wave Equation |
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| Guo Wang CHEN.Cauchy Problem for an N-Dimensional Nonlinear Wave Equation[J].Acta Mathematica Sinica,2012(5):797-810. |
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| Authors: | Guo Wang CHEN |
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| Institution: | Guo Wang CHEN Department of Mathematics,Zhengzhou University,Zhengzhou 450052,P.R.China |
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| Abstract: | We prove that the Cauchy problem for the nonlinear wave equationv_(tt) -αΔv_(tt) -Δv = g(v) -αΔg(v),x∈R~N,t>0,(1)v(x,0)=v_0(x),v_t(x,0) = v_1(x),x∈R~N(2)has a unique global generalized solution v in C~2(0,∞);H~s(R~N))(s>N/2) and a uniqueglobal classical solution v in C~2(0,∞);H~s(R~N))(s>2+N/2),i.e.,v∈C~2(0,∞);C_B~2(R~n)).We also prove that the Cauchy problem(1),(2) admits a unique global generalizedsolution v in C~3(0,∞);W~(m,p)(R~N)∩L~∞(R~N))(m≥0,1≤p≤∞) and a uniqueglobal classical solution v in C~3{0,∞);W~(m,p)(R~N) D L~∞(R~N))(m>2 + N/P),i.e.,v∈C~3(0,∞);C~2(R~N)∩L~∞(R~N)). |
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| Keywords: | N-dimensional nonlinear wave equation Cauchy problem global existence of solution |
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