证明下列非线性波动方程的Cauchy问题 υtt-αΔυtt-Δυ=g(υ)-αΔg(υ), χ∈RN, t>0, (1) υ(χ, 0)=υ0(χ), υt(χ, 0)=υ1(χ), χ∈RN (2) 在空间C2([0, ∞);Hs(RN))(s>N/2)中存在唯一整体广义解υ和在空间C2([0, ∞);Hs(RN))(s>2+N/2)中存在唯一整体古典解υ, 即υ∈C2([0, ∞);CB2(RN)).还证明Cauchy问题(1), (2)在C3([0, ∞);W m, p∩L∞(RN))(m≥0, 1≤p≤∞)中有唯一整体广义解υ和在C3(0, ∞);W m, p(RN)∩L∞(RN))(m>2+N/P)中有唯一整体古典解υ, 即υ∈C3([0, ∞;C2(RN)∩L∞(RN)).
Abstract
We prove that the Cauchy problem for the nonlinear wave equation υtt-αΔυtt-Δυ=g(υ)-αΔg(υ),χ∈RN,t>0, (1) υ(χ,0)=υ0(χ),υt(χ,0)=υ1(χ),χ∈RN (2) has a unique global generalized solution υ in C2([0,∞);Hs(RN))(s>N/2) and a unique global classical solution υ in C2([0,∞);Hs(RN))(s>2+N/2),i.e.,υ∈C2([0,∞);CB2(RN)). We also prove that the Cauchy problem(1),(2)admits a unique global generalized solution υ in C 3(0,∞);W m,p(RN)∩L∞(RN))(m≥0,1≤p≤∞)and a unique global classical solution υ in C 3(0,∞);W m,p(RN)∩L∞(RN))(m>2+N/P,i.e.,υ∈C3([0,∞;C2(RN)∩L∞(RN)).
关键词
N维非线性波动方程 /
Cauchy问题 /
解的整体存在性
{{custom_keyword}} /
Key words
N-dimensional nonlinear wave equation /
Cauchy problem /
global existence of solution
{{custom_keyword}} /
{{custom_sec.title}}
{{custom_sec.title}}
{{custom_sec.content}}
参考文献
[1] Kevrekidis P.G.,Kevrekidis I.G.,Bishop A.R.,Titi E.S.,Continuum approach to discreteness,Physical Review E,2002,65(4):046613-1.
[2] Makhankov V.G.,Dynamics of classical solitions(in nonintegrable systems),Physics Reports,A Review Section of Phys.Lett.Section C,1978,35(1):1-128.
[3] Peter A.C.,Randall J.L.,Ralph S.,Solitary-wave interactions in elastic rods,Studies in Applied Mathe-matics,1986,75:95-122.
[4] Chen G.,Wang S.,Cauchy problem for generalized IMBq equation,In:Proceeding of the Conference on Nonlinear Partial differential Equations and Applications(Guo Boling,Yang Dadi eds),World Science, Singapore,1998.
[5] Liu Y.,Existence and blow-up of solutions of a nonlinear Pochhammer-Chree equation,Indiana University Mathematics Journal,1996,45:797-816.
[6] Akmel Dé G.,Blow-up of solutions of a generalized Boussinesq equation,IMA Journal of Applied Mathe- matics,1998,60:123-138.
[7] Chen G.,Xing J.,Yang Z.,Cauchy problem for generalized IMBq equation with several variables,Nonlinear Analysis TMA,1996,26:1255-1270.
[8] Wang S.,Chen G.,The Cauchy problem for generalized IMBq equation in Ws,p(Rn),J.Math.Anal.Appl., 2002,266:38-54.
[9] Stein E.M.,Singular Integrals and differentiability Properties of Functions,Princeton University Press, Princeton,New Jersey,1970.
[10] Donoghue W.F.,Distributions and Fourier Transforms,Academic Press,New York,1969.
[11] Wang S.,Chen G.,Small amplitude solution of the generalized IMBq equation,J.Math.Anal.Appl.,2002, 274:846-866.
[12] Kato T.,Ponce G.,Commutator estimates and the Euler and Navier-Stokes equations,Comm.Pure Appl. Math.,1988,41:891-907.
[13] Li T.T.,Chen Y.,Global Classical Solutions for Nonlinear Evolution Equations,Pitman Monographs and Surveys in Pure and Applied Mathematics,Vol.45,Longman Scientific & Technical,New York,1992.
{{custom_fnGroup.title_cn}}
脚注
{{custom_fn.content}}
基金
国家自然科学基金资助项目(10971199, 11171311);河南省教育厅基金(2009C110006)
{{custom_fund}}
PDF(506 KB)
