拟线性双曲组在半有界初始轴上的柯西问题
|
摘要
本文系统地研究了拟线性双曲组在半有界初始轴上的柯西问题,分别对其经典解的整体存在性、渐近性态(在整体经典解存在的前提下)、破裂现象(生命跨度及奇性形成)等进行了讨论,并举例说明其应用.
本文的结构如下:
第一章,简要介绍了有关拟线性双曲组柯西问题(包括在整个初始轴和半有界初始轴上)经典解的研究历史、现状和本文的主要结果.
为完整起见,在第二章,我们给出了一些预备知识,包括广义标准化坐标、波的分解公式以及本文所要用到的三个重要引理.
第三章考虑了非齐次拟线性双曲组在半有界初始轴上柯西问题的整体经典解.在最大(相应地,最小)的特征弱线性退化及非齐次项满足相应于此特征的匹配条件的假设下,对于小而衰减的初值,得到了柯西问题经典解的整体存在唯一性.
第四章在第三章的基础上,讨论了拟线性双曲组在半有界初始轴上柯西问题经典解当t→+∞时的渐近性态.证明了当时间t趋于无穷大时,只要初始数据当x→+∞(相应地,x→-∞)时以速率(1+x)~(-(1+μ))(相应地,(1—x)~(-(1+μ))衰减,柯西问题的经典解就以速率(1+t)~(-μ)逼近于C~1行波解的组合,其中μ是一个正常数.
第五章主要研究拟线性双曲组在半有界初始轴上柯西问题经典解的破裂现象.在最大(相应地,最小)的特征非弱线性退化及非齐次项满足相应于此特征的匹配条件的假设下,对于小而衰减并满足特定条件的初值,我们得到了经典解生命跨度的精确估计及奇性形成机制.
第六章将举例说明第三、四、五章所研究的理论的应用.
In this Ph.D.thesis,we systematically study the Cauchy problem on a semibounded initial axis for quasilinear hyperbolic systems.The global existence,the asymptotic behavior(based on the global existence),the blow-up phenomenon(the lifespan and formation of singularities) of classical solutions to the Cauchy problem are discussed respectively.We also give some examples to show the applications of the theory.
The structure of the thesis is as follows:
A brief introduction on the research history of classical solutions to the Cauchy problem(both on the whole and a semi-bounded initial axis) for quasilinear hyperbolic systems is given in Chapter 1.The main results of this thesis will be also included in this chapter.
For the sake of completeness,in Chapter 2,we give some preliminaries,which contain generalized normalized coordinates,John's formula on the decomposition of waves and three important lemmas to be used in this thesis.
In Chapter 3,we consider the global existence of the Cauchy problem on a semi-bounded initial axis for quasilinear hyperbolic systems.Under the assumptions that the rightmost(resp.leftmost) eigenvalue is weakly linearly degenerate and the inhomogeneous term satisfies the corresponding matching condition,we obtain the global existence and uniqueness of classical solution with small and decaying initial data.
Based on Chapter 3,Chapter 4 deals with the asymptotic behavior of classical solution to the Cauchy problem on a semi-bounded initial axis for quasilinear hyperbolic systems.We prove that,when t tends to the infinity,the solution approaches a combination of C~I travelling wave solutions with algebraic rate(1 + t)~(-μ),provided the initial data decay with the rate(1 + x)~(-(1+μ))(resp.(1 - x)~(-(1+μ))) as x tends to +∞(resp.-∞),whereμis a positive constant.
Chapter 5 is devoted to the study of the blow-up phenomenon of classical solutions to the Cauchy problem on a semi-bounded initial axis for quasilinear hyperbolic systems.Under the assumptions that the rightmost(resp.leftmost) eigenvalue is not weakly linearly degenerate and the inhomogeneous term satisfies the corresponding matching condition,for small and decaying initial data satisfying a special condi-tion,we obtain a sharp estimate and the mechanism of formation of singularities of classical solutions.
In Chapter 6,we will give some examples to show applications of the theory we study in Chapter 3,Chapter 4 and Chapter 5.
本文的结构如下:
第一章,简要介绍了有关拟线性双曲组柯西问题(包括在整个初始轴和半有界初始轴上)经典解的研究历史、现状和本文的主要结果.
为完整起见,在第二章,我们给出了一些预备知识,包括广义标准化坐标、波的分解公式以及本文所要用到的三个重要引理.
第三章考虑了非齐次拟线性双曲组在半有界初始轴上柯西问题的整体经典解.在最大(相应地,最小)的特征弱线性退化及非齐次项满足相应于此特征的匹配条件的假设下,对于小而衰减的初值,得到了柯西问题经典解的整体存在唯一性.
第四章在第三章的基础上,讨论了拟线性双曲组在半有界初始轴上柯西问题经典解当t→+∞时的渐近性态.证明了当时间t趋于无穷大时,只要初始数据当x→+∞(相应地,x→-∞)时以速率(1+x)~(-(1+μ))(相应地,(1—x)~(-(1+μ))衰减,柯西问题的经典解就以速率(1+t)~(-μ)逼近于C~1行波解的组合,其中μ是一个正常数.
第五章主要研究拟线性双曲组在半有界初始轴上柯西问题经典解的破裂现象.在最大(相应地,最小)的特征非弱线性退化及非齐次项满足相应于此特征的匹配条件的假设下,对于小而衰减并满足特定条件的初值,我们得到了经典解生命跨度的精确估计及奇性形成机制.
第六章将举例说明第三、四、五章所研究的理论的应用.
In this Ph.D.thesis,we systematically study the Cauchy problem on a semibounded initial axis for quasilinear hyperbolic systems.The global existence,the asymptotic behavior(based on the global existence),the blow-up phenomenon(the lifespan and formation of singularities) of classical solutions to the Cauchy problem are discussed respectively.We also give some examples to show the applications of the theory.
The structure of the thesis is as follows:
A brief introduction on the research history of classical solutions to the Cauchy problem(both on the whole and a semi-bounded initial axis) for quasilinear hyperbolic systems is given in Chapter 1.The main results of this thesis will be also included in this chapter.
For the sake of completeness,in Chapter 2,we give some preliminaries,which contain generalized normalized coordinates,John's formula on the decomposition of waves and three important lemmas to be used in this thesis.
In Chapter 3,we consider the global existence of the Cauchy problem on a semi-bounded initial axis for quasilinear hyperbolic systems.Under the assumptions that the rightmost(resp.leftmost) eigenvalue is weakly linearly degenerate and the inhomogeneous term satisfies the corresponding matching condition,we obtain the global existence and uniqueness of classical solution with small and decaying initial data.
Based on Chapter 3,Chapter 4 deals with the asymptotic behavior of classical solution to the Cauchy problem on a semi-bounded initial axis for quasilinear hyperbolic systems.We prove that,when t tends to the infinity,the solution approaches a combination of C~I travelling wave solutions with algebraic rate(1 + t)~(-μ),provided the initial data decay with the rate(1 + x)~(-(1+μ))(resp.(1 - x)~(-(1+μ))) as x tends to +∞(resp.-∞),whereμis a positive constant.
Chapter 5 is devoted to the study of the blow-up phenomenon of classical solutions to the Cauchy problem on a semi-bounded initial axis for quasilinear hyperbolic systems.Under the assumptions that the rightmost(resp.leftmost) eigenvalue is not weakly linearly degenerate and the inhomogeneous term satisfies the corresponding matching condition,for small and decaying initial data satisfying a special condi-tion,we obtain a sharp estimate and the mechanism of formation of singularities of classical solutions.
In Chapter 6,we will give some examples to show applications of the theory we study in Chapter 3,Chapter 4 and Chapter 5.
引文
[1]S.Alinhac,Blowup for Nonlinear Hyperbolic Equations,Progress in Nonlinear Partial Differential Equations and Their Applications 17,Birkhauser,1995.
[2]S.S.Antman,Nonlinear Problems of Elasticity,Applied Mathematical Sciences 107,Springer-Verlag,1995.
[3]A.Aw and M.Rascle,Resurrection of "second order" models of traffic flow,SIAM J.Appl.Math.60(2000),916 - 938.
[4]A.Bamberger,C.Flytzanis et D.Jennevé Etude mathématique et numérique d'un module de propagation d'ondes lumineuses dans une fibre optique non centrosymétrique,Ecole Eolytechnique,Mathématiques appliquées 149,(juin 1986).
[5]A.J.Chorin and J.Marsden,A Mathematical Intrduction to Fluid Mechanics,Springer-Verlag,New York,1979.
[6]R.Courant and K.O.Friedrichs,Supersonic flow and shock waves,Sprigner,Berlin,1976.
[7]Dai Wenrong,Asymptotic behavior of global classical solutions of quasilinear nonstrictly hyperbolic systems with weakly linear degeneracy,Chinese Annals of Mathematics,27B(2006),263-286.
[8]Dai Wenrong & Kong Dexing,Global existence and asymptotic behavior of classical solutions of quasilinear hyperbolic systems with linearly degenerate characteristic fields,Journal of Differential Equations,235(2007),127-165.
[9]Lawrence C.Evans,Partial Differential Equations,Graduate Studies in Mathematics,Vol.14,American Mathematical Socity,Providence,Rhode Island,1998.
[10]谷超豪,李大潜等,数学物理方程(第二版),高等教育出版社,北京,2002.
[11]Han Weiwei,Global existence of classical solutions to the Cauchy problem on a semibounded initial axis for inhomogeneous quasilinear hyperbolic systems,Journal of Partial Differential Equations,20(2007),273-288.
[12]Han Weiwei,Breakdown of classical solutions to the Cauchy problem on a semibounded initial axis for quasilinear hyperbolic systems,Nonlinear Analysis(2007),doi:10.1016/j.na.2007.07.027.
[13]Han Weiwei,Asymptotic behavior of global classical solutions to the Cauchy problem on a semi-bounded initial axis for quasilinear hyperbolic systems,to appear in Journal of Mathematical Research and Exposition.
[14]L.Hormander,The Lifespan of Classical Solutions of Nonlinear Hyperbolic Equations,Lecture Notes in Math.,vol,1256,Springer,Berlin,1987.
[15]L.Hormander,Lectures on Nonlinear Hyperbolic Equations,Math.Appl.,vol.26,Springer,Berlin,1997.
[16]A.Jefferey,Quasilinear Hyperbolic Systems and Waves,Pitman Publishing,1976.
[17]F.John,Formation of singularities in one-dimensional nonlinear wave propagation,Communications on Pure and Applied Mathematics,27(1974),377-405.
[18]孔德兴,一阶拟线性双曲型方程组的CauChy问题,复旦学报(自然科学版),33(1994),705-708.
[19]Kong Dexing,Cauchy Problem for Quasilinear Hyperbolic Systems:MSJ Memoirs 6,Tokyo:The Mathematical Society of Japan,2000.
[20]Kong Dexing,Life-span of classical solutions to quasilinear hyperbolic systems with slow decay initial data,Chinese Annals of Mathematics,21B(2000),413-440.
[21]Kong Dexing,Sun Qingyou & Zhou Yi,The equation for time-like extremal surfaces in Minkowski space R~(2+n),Journal of Mathematical Physics,(47)2006,013503:16pp.
[22] Kong Dexing. Sun Qingyou & Zhou Yi, Erratum: "The equation for time-like extremal surfaces in Minkowski space R2+n [J. Math. Phys. 47, 013503](2006), Journal of Mathematical Physics, (47)2006, 069901: 1pp.
[23] D.J. Kaup, A.Reiman et A. Bers, Space-time evolution of nonlinear three-wave interaction I: Interaction in a homogeneous medium, Reviews of Modern Physics 51, (avril 1979).
[24] Kong Dexing & Yang Tong, Asymptotic behavior of Global classical solutions of quasilinear hyperbolic systems, Communications in Partial Differential Equations, 28(2003), 1203-1220.
[25] L.D.Landau and E.M.Lifschitz, Fluid Mechanics, Addison Wesley, Reading MA.1971.
[26] P. D. Lax, Hyperbolic systems of conservation laws II, Communications on Pure and Applied Mathematics, 10(1957), 537-566.
[27] Li Shumin, Cauchy problem for general first order inhomogeneous quasilinear hyperbolic system, Journal of Partial Differential Equations, 15(2002), 46-68.
[28] Li Tatsien, Global Classical Solutions for Quasilinear Hyperbolic Systems, Research in Applied Mathematics 32, J. Wiley/Masson, 1994.
[29] Li Tatsien, Une remarque sur les coordonnees normalisees et ses applications aux systemes hyperboliques quasi lineaires, Comptes Rendus de I'Academie des Sciences, Seris I Mathematic, 331(2000), 447-452.
[30] Li Tatsien, A remark on the normalized coordinates and its applications to quasilinear hyperbolic systems, in Optimal Control and Partial Differential Equations, J. L. Menaldi et al. (eds.), IOS Press, 2001, 181-187.
[31] 李大潜,拟线性双曲组选讲,2005/2-2006/5．
[32] Li Tatsien & Kong Dexing, Breakdown of classical solutions to quasilinear hyperbolic systems, Nonlinear Analysis, 40(2000), 407-437.
[33]Li Tatsien & Kong Dexing,Initial value problem for general quasilinear hyperbolic systems with characteristics with constant multiplicity,Journal of Partial Differential Equations,10(1997),299-322.
[34]Li Tatsien,Kong Dexing & Zhou Yi,Global classical solutions for general quasilinear non-strictly hyperbolic systems with decay initial data,Nonlinear Studies,3(1996),203-229.
[35]李大潜,秦铁虎,物理学与偏微分方程(上册,第一版),高等教育出版社,北京,1997.
[36]李大潜,秦铁虎,物理学与偏微分方程(下册,第一版),高等教育出版社,北京,2000.
[37]Li Tatsien & Wang Libin,Global existence of weakly discontinuous solutions to the Cauchy problem with a kind of non-smooth initial data for quasilinear hyperbolic systems,Chinese Annals of Mathematics,25B(2004),319-334.
[38]Li Tatsien & Wang Libin,Global existence of classical solutions to the Cauchy prolem on a semi-bounded initial axis for quasilinear hyperbolic systems,Nonlinear Analysis,56(2004),961-974.
[39]Li Tatsien & Wang Libin,Blow-up mechanism of classical solutions to quasilinear hyperbolic systems in the critical case,Chinese Annals of Mathematics,27B(2006),53-66.
[40]Li Tatsien & Wang Libin,Existence and uniqueness of global solution to an inverse piston problem,Inverse Problems,23(2007),683-694.
[41]Li Tatsien & Yu Wenci,Boundary Value Problems for Quasilinear Hyperbolic Systems,Duke University Mathematics Series V,1985.
[42]Li Tatsien,Zhou Yi & Kong Dexing,Weak linear degeneracy and global classical solutions for general quasilinear hyperbolic systems,Communications in Partial Differential Equations,19(1994),1263-1317.
[43]Li Tatsien,Zhou Yi & Kong Dexing,Global classical solutions for general quasilinear hyperbolic systems with decay initial data,Nonlinear Analysis,28(1997),1299-1322.
[44]Liu Jianli & Zhou Yi,Asymptotic behaviour of global classical solutions of diagonalizable quasilinear hyperbolic systems,Mathematical Methods in the Applied Sciences,30(2007),479-500.
[45]Liu Taiping,Development of singularities in the nonlinear waves for quasilinear hyperbolic partial differential equation,Journal of DifferentiaI Equations,33(1979),92-111.
[46]Liu Taiping,Quasilinear hyperbolic systems,Communications in Mathematical Physics,68(1979),141-172.
[47]A.Majda,Compressible Fluid Flow and System of Conservation Laws in Several Space Variables,Applied Mathematical Sciences 53,Springer-Verlag,1984.
[48]J.Serrin,Mathematical Principles of Classical Fluid Mechanics,in Handbuch der Physik 8,Springer-Verlag,1959.
[49]A.H.Taub,Relativistic fluid mechanics,Annu.Rev.Fluid Mech.10,301-332(1978).
[50]Wang Libin,Formation of singularities for a kind of quasilinear non-strictly hyperbolic systems,Chinese Annals of Mathematics,23B(2002),439-454.
[5]王利彬,关于具有常重特征的拟线性双曲组Cauchy问题的一个注记,复旦学报(自然科学版),40(2001),633-636.
[52]Wang Libin,Blow-up mechanism of classical solutions to quasilinear hyperbolic systems,Nonlinear Analysis,67(2007),1068-1081.
[53]J.C.Weiland et H.Wilhelmsson,Coherent Non Linear Interaction of Waves in Plasmas,Pergamon Press,1977.
[2]S.S.Antman,Nonlinear Problems of Elasticity,Applied Mathematical Sciences 107,Springer-Verlag,1995.
[3]A.Aw and M.Rascle,Resurrection of "second order" models of traffic flow,SIAM J.Appl.Math.60(2000),916 - 938.
[4]A.Bamberger,C.Flytzanis et D.Jennevé Etude mathématique et numérique d'un module de propagation d'ondes lumineuses dans une fibre optique non centrosymétrique,Ecole Eolytechnique,Mathématiques appliquées 149,(juin 1986).
[5]A.J.Chorin and J.Marsden,A Mathematical Intrduction to Fluid Mechanics,Springer-Verlag,New York,1979.
[6]R.Courant and K.O.Friedrichs,Supersonic flow and shock waves,Sprigner,Berlin,1976.
[7]Dai Wenrong,Asymptotic behavior of global classical solutions of quasilinear nonstrictly hyperbolic systems with weakly linear degeneracy,Chinese Annals of Mathematics,27B(2006),263-286.
[8]Dai Wenrong & Kong Dexing,Global existence and asymptotic behavior of classical solutions of quasilinear hyperbolic systems with linearly degenerate characteristic fields,Journal of Differential Equations,235(2007),127-165.
[9]Lawrence C.Evans,Partial Differential Equations,Graduate Studies in Mathematics,Vol.14,American Mathematical Socity,Providence,Rhode Island,1998.
[10]谷超豪,李大潜等,数学物理方程(第二版),高等教育出版社,北京,2002.
[11]Han Weiwei,Global existence of classical solutions to the Cauchy problem on a semibounded initial axis for inhomogeneous quasilinear hyperbolic systems,Journal of Partial Differential Equations,20(2007),273-288.
[12]Han Weiwei,Breakdown of classical solutions to the Cauchy problem on a semibounded initial axis for quasilinear hyperbolic systems,Nonlinear Analysis(2007),doi:10.1016/j.na.2007.07.027.
[13]Han Weiwei,Asymptotic behavior of global classical solutions to the Cauchy problem on a semi-bounded initial axis for quasilinear hyperbolic systems,to appear in Journal of Mathematical Research and Exposition.
[14]L.Hormander,The Lifespan of Classical Solutions of Nonlinear Hyperbolic Equations,Lecture Notes in Math.,vol,1256,Springer,Berlin,1987.
[15]L.Hormander,Lectures on Nonlinear Hyperbolic Equations,Math.Appl.,vol.26,Springer,Berlin,1997.
[16]A.Jefferey,Quasilinear Hyperbolic Systems and Waves,Pitman Publishing,1976.
[17]F.John,Formation of singularities in one-dimensional nonlinear wave propagation,Communications on Pure and Applied Mathematics,27(1974),377-405.
[18]孔德兴,一阶拟线性双曲型方程组的CauChy问题,复旦学报(自然科学版),33(1994),705-708.
[19]Kong Dexing,Cauchy Problem for Quasilinear Hyperbolic Systems:MSJ Memoirs 6,Tokyo:The Mathematical Society of Japan,2000.
[20]Kong Dexing,Life-span of classical solutions to quasilinear hyperbolic systems with slow decay initial data,Chinese Annals of Mathematics,21B(2000),413-440.
[21]Kong Dexing,Sun Qingyou & Zhou Yi,The equation for time-like extremal surfaces in Minkowski space R~(2+n),Journal of Mathematical Physics,(47)2006,013503:16pp.
[22] Kong Dexing. Sun Qingyou & Zhou Yi, Erratum: "The equation for time-like extremal surfaces in Minkowski space R2+n [J. Math. Phys. 47, 013503](2006), Journal of Mathematical Physics, (47)2006, 069901: 1pp.
[23] D.J. Kaup, A.Reiman et A. Bers, Space-time evolution of nonlinear three-wave interaction I: Interaction in a homogeneous medium, Reviews of Modern Physics 51, (avril 1979).
[24] Kong Dexing & Yang Tong, Asymptotic behavior of Global classical solutions of quasilinear hyperbolic systems, Communications in Partial Differential Equations, 28(2003), 1203-1220.
[25] L.D.Landau and E.M.Lifschitz, Fluid Mechanics, Addison Wesley, Reading MA.1971.
[26] P. D. Lax, Hyperbolic systems of conservation laws II, Communications on Pure and Applied Mathematics, 10(1957), 537-566.
[27] Li Shumin, Cauchy problem for general first order inhomogeneous quasilinear hyperbolic system, Journal of Partial Differential Equations, 15(2002), 46-68.
[28] Li Tatsien, Global Classical Solutions for Quasilinear Hyperbolic Systems, Research in Applied Mathematics 32, J. Wiley/Masson, 1994.
[29] Li Tatsien, Une remarque sur les coordonnees normalisees et ses applications aux systemes hyperboliques quasi lineaires, Comptes Rendus de I'Academie des Sciences, Seris I Mathematic, 331(2000), 447-452.
[30] Li Tatsien, A remark on the normalized coordinates and its applications to quasilinear hyperbolic systems, in Optimal Control and Partial Differential Equations, J. L. Menaldi et al. (eds.), IOS Press, 2001, 181-187.
[31] 李大潜,拟线性双曲组选讲,2005/2-2006/5．
[32] Li Tatsien & Kong Dexing, Breakdown of classical solutions to quasilinear hyperbolic systems, Nonlinear Analysis, 40(2000), 407-437.
[33]Li Tatsien & Kong Dexing,Initial value problem for general quasilinear hyperbolic systems with characteristics with constant multiplicity,Journal of Partial Differential Equations,10(1997),299-322.
[34]Li Tatsien,Kong Dexing & Zhou Yi,Global classical solutions for general quasilinear non-strictly hyperbolic systems with decay initial data,Nonlinear Studies,3(1996),203-229.
[35]李大潜,秦铁虎,物理学与偏微分方程(上册,第一版),高等教育出版社,北京,1997.
[36]李大潜,秦铁虎,物理学与偏微分方程(下册,第一版),高等教育出版社,北京,2000.
[37]Li Tatsien & Wang Libin,Global existence of weakly discontinuous solutions to the Cauchy problem with a kind of non-smooth initial data for quasilinear hyperbolic systems,Chinese Annals of Mathematics,25B(2004),319-334.
[38]Li Tatsien & Wang Libin,Global existence of classical solutions to the Cauchy prolem on a semi-bounded initial axis for quasilinear hyperbolic systems,Nonlinear Analysis,56(2004),961-974.
[39]Li Tatsien & Wang Libin,Blow-up mechanism of classical solutions to quasilinear hyperbolic systems in the critical case,Chinese Annals of Mathematics,27B(2006),53-66.
[40]Li Tatsien & Wang Libin,Existence and uniqueness of global solution to an inverse piston problem,Inverse Problems,23(2007),683-694.
[41]Li Tatsien & Yu Wenci,Boundary Value Problems for Quasilinear Hyperbolic Systems,Duke University Mathematics Series V,1985.
[42]Li Tatsien,Zhou Yi & Kong Dexing,Weak linear degeneracy and global classical solutions for general quasilinear hyperbolic systems,Communications in Partial Differential Equations,19(1994),1263-1317.
[43]Li Tatsien,Zhou Yi & Kong Dexing,Global classical solutions for general quasilinear hyperbolic systems with decay initial data,Nonlinear Analysis,28(1997),1299-1322.
[44]Liu Jianli & Zhou Yi,Asymptotic behaviour of global classical solutions of diagonalizable quasilinear hyperbolic systems,Mathematical Methods in the Applied Sciences,30(2007),479-500.
[45]Liu Taiping,Development of singularities in the nonlinear waves for quasilinear hyperbolic partial differential equation,Journal of DifferentiaI Equations,33(1979),92-111.
[46]Liu Taiping,Quasilinear hyperbolic systems,Communications in Mathematical Physics,68(1979),141-172.
[47]A.Majda,Compressible Fluid Flow and System of Conservation Laws in Several Space Variables,Applied Mathematical Sciences 53,Springer-Verlag,1984.
[48]J.Serrin,Mathematical Principles of Classical Fluid Mechanics,in Handbuch der Physik 8,Springer-Verlag,1959.
[49]A.H.Taub,Relativistic fluid mechanics,Annu.Rev.Fluid Mech.10,301-332(1978).
[50]Wang Libin,Formation of singularities for a kind of quasilinear non-strictly hyperbolic systems,Chinese Annals of Mathematics,23B(2002),439-454.
[5]王利彬,关于具有常重特征的拟线性双曲组Cauchy问题的一个注记,复旦学报(自然科学版),40(2001),633-636.
[52]Wang Libin,Blow-up mechanism of classical solutions to quasilinear hyperbolic systems,Nonlinear Analysis,67(2007),1068-1081.
[53]J.C.Weiland et H.Wilhelmsson,Coherent Non Linear Interaction of Waves in Plasmas,Pergamon Press,1977.