A NOTE ON “THE CAUCHY PROBLEM FOR COUPLED IMBQ EQUATIONS”

doi:10.1016/S0252-9602(13)60005-3

数学物理学报(英文版) ›› 2013, Vol. 33 ›› Issue (2): 375-392.doi: 10.1016/S0252-9602(13)60005-3

A NOTE ON “THE CAUCHY PROBLEM FOR COUPLED IMBQ EQUATIONS”

郭红霞|陈国旺

Department of Mathematics, Zhengzhou University, Zhengzhou 450052, China

收稿日期:2011-10-10 修回日期:2012-06-29 出版日期:2013-03-20 发布日期:2013-03-20
基金资助:
The authors are supported by Tianyuan Youth Foun-dation of Mathematics (11226177), the National Natural Science Foundation of China (11271336 and 11171311), and Foundation of He’nan Educational Committee (2009C110006).

A NOTE ON “THE CAUCHY PROBLEM FOR COUPLED IMBQ EQUATIONS”

GUO Hong-Xia, CHEN Guo-Wang

Department of Mathematics, Zhengzhou University, Zhengzhou 450052, China

Received:2011-10-10 Revised:2012-06-29 Online:2013-03-20 Published:2013-03-20
Supported by:
The authors are supported by Tianyuan Youth Foun-dation of Mathematics (11226177), the National Natural Science Foundation of China (11271336 and 11171311), and Foundation of He’nan Educational Committee (2009C110006).

摘要/Abstract

摘要：

In this article, we prove that the Cauchy problem for a N-dimensional system of nonlinear wave equations
u_tt − aΔu_tt = Δf(u, v), x ∈ R^N, t > 0,
v_tt − aΔv_tt = Δg(u, v), x ∈ R^N, t > 0
admits a unique global generalized solution in C³([0, ∞); W ^{m, p}(R^N) ∩L^∞(R^N)∩L²(R^N))(m ≥ 0 is an integer, 1 ≤ p < 1 ) and a unique global classical solution in C³([0, ∞); W ^m,p∩L^∞ ∩L²) (m > 2 + N/p ), the sufficient conditions of the blow up of the solution in finite time are given, and also two examples are given.

关键词: N-dimensional system of nonlinear wave equations, Cauchy problem, global solution, blow up of solution

Abstract:

Key words: N-dimensional system of nonlinear wave equations, Cauchy problem, global solution, blow up of solution

中图分类号:

35L30

郭红霞, 陈国旺. A NOTE ON “THE CAUCHY PROBLEM FOR COUPLED IMBQ EQUATIONS”[J]. 数学物理学报(英文版), 2013, 33(2): 375-392.

GUO Hong-Xia, CHEN Guo-Wang. A NOTE ON “THE CAUCHY PROBLEM FOR COUPLED IMBQ EQUATIONS”[J]. Acta mathematica scientia,Series B, 2013, 33(2): 375-392.

参考文献

[1] Wang S, Li M. The Cauchy problem for coupled IMBq equations. IMA Journal of Applied Mathematics, 2009: 1–15

[2] Dé Godefroy A. Blow up of solutions of a generalized Boussinesq equation. IMA J Appl Math, 1998, 60: 123–138

[3] Muto V. Nonlinear models for DNA dynamics. Ph D Thesis. The Technical University of Denmark: Labo-ratory of Applied Mathematical Physics, 1988

[4] Muto V, Scott A C, Christiansen P L. A Toda lattice model for DNA: thermally generated solitions. Physica D, 1990, 44: 75–91

[5] Muto V, Scott A C, Christiansen P L. Thermally generated solition in a Toda lattice model of DNA. Phys Lett, 1989, 136A: 33–36

[6] Rosenau D. Dynamics of dense lattice. Phys Rev B, 1987, 36: 5868–5876

[7] Christiansen P L, Lomdahl P S, Muto V. On a Toda lattice model with a transversal degree of freedom. Nonlinearity, 1990, 4: 477–501

[8] Sergei K T. On a Toda lattice model with a transversal degree of freedom, sufficient criterion of blow up in the continuum limit. Phys Lett, 1993, A173: 267–269

[9] Fan X, Tian L. Compaction solutions and multiple compaction solutions for a continuum Toda lattice model. Chaos Solitons Fractals, 2006, 29: 882–894

[10] Pego R L, Simereka P, Weinstein M I. Oscillatory instability of solitary waves in a continuum model of lattice vibrations. Nonlinearity, 1995, 8: 921–941

[11] Chen G, Guo H, Zhang H. Global existence of solutions of Cauchy problem for generalized system of nonlinear evolution equations arising from DNA. Jouranl of Mathematical Physics, 2009, 50: 083514–23

[12] Chen G, Zhang H. Initial boundary value problem for a system of generalized IMBq equations. Math Methods Appl Sci, 2004, 27: 497–518

[13] Chen G, Wang S. Existence ang nonexistence of global solutions for the generalized IMBq equations. Nonliear Analysis TMA, 1999, 36: 961–986

[14] Cho Y, Ozawa T. Remarks on modified improved Boussinesq equations in one space dimension. Proc R Soc A, 2006, 462: 1949–1963

[15] Cho Y, Ozawa T. On small amplitude solutions to the generalized Boussinesq equations. Discrete Contin Dyn Syst, 2007, 17: 691–711

[16] Liu Y. Existence and blow up of solutions of a nonlinear Pochhammer-Chree equation. Indiana Univ Math J, 1996, 45: 797–816

[17] Turitsyn S K. Blow-up in the Boussinesq equation. Phys Rev E, 1993, 47: 795–796

[18] Wang S, Chen G. The Cauchy problem for the generalized IMBq equation in Ws,p(Rn). J Math Anal Appl, 2002, 266: 38–54

[19] Wang S, Chen G. Small amplitude solutions of the generalized IMBq equation. J Math Anal Appl, 2002, 274: 846–866

[20] Zhu Y. Global existence of small amplitude solution for the generalized IMBq equtaion. J Math Anal Appl, 2008, 340: 304–321

[21] Li Tatsien, Chen Yunmei. Global Classical Solutions for Nonlinear Evolution Equations//Pitman Mono-graphs and Surveys in Pure and Applied Mathematics. Vol 45. New York: Longman Scientific & Technical, 1992

[22] Levine H A. Some additional remarks on the nonexistence of global solutions to nonlinear wave equations. SIAM J MathAnal, 1974, 5: 138–146

A NOTE ON “THE CAUCHY PROBLEM FOR COUPLED IMBQ EQUATIONS”

A NOTE ON “THE CAUCHY PROBLEM FOR COUPLED IMBQ EQUATIONS”

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