Acta mathematica scientia, Series B >
A NOTE ON “THE CAUCHY PROBLEM FOR COUPLED IMBQ EQUATIONS”
Received date: 2011-10-10
Revised date: 2012-06-29
Online 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).
In this article, we prove that the Cauchy problem for a N-dimensional system of nonlinear wave equations
utt − aΔutt = Δf(u, v), x ∈ RN, t > 0,
vtt − aΔvtt = Δg(u, v), x ∈ RN, t > 0
admits a unique global generalized solution in C3([0, ∞); W m, p(RN) ∩L∞(RN)∩L2(RN))(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.
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 . DOI: 10.1016/S0252-9602(13)60005-3
[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
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