Existence and blow-up of solution of Cauchy problem for the sixth order damped Boussinesq equation

Document Type : Research Paper

Author

School of Mathematical Sciences‎, ‎University of Electronic Science and Technology of China‎, ‎Chengdu 611731‎, ‎P.R‎. ‎China.

Abstract

‎In this paper‎, ‎we consider the existence and uniqueness of the global solution for the sixth-order damped Boussinesq equation‎. ‎Moreover‎, ‎the finite-time blow-up of the solution for the equation is investigated by the concavity method‎.

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Main Subjects


A. Constantin and L. Molinet, The initial value problem for a generalized Boussinesq equation, Differential Integral Equations 15 (2002), no. 9, 1061--1072.
V.K. Kalantarov and O.A. Ladyzhenskaya, The occurrence of collapse for quasilinear equations of parabolic and hyperbolic type, J. Soviet Math. 10 (1978) 53--70.
S.Y. Lai and Y.H. Wu, The asymptotic solution of the Cauchy problem for a generalized Boussinesq equation, Discrete Contin. Dyn. Syst. Ser. B 3 (2003), no. 3, 401--408.
S. Lai, Y. Wang, Y. Wu and Q. Lin, An initial-boundary value problem for a generalized Boussinesq water system in a ball, Int. J. Appl. Math. Sci. 3 (2006), no. 2, 117--133.
H. A. Levine, Instability and nonexisence of global solutions of nonlinear wave equations of the form Putt = Au + F(u), Trans. Amer. Math. Soc. 192 (1974) 1--21.
 H.A. Levine, Some additional remarks on the nonexistence of global solutions to non-linear euqations, SIAM J. Math. Anal. 5 (1974) 138--146.
E. Piskin and N. Polat, Existence, global nonexistence, and asymptotic behavior of solutions for the Cauchy problem of a multidimensional generalized damped Boussinesq-type equation, Turkish J. Math. 38 (2014), no. 4, 706--727.
N. Polat and A. Ertas, Existence and blow-up of solution of Cauchy problem for the generalized damped multidimensional Boussinesq equation, J. math. Anal. Appl. 349 (2009), no. 1, 10--20.
N. Polat, E. Piskin, Asymptotic behavior of a solution of the Cauchy problem for the generalized damped multidimensional Boussinesq equation, Appl. Math. Lett. 25 (2012), no. 11, 1871--1874.
G. Schneider and C.W. Eugene, Kawahara dynamics in dispersive media. Phys. D 152/153 (2001) 384--394.
S. Selberg, Lecture Notes Mat., 632, PDE, http://www.math.ntnu.no/ sselberg, 2011.
V.V. Varlamov, On the Cauchy problem for the damped Boussinesq equation, Differential Integral Equations 9 (1996), no. 3, 619--634.
V.V. Varlamov, On the initial boundary Value Problem for the Damped Boussinesq Equation, Discrete Contin. Dynam. Systems 4 (1998), no. 3, 431--444.
V.V. Varlamov, On the damped Boussinesq equation in a circle, Nonlinear Anal. 38 (1999), no. 4, 447--470.
S.B. Wang and G.W. Chen, Cauchy problem of the generalized double dispersion equation, Nonlinear Anal. 64 (2006), no. 1, 159--173.
S.B.Wang and G.W. Chen, Small amplitude solutions of the generalized IMBq equation, J. Math. Anal. Appl. 274 (2002), no. 2, 846--866.
Y. Wang and C. L. Mu, Blow-up and Scattering of Solution for a Generalized Boussinesq Equation, Appl. Math. Comput. 188 (2007), no. 2, 1131--1141.
Y. Wang and C.L. Mu, Global existence and blow-up of the solutions for the multidimensional generalized Boussinesq equation, Math. Methods Appl. Sci. 30 (2007), no. 12, 1403--1417.
Y.X. Wang, Existence and asymptotic behavior of solutions to the generalized damped Boussinesq equation, Electron. J. Differential Equations (2012), no. 96, 11 pages.
Y.Z. Wang and K.Y. Wang, Decay estimate of solutions to the sixth order damped Boussinesq equation, Appl. Math. Comput. 239 (2014), 171--179.
Y. Zhang, Q. Lin and S. Lai, Long time asymptotic for the damped Boussinesq equation in a circle, J. Partial Differential Equations 18 (2005), no. 2, 97--113.