Document Type : Research Paper

**Authors**

School of Mathematics and Statistics, North China University of Water Resources and Electric Power, Zhengzhou 450011, China.

**Abstract**

In this paper we focus on the Cauchy problem for the generalized IBq equation with damped term in $n$-dimensional space. We establish the global existence and decay estimates of solution with $L^q(1\leq q\leq 2)$ initial value, provided that the initial value is suitably small. Moreover, we also show that the solution is asymptotic to the solution $u_L$ to the corresponding linear equation as time tends to infinity. Finally, asymptotic profile of the solution $u_L$ to the linearized problem is also discussed.

**Keywords**

**Main Subjects**

[1] E. Arevalo, Y. Gaididei and F. Mertens, Soliton dynamics in damped and forced Boussinesq equations, *Eur. Phys. J. B ***27 **(2002) 63-74.

[2] H. Bahouri, J.Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren Math. Wiss. 343, Springer-Verlag, Berlin-Heidelberg, 2011.

[3] D.E. Edmunds and H. Triebel, Function Spaces, Entropy Numbers, Differential Operators, Cambridge Univ. Press, Cambridge, 2008.

[4] M. Kato, Y. Wang, and S. Kawashima, Asymptotic behavior of solutions to the generalized cubic double dispersion equation in one space dimension, *Kinet. Relat. Models ***6 **(2013), no. 4, 969--987.

[5] S. Kawashima and Y. Wang, Global existence and asymptotic behavior of solutions to the generalized cubic double dispersion equation, *Anal. Appl. (Singap.) ***13 **(2015), no. 3, 233--254.

[6] T. Li and Y. Chen, Nonlinear Evolution Equations (Chinese), Scientific Press, r 1989.

[7] Y. Liu and S. Kawashima, Global existence and asymptotic behavior of solutions for quasi-linear dissipative plate equation, *Discrete Contin. Dyn. Syst. ***29 **(2011) 1113--1139.

[8] M. Nakao and K. Ono, Existence of global solutions to the Cauchy problem for the semilinear dissipative wave equations, *Math. Z. ***214 **(1993), no. 2, 325--342.

[9] K. Nishihara, *L**p*-*L**q *estimates of solutions to the damped wave equation in 3-dimensional space and their applications, *Math. Z. ***244 **(2003), no. 3, 631--649.

[10] K. Ono, Global existence and asymptotic behavior of small solutions for semilinear dissipative wave equations, *Discrete Contin. Dyn. Syst. ***9 **(2003), no. 3, 651--662.

[11] N. Polat, Existence and blow up of solution of Cauchy problem of the generalized damped multidimensional improved modified Boussinesq equation, *Z. Naturforsch. A ***63 **(2008) 543--552.

[12] Y. Sugitani and S. Kawashima, Decay estimates of solution to a semi-linear dissipative plate equation, *J. Hyperbolic Differ. Equ. ***7 **(2010) 471--501.

[13] T. Umeda, S. Kawashima and Y. Shizuta, On the decay of solutions to the linearized equations of electro-magneto-uid dynamics, *Japan J. Appl. Math. ***1 **(1984) 435-457.

[14] Y. Wang, Global existence and asymptotic behaviour of solutions for the generalized Boussinesq equation, *Nonlinear Anal. ***70 **(2009), no. 1, 465--482.

[15] Y.X. Wang, Existence and asymptotic behavior of solutions to the generalized damped Boussinesq equation, *Electron. J. Differential Equations *(2012), no. 96, 11 pp.

[16] Y.X. Wang, On the Cauchy problem for one dimension generalized Boussinesq equation, *Internat. J. Math. ***26 **(2015), no. 3, Article ID 1550023, 22 pages.

[17] S. Wang and G. Chen, The Cauchy problem for the generalized IMBq equation in *W**s;p*(R*n*), *J. Math. Anal. Appl. ***266 **(2002), no. 1, 38--54.

[18] S. Wang and G. Chen, Small amplitude solutions of the generalized IMBq equation, *J. Math. Anal. Appl. ***274 **(2002), no. 2, 846--866.

[19] S. Wang and F. Da, On the asymptotic behaviour of solution for the generalized double dispersion equation, *Appl. Anal. ***92 **(2013), no. 6, 1179--1193.

[20] S. Wang and H. Xu, On the asymptotic behavior of solution for the generalized IBq equation with hydrodynamical damped term, *J. Differential Equations ***252 **(2012), no. 7, 4243--4258.

[21] W. Wang and W. Wang, The pointwise estimates of solutions for semilinear dissipative wave equation in multi-dimensions, *J. Math. Anal. Appl. ***368 **(2010), no. 1, 226--241.

[22] Y. Wang, F. Liu and Y. Zhang, Global existence and asymptotic of solutions for a semi-linear wave equation, *J. Math. Anal. Appl. ***385 **(2012) 836--853.

[23] Y. Wang and Y.X. Wang, Global existence and asymptotic behavior of solutions to a nonlinear wave equation of fourth-order, *J. Math. Phys. ***53 **(2012), Article ID 013512, 13 pages.

[24] S. Zheng, Nonlinear Evolution Equations, Monographs and Surveys in Pure and Applied Mathematics 133, Chapman & Hall/CRC, 2004.

[25] Z. Zhuang and Y.Z. Zhang, Global existence and asymptotic behavior of solutions to a class of fourth-order wave equations, *Bound. Value Probl. ***2013 **(2013), no. 168, 15 pages.

November 2017

Pages 1585-1600

**Receive Date:**26 November 2015**Revise Date:**12 June 2016**Accept Date:**16 July 2016