American Institute of Mathematical Sciences

Advanced
October  2007, 17(4): 691-711. doi: 10.3934/dcds.2007.17.691

On small amplitude solutions to the generalized Boussinesq equations

Yonggeun Cho 1, and  Tohru Ozawa 1,

1. 

Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan

Received  March 2006 Revised  October 2006 Published  January 2007

We study the existence and scattering of global small amplitude solutions to generalized Boussinesq (Bq) and improved modified Boussinesq (imBq) equations with nonlinear term $f(u)$ behaving as a power $u^p$ as $u \to 0$ in $\mathbb{R}^n, n \ge 1$.
Keywords: global existence, generalized Bq and imBq equations, small amplitude solution, scattering..
Mathematics Subject Classification: Primary: 35Q53; Secondary: 47J3.
Citation: Yonggeun Cho, Tohru Ozawa. On small amplitude solutions to the generalized Boussinesq equations. Discrete & Continuous Dynamical Systems, 2007, 17 (4) : 691-711. doi: 10.3934/dcds.2007.17.691
[1]

J. Colliander, Justin Holmer, Monica Visan, Xiaoyi Zhang. Global existence and scattering for rough solutions to generalized nonlinear Schrödinger equations on $R$. Communications on Pure & Applied Analysis, 2008, 7 (3) : 467-489. doi: 10.3934/cpaa.2008.7.467
[2]

Walter A. Strauss, Kimitoshi Tsutaya. Existence and blow up of small amplitude nonlinear waves with a negative potential. Discrete & Continuous Dynamical Systems, 1997, 3 (2) : 175-188. doi: 10.3934/dcds.1997.3.175
[3]

Guy Métivier, K. Zumbrun. Existence and sharp localization in velocity of small-amplitude Boltzmann shocks. Kinetic & Related Models, 2009, 2 (4) : 667-705. doi: 10.3934/krm.2009.2.667
[4]

Kosuke Ono. Global existence and asymptotic behavior of small solutions for semilinear dissipative wave equations. Discrete & Continuous Dynamical Systems, 2003, 9 (3) : 651-662. doi: 10.3934/dcds.2003.9.651
[5]

Reinhard Racke, Jürgen Saal. Hyperbolic Navier-Stokes equations II: Global existence of small solutions. Evolution Equations & Control Theory, 2012, 1 (1) : 217-234. doi: 10.3934/eect.2012.1.217
[6]

Ling Liu, Jiashan Zheng. Global existence and boundedness of solution of a parabolic-parabolic-ODE chemotaxis-haptotaxis model with (generalized) logistic source. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3357-3377. doi: 10.3934/dcdsb.2018324
[7]

Yonggeun Cho, Gyeongha Hwang, Tohru Ozawa. On small data scattering of Hartree equations with short-range interaction. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1809-1823. doi: 10.3934/cpaa.2016016
[8]

Changhun Yang. Scattering results for Dirac Hartree-type equations with small initial data. Communications on Pure & Applied Analysis, 2019, 18 (4) : 1711-1734. doi: 10.3934/cpaa.2019081
[9]

Shubin Wang, Guowang Chen. Cauchy problem for the nonlinear Schrödinger-IMBq equations. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 203-214. doi: 10.3934/dcdsb.2006.6.203
[10]

Laiqing Meng, Jia Yuan, Xiaoxin Zheng. Global existence of almost energy solution to the two-dimensional chemotaxis-Navier-Stokes equations with partial diffusion. Discrete & Continuous Dynamical Systems, 2019, 39 (6) : 3413-3441. doi: 10.3934/dcds.2019141
[11]

Victor Wasiolek. Uniform global existence and convergence of Euler-Maxwell systems with small parameters. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2007-2021. doi: 10.3934/cpaa.2016025
[12]

Roger Grimshaw, Dmitry Pelinovsky. Global existence of small-norm solutions in the reduced Ostrovsky equation. Discrete & Continuous Dynamical Systems, 2014, 34 (2) : 557-566. doi: 10.3934/dcds.2014.34.557
[13]

Anudeep Kumar Arora. Scattering of radial data in the focusing NLS and generalized Hartree equations. Discrete & Continuous Dynamical Systems, 2019, 39 (11) : 6643-6668. doi: 10.3934/dcds.2019289
[14]

Dimitra Antonopoulou, Georgia Karali. Existence of solution for a generalized stochastic Cahn-Hilliard equation on convex domains. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 31-55. doi: 10.3934/dcdsb.2011.16.31
[15]

Rodica Toader. Scattering in domains with many small obstacles. Discrete & Continuous Dynamical Systems, 1998, 4 (2) : 321-338. doi: 10.3934/dcds.1998.4.321
[16]

Fabio Scalco Dias, Luis Fernando Mello. The center--focus problem and small amplitude limit cycles in rigid systems. Discrete & Continuous Dynamical Systems, 2012, 32 (5) : 1627-1637. doi: 10.3934/dcds.2012.32.1627
[17]

Ling-Jun Wang. The dynamics of small amplitude solutions of the Swift-Hohenberg equation on a large interval. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1129-1156. doi: 10.3934/cpaa.2012.11.1129
[18]

Andrew Comech, David Stuart. Small amplitude solitary waves in the Dirac-Maxwell system. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1349-1370. doi: 10.3934/cpaa.2018066
[19]

Deconinck Bernard, Olga Trichtchenko. High-frequency instabilities of small-amplitude solutions of Hamiltonian PDEs. Discrete & Continuous Dynamical Systems, 2017, 37 (3) : 1323-1358. doi: 10.3934/dcds.2017055
[20]

Ramon Plaza, K. Zumbrun. An Evans function approach to spectral stability of small-amplitude shock profiles. Discrete & Continuous Dynamical Systems, 2004, 10 (4) : 885-924. doi: 10.3934/dcds.2004.10.885

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (155)
  • HTML views (0)
  • Cited by (31)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences