May  2014, 13(3): 1203-1222. doi: 10.3934/cpaa.2014.13.1203

Global existence and pointwise estimates of solutions for the multidimensional generalized Boussinesq-type equation

1. 

Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, China

2. 

Department of Mathematics, Shanghai Jiao Tong University, 800 Dong Chuan Road, 200240, Shanghai

Received  June 2013 Revised  November 2013 Published  December 2013

In this paper, we study the Cauchy problem for the Boussinesq-type equation \begin{eqnarray} \partial^2_t u-\varepsilon \partial_t \Delta u=-\Delta ^2 u+\Delta u+\Delta g(u), \end{eqnarray} where $g(u)=O(u^\rho),$ $\rho \geq 2.$ By means of long wave-short wave decomposition, Green's function method and energy method, we show that the Cauchy Problem admits a global classical solution in multi dimension. We also show the pointwise estimate of the time asymptotic shape of the solutions in odd dimensional space.
Citation: Miao Liu, Weike Wang. Global existence and pointwise estimates of solutions for the multidimensional generalized Boussinesq-type equation. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1203-1222. doi: 10.3934/cpaa.2014.13.1203
References:
[1]

J. Boussinesq, Theorie des ondes et des remous qui se propagent le long d’un canal rectangulaire horizontal, en communiquant au liquide contene dans ce canal des vitesses sensiblement pareilles de la surface au fond ,, \emph{J. Math. Pures Appl. Ser.}, 17 (1872), 55. Google Scholar

[2]

J. Bona and R. Sachs, Global existence of smooth solutions and stability of solitary waves for a generalized Boussinesq equation ,, \emph{Comm. Math. Phys.}, 118 (1988). doi: 10.1007/BF01218475. Google Scholar

[3]

M. Tsutsumi and T. Matahashi, On the Cauchy problem for the Boussinesq type equation ,, \emph{Math. Japon.}, 36 (1991), 371. Google Scholar

[4]

F. Linares, Global existence of small solutions for a generalized Boussinesq equation ,, \emph{J. Differential Equations}, 106 (1993), 257. doi: 10.1006/jdeq.1993.1108. Google Scholar

[5]

Se. Oh and A. Stefanov, Improved local well-posedness for the periodic "good" Boussinesq equation ,, \emph{J. Differential Equations}, 254 (2013), 4047. doi: 10.1016/j.jde.2013.02.006. Google Scholar

[6]

G. Farah and M. Scialom, On the periodic "good" Boussinesq equation ,, \emph{Proc. Amer. Math. Soc.}, 138 (2010), 953. doi: 10.1090/S0002-9939-09-10142-9. Google Scholar

[7]

Z. Yang and B. Guo, Cauchy problem for the multi-dimensional Boussinesq type equation ,, \emph{J. Math. Anal. Appl.}, 340 (2008), 64. doi: 10.1016/j.jmaa.2007.08.017. Google Scholar

[8]

S. Lai and Y. Wu, The asymptotic solution of the Cauchy problem for a generalized Boussinesq equation ,, \emph{Discrete Contin. Dyn. Syst. Ser. B}, 3 (2003), 401. Google Scholar

[9]

Y. Wang, C. Mu and Y. Wu, Decay and scattering of solutions for a generalized Boussinesq equation ,, \emph{J. Differential Equations}, 247 (2009), 2380. doi: 10.1016/j.jde.2009.07.022. Google Scholar

[10]

N. Polat and A. Ertaş, Existence and blow-up of solution of Cauchy problem for the generalized damped multidimensional Boussinesq equation ,, \emph{J. Math. Anal. Appl.}, 349 (2009), 10. doi: 10.1016/j.jmaa.2008.08.025. Google Scholar

[11]

N. Polat and E. Pişkin, Asymptotic behavior of a solution of the Cauchy problem for the generalized damped multidimensional Boussinesq equation ,, \emph{Appl. Math. Lett.}, 25 (2012), 1871. doi: 10.1016/j.aml.2012.02.051. Google Scholar

[12]

V. Varlamov, Existence and uniqueness of a solution to the Cauchy problem for the damped Boussinesq equation ,, \emph{Math. Methods Appl. Sci.}, 19 (1996), 639. Google Scholar

[13]

V. Varlamov, Asymptotics as $t\rightarrow+\oo$ of a solution to the periodic Cauchy problem for the damped Boussinesq equation ,, \emph{Math. Methods Appl. Sci.}, 20 (1997), 805. Google Scholar

[14]

V. Varlamov, On the initial-boundary value problem for the damped Boussinesq equation ,, \emph{Discrete Contin. Dynam. Systems}, 4 (1998), 431. Google Scholar

[15]

V. Varlamov, On spatially periodic solutions of the damped Boussinesq equation ,, \emph{Differential Integral Equations}, 10 (1997), 1197. Google Scholar

[16]

V. Varlamov, Eigenfunction expansion method and the long-time asymptotics for the damped Boussinesq equation ,, \emph{Discrete Contin. Dynam. Systems}, 7 (2001), 675. Google Scholar

[17]

V. Varlamov, On the spatially two-dimensional Boussinesq equation in a circular domain ,, \emph{Nonlinear Anal. Ser. A: Theory Methods}, 46 (2001), 699. Google Scholar

[18]

V. Varlamov and A. Balogh, Forced nonlinear oscillations of elastic membranes ,, \emph{Nonlinear Anal. Real World Appl.}, 7 (2006), 1005. doi: 10.1016/j.nonrwa.2005.09.006. Google Scholar

[19]

W. J. Wang and W. K. Wang, The pointwise estimates of solutions for semilinear dissipative wave equation in multi-dimensions ,, \emph{J. Math. Anal. Appl.}, 366 (2010), 226. doi: 10.1016/j.jmaa.2009.12.013. Google Scholar

[20]

W. K. Wang and T. Yang, The pointwise estimates of solutions for Euler equations with damping in multi-dimensions ,, \emph{J. Differential Equations}, 173 (2001), 410. doi: 10.1006/jdeq.2000.3937. Google Scholar

[21]

T.-P. Liu and W. K. Wang, The pointwise estimates of diffusion wave for the Navier-Stokes systems in odd multi-dimensions ,, \emph{Comm. Math. Phys.}, 169 (1998), 145. Google Scholar

[22]

N. Kutev, N. Kolkovska and M. Dimova, Global existence of Cauchy problem for Boussinesq paradigm equation ,, \emph{Comput. Math. Appl.}, 65 (2013), 500. doi: 10.1016/j.camwa.2012.05.024. Google Scholar

[23]

Z. Yang, Longtime dynamics of the damped Boussinesq equation ,, \emph{J. Math. Anal. Appl.}, 399 (2013), 180. doi: 10.1016/j.jmaa.2012.09.042. Google Scholar

[24]

D. Li and Y. Chen, Nonlinear Evolution Equation,, Publication of Science, (1989). Google Scholar

show all references

References:
[1]

J. Boussinesq, Theorie des ondes et des remous qui se propagent le long d’un canal rectangulaire horizontal, en communiquant au liquide contene dans ce canal des vitesses sensiblement pareilles de la surface au fond ,, \emph{J. Math. Pures Appl. Ser.}, 17 (1872), 55. Google Scholar

[2]

J. Bona and R. Sachs, Global existence of smooth solutions and stability of solitary waves for a generalized Boussinesq equation ,, \emph{Comm. Math. Phys.}, 118 (1988). doi: 10.1007/BF01218475. Google Scholar

[3]

M. Tsutsumi and T. Matahashi, On the Cauchy problem for the Boussinesq type equation ,, \emph{Math. Japon.}, 36 (1991), 371. Google Scholar

[4]

F. Linares, Global existence of small solutions for a generalized Boussinesq equation ,, \emph{J. Differential Equations}, 106 (1993), 257. doi: 10.1006/jdeq.1993.1108. Google Scholar

[5]

Se. Oh and A. Stefanov, Improved local well-posedness for the periodic "good" Boussinesq equation ,, \emph{J. Differential Equations}, 254 (2013), 4047. doi: 10.1016/j.jde.2013.02.006. Google Scholar

[6]

G. Farah and M. Scialom, On the periodic "good" Boussinesq equation ,, \emph{Proc. Amer. Math. Soc.}, 138 (2010), 953. doi: 10.1090/S0002-9939-09-10142-9. Google Scholar

[7]

Z. Yang and B. Guo, Cauchy problem for the multi-dimensional Boussinesq type equation ,, \emph{J. Math. Anal. Appl.}, 340 (2008), 64. doi: 10.1016/j.jmaa.2007.08.017. Google Scholar

[8]

S. Lai and Y. Wu, The asymptotic solution of the Cauchy problem for a generalized Boussinesq equation ,, \emph{Discrete Contin. Dyn. Syst. Ser. B}, 3 (2003), 401. Google Scholar

[9]

Y. Wang, C. Mu and Y. Wu, Decay and scattering of solutions for a generalized Boussinesq equation ,, \emph{J. Differential Equations}, 247 (2009), 2380. doi: 10.1016/j.jde.2009.07.022. Google Scholar

[10]

N. Polat and A. Ertaş, Existence and blow-up of solution of Cauchy problem for the generalized damped multidimensional Boussinesq equation ,, \emph{J. Math. Anal. Appl.}, 349 (2009), 10. doi: 10.1016/j.jmaa.2008.08.025. Google Scholar

[11]

N. Polat and E. Pişkin, Asymptotic behavior of a solution of the Cauchy problem for the generalized damped multidimensional Boussinesq equation ,, \emph{Appl. Math. Lett.}, 25 (2012), 1871. doi: 10.1016/j.aml.2012.02.051. Google Scholar

[12]

V. Varlamov, Existence and uniqueness of a solution to the Cauchy problem for the damped Boussinesq equation ,, \emph{Math. Methods Appl. Sci.}, 19 (1996), 639. Google Scholar

[13]

V. Varlamov, Asymptotics as $t\rightarrow+\oo$ of a solution to the periodic Cauchy problem for the damped Boussinesq equation ,, \emph{Math. Methods Appl. Sci.}, 20 (1997), 805. Google Scholar

[14]

V. Varlamov, On the initial-boundary value problem for the damped Boussinesq equation ,, \emph{Discrete Contin. Dynam. Systems}, 4 (1998), 431. Google Scholar

[15]

V. Varlamov, On spatially periodic solutions of the damped Boussinesq equation ,, \emph{Differential Integral Equations}, 10 (1997), 1197. Google Scholar

[16]

V. Varlamov, Eigenfunction expansion method and the long-time asymptotics for the damped Boussinesq equation ,, \emph{Discrete Contin. Dynam. Systems}, 7 (2001), 675. Google Scholar

[17]

V. Varlamov, On the spatially two-dimensional Boussinesq equation in a circular domain ,, \emph{Nonlinear Anal. Ser. A: Theory Methods}, 46 (2001), 699. Google Scholar

[18]

V. Varlamov and A. Balogh, Forced nonlinear oscillations of elastic membranes ,, \emph{Nonlinear Anal. Real World Appl.}, 7 (2006), 1005. doi: 10.1016/j.nonrwa.2005.09.006. Google Scholar

[19]

W. J. Wang and W. K. Wang, The pointwise estimates of solutions for semilinear dissipative wave equation in multi-dimensions ,, \emph{J. Math. Anal. Appl.}, 366 (2010), 226. doi: 10.1016/j.jmaa.2009.12.013. Google Scholar

[20]

W. K. Wang and T. Yang, The pointwise estimates of solutions for Euler equations with damping in multi-dimensions ,, \emph{J. Differential Equations}, 173 (2001), 410. doi: 10.1006/jdeq.2000.3937. Google Scholar

[21]

T.-P. Liu and W. K. Wang, The pointwise estimates of diffusion wave for the Navier-Stokes systems in odd multi-dimensions ,, \emph{Comm. Math. Phys.}, 169 (1998), 145. Google Scholar

[22]

N. Kutev, N. Kolkovska and M. Dimova, Global existence of Cauchy problem for Boussinesq paradigm equation ,, \emph{Comput. Math. Appl.}, 65 (2013), 500. doi: 10.1016/j.camwa.2012.05.024. Google Scholar

[23]

Z. Yang, Longtime dynamics of the damped Boussinesq equation ,, \emph{J. Math. Anal. Appl.}, 399 (2013), 180. doi: 10.1016/j.jmaa.2012.09.042. Google Scholar

[24]

D. Li and Y. Chen, Nonlinear Evolution Equation,, Publication of Science, (1989). Google Scholar

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