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.

[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.

[3]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[14]

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

[15]

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

[16]

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

[17]

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

[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.

[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.

[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.

[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.

[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.

[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.

[24]

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

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.

[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.

[3]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[14]

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

[15]

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

[16]

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

[17]

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

[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.

[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.

[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.

[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.

[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.

[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.

[24]

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

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