# American Institute of Mathematical Sciences

December  2017, 6(4): 629-645. doi: 10.3934/eect.2017032

## Asymptotic profile of solutions to the linearized double dispersion equation on the half space $\mathbb{R}^{n}_{+}$

 1 School of Mathematics and Statistics, North China University of Water Resources and Electric Power, Zhengzhou 450011, China 2 School of Sciences, Zhejiang Tongji Vocational College of Science and Technology, Hangzhou 310001, China 3 School of Mathematics, Luoyang Normal University, Luoyang 471934, China

* Corresponding author: Yu-Zhu Wang

Received  May 2017 Revised  July 2017 Published  September 2017

Fund Project: The first author is supported by Innovation Scientists and Technicians Troop Construction Projects of Henan Province(Grant No.14HASTIT041) and the third author is supported by National Natural Science Foundation of China(Grant No. 11671188).

In this paper, we investigate the initial boundary value problem for the linearized double dispersion equation on the half space $\mathbb{R}^{n}_{+}$. We convert the initial boundary value problem into the initial value problem by odd reflection. The asymptotic profile of solutions to the initial boundary value problem is derived by establishing the asymptotic profile of solutions to the initial value problem. More precisely, the asymptotic profile of solutions is associated with the convolution of the partial derivative of the fundamental solutions of heat equation and the fundamental solutions of free wave equation.

Citation: Yu-Zhu Wang, Si Chen, Menglong Su. Asymptotic profile of solutions to the linearized double dispersion equation on the half space $\mathbb{R}^{n}_{+}$. Evolution Equations and Control Theory, 2017, 6 (4) : 629-645. doi: 10.3934/eect.2017032
##### References:
 [1] G. Chen, Y. Wang and S. Wang, Initial boundary value problem of the generalized cubic double dispersion equation, J. Math. Anal. Appl., 299 (2004), 563-577.  doi: 10.1016/j.jmaa.2004.05.044. [2] G. Chen and H. Xue, Periodic boundary value problem and Cauchy problem of the generalized cubic double dispersion equation, Acta Math. Scientia, 28 (2008), 573-587.  doi: 10.1016/S0252-9602(08)60060-0. [3] D. Hoff and K. Zumbrum, Multi-dimensional diffusion waves for the Navier-Stokes equations of compressible flow, Indiana Univ. Math. J., 44 (1995), 603-676.  doi: 10.1512/iumj.1995.44.2003. [4] X. Hu and Y. Z. Wang, Asymptotic profiles of solutions to two dimensional degenerated wave equation with damping, submitted. [5] R. Ikehata, New decay estimates for linear damped wave equations and its application to nonlinear problem, Math. Meth. Appl. Sci., 27 (2004), 865-889.  doi: 10.1002/mma.476. [6] R. Ikehata, Asymptotic profiles for wave equations with strong damping, J. Differential Equations, 257 (2014), 2159-2177.  doi: 10.1016/j.jde.2014.05.031. [7] R. Ikehata, Some remarks on the asymptotic profiles of solutions for strongly damped wave equations on the 1-D half space, J. Math. Anal. Appl., 421 (2015), 905-916.  doi: 10.1016/j.jmaa.2014.07.055. [8] M. Kato, Y.-Z. Wang and S. Kawashima, Asymptotic behavior of solutions to the generalized cubic double dispersion equation in one space dimension, Kinet. Relat. Models, 6 (2013), 969-987.  doi: 10.3934/krm.2013.6.969. [9] S. Kawashima and Y.-Z. Wang, Global existence and asymptotic behavior of solutions to the generalized cubic double dispersion equation, Analysis and Applications, 13 (2015), 233-254.  doi: 10.1142/S021953051450002X. [10] T. Liu and W. Wang, The pointwise estimates of diffusion wave for the Navier-Stokes systems in odd multi-dimensions, Commun. Math. Phys., 196 (1998), 145-173.  doi: 10.1007/s002200050418. [11] M. Liu and W. Wang, Global existence and pointwise estimates of solutions for the multidimensional generalized Boussinesq equation, Comm. Pure Appl. Anal., 13 (2014), 1203-1222.  doi: 10.3934/cpaa.2014.13.1203. [12] N. Polat and A. Ertaş, Existence and blow-up of solution of Cauchy problem for the generalized damped multidimensional Boussinesq equation, J. Math. Anal. Appl., 349 (2009), 10-20.  doi: 10.1016/j.jmaa.2008.08.025. [13] A. M. Samsonov, Nonlinear strain waves in elastic waveguides, in: Nonlinear Waves in Solids, CISM Coyrses and Lecture, Springer, 341 (1994), 349-382. [14] A. M. Samsonov and E. V. Sokurinskaya, Energy exchange between nonlinear waves in elastic waveguides and external media, in: Nonlinear Waves in Active Media, Springer, Berlin, Heidelberg, 1989, 99-104. [15] W. Wang, Nonlinear evolution systems and Green's function, Acta Math. Sci. Ser. B Engl. Ed., 30 (2010), 2051-2063.  doi: 10.1016/S0252-9602(10)60190-7. [16] S. Wang and G. Chen, Cauchy problem of the generalized double dispersion equation, Nonlinear Anal., 64 (2006), 159-173.  doi: 10.1016/j.na.2005.06.017. [17] S. Wang and F. Da, On the asymptotic behavior of solution for the generalized double dipersion equation, Appl. Anal., 92 (2013), 1179-1193.  doi: 10.1080/00036811.2012.661044. [18] Y.-Z. Wang and S. Chen, Asymptotic profile of solutions to the double dispersion equation, Nonlinear Anal., 134 (2016), 236-254.  doi: 10.1016/j.na.2016.01.009. [19] Y.-Z. Wang and H. Zhao, Pointwise estimates of global small solutions to the generalized double dispersion equation, J. Math. Anal. Appl., 448 (2017), 672-690.  doi: 10.1016/j.jmaa.2016.11.030. [20] Y. -Z. Wang, S. Kawashima and J. Xu, Optimal decay estimates of solutions to the generalized double dispersion equation, preprint. [21] Z. Wu and W. Wang, Refined pointwise estimates for the Navier-Stokes-Poisson equations, Anal. Appl., 14 (2016), 739-762.  doi: 10.1142/S0219530515500153. [22] R. Xu, Y. Liu and T. Yu, Global existence of solution for Cauchy problem of multidimensional generalized double dispersion equations, Nonlinear Anal., 71 (2009), 4977-4983.  doi: 10.1016/j.na.2009.03.069. [23] Z. Yang, Global attractor for a nonlinear wave equation arising in elastic waveguide model, Nonlinear Anal., 70 (2009), 2132-2142.  doi: 10.1016/j.na.2008.02.114. [24] Z. Yang, A global attractor for the elastic waveguide model in $\mathbb{R}^n$, Nonlinear Anal., 74 (2011), 6640-6661.  doi: 10.1016/j.na.2011.06.045. [25] Z. Yang and K. Li, Longtime dynamics for a elastic waveguide model, Discrete Contin. Dyn. Syst., (2013), 797-806.  doi: 10.3934/proc.2013.2013.797. [26] Z. Yang, N. Feng and T. Ma, Global attractor for the generalized double dispersion equation, Nonlinear Anal., 115 (2015), 103-116.  doi: 10.1016/j.na.2014.12.006.

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##### References:
 [1] G. Chen, Y. Wang and S. Wang, Initial boundary value problem of the generalized cubic double dispersion equation, J. Math. Anal. Appl., 299 (2004), 563-577.  doi: 10.1016/j.jmaa.2004.05.044. [2] G. Chen and H. Xue, Periodic boundary value problem and Cauchy problem of the generalized cubic double dispersion equation, Acta Math. Scientia, 28 (2008), 573-587.  doi: 10.1016/S0252-9602(08)60060-0. [3] D. Hoff and K. Zumbrum, Multi-dimensional diffusion waves for the Navier-Stokes equations of compressible flow, Indiana Univ. Math. J., 44 (1995), 603-676.  doi: 10.1512/iumj.1995.44.2003. [4] X. Hu and Y. Z. Wang, Asymptotic profiles of solutions to two dimensional degenerated wave equation with damping, submitted. [5] R. Ikehata, New decay estimates for linear damped wave equations and its application to nonlinear problem, Math. Meth. Appl. Sci., 27 (2004), 865-889.  doi: 10.1002/mma.476. [6] R. Ikehata, Asymptotic profiles for wave equations with strong damping, J. Differential Equations, 257 (2014), 2159-2177.  doi: 10.1016/j.jde.2014.05.031. [7] R. Ikehata, Some remarks on the asymptotic profiles of solutions for strongly damped wave equations on the 1-D half space, J. Math. Anal. Appl., 421 (2015), 905-916.  doi: 10.1016/j.jmaa.2014.07.055. [8] M. Kato, Y.-Z. Wang and S. Kawashima, Asymptotic behavior of solutions to the generalized cubic double dispersion equation in one space dimension, Kinet. Relat. Models, 6 (2013), 969-987.  doi: 10.3934/krm.2013.6.969. [9] S. Kawashima and Y.-Z. Wang, Global existence and asymptotic behavior of solutions to the generalized cubic double dispersion equation, Analysis and Applications, 13 (2015), 233-254.  doi: 10.1142/S021953051450002X. [10] T. Liu and W. Wang, The pointwise estimates of diffusion wave for the Navier-Stokes systems in odd multi-dimensions, Commun. Math. Phys., 196 (1998), 145-173.  doi: 10.1007/s002200050418. [11] M. Liu and W. Wang, Global existence and pointwise estimates of solutions for the multidimensional generalized Boussinesq equation, Comm. Pure Appl. Anal., 13 (2014), 1203-1222.  doi: 10.3934/cpaa.2014.13.1203. [12] N. Polat and A. Ertaş, Existence and blow-up of solution of Cauchy problem for the generalized damped multidimensional Boussinesq equation, J. Math. Anal. Appl., 349 (2009), 10-20.  doi: 10.1016/j.jmaa.2008.08.025. [13] A. M. Samsonov, Nonlinear strain waves in elastic waveguides, in: Nonlinear Waves in Solids, CISM Coyrses and Lecture, Springer, 341 (1994), 349-382. [14] A. M. Samsonov and E. V. Sokurinskaya, Energy exchange between nonlinear waves in elastic waveguides and external media, in: Nonlinear Waves in Active Media, Springer, Berlin, Heidelberg, 1989, 99-104. [15] W. Wang, Nonlinear evolution systems and Green's function, Acta Math. Sci. Ser. B Engl. Ed., 30 (2010), 2051-2063.  doi: 10.1016/S0252-9602(10)60190-7. [16] S. Wang and G. Chen, Cauchy problem of the generalized double dispersion equation, Nonlinear Anal., 64 (2006), 159-173.  doi: 10.1016/j.na.2005.06.017. [17] S. Wang and F. Da, On the asymptotic behavior of solution for the generalized double dipersion equation, Appl. Anal., 92 (2013), 1179-1193.  doi: 10.1080/00036811.2012.661044. [18] Y.-Z. Wang and S. Chen, Asymptotic profile of solutions to the double dispersion equation, Nonlinear Anal., 134 (2016), 236-254.  doi: 10.1016/j.na.2016.01.009. [19] Y.-Z. Wang and H. Zhao, Pointwise estimates of global small solutions to the generalized double dispersion equation, J. Math. Anal. Appl., 448 (2017), 672-690.  doi: 10.1016/j.jmaa.2016.11.030. [20] Y. -Z. Wang, S. Kawashima and J. Xu, Optimal decay estimates of solutions to the generalized double dispersion equation, preprint. [21] Z. Wu and W. Wang, Refined pointwise estimates for the Navier-Stokes-Poisson equations, Anal. Appl., 14 (2016), 739-762.  doi: 10.1142/S0219530515500153. [22] R. Xu, Y. Liu and T. Yu, Global existence of solution for Cauchy problem of multidimensional generalized double dispersion equations, Nonlinear Anal., 71 (2009), 4977-4983.  doi: 10.1016/j.na.2009.03.069. [23] Z. Yang, Global attractor for a nonlinear wave equation arising in elastic waveguide model, Nonlinear Anal., 70 (2009), 2132-2142.  doi: 10.1016/j.na.2008.02.114. [24] Z. Yang, A global attractor for the elastic waveguide model in $\mathbb{R}^n$, Nonlinear Anal., 74 (2011), 6640-6661.  doi: 10.1016/j.na.2011.06.045. [25] Z. Yang and K. Li, Longtime dynamics for a elastic waveguide model, Discrete Contin. Dyn. Syst., (2013), 797-806.  doi: 10.3934/proc.2013.2013.797. [26] Z. Yang, N. Feng and T. Ma, Global attractor for the generalized double dispersion equation, Nonlinear Anal., 115 (2015), 103-116.  doi: 10.1016/j.na.2014.12.006.
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