    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 & Control Theory, 2017, 6 (4) : 629-645. doi: 10.3934/eect.2017032
##### References:
  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.  Google Scholar  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.  Google Scholar  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. Google Scholar  X. Hu and Y. Z. Wang, Asymptotic profiles of solutions to two dimensional degenerated wave equation with damping, submitted. Google Scholar  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.  Google Scholar  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.  Google Scholar  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. Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  A. M. Samsonov, Nonlinear strain waves in elastic waveguides, in: Nonlinear Waves in Solids, CISM Coyrses and Lecture, Springer, 341 (1994), 349-382. Google Scholar  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. Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  Y. -Z. Wang, S. Kawashima and J. Xu, Optimal decay estimates of solutions to the generalized double dispersion equation, preprint. Google Scholar  Z. Wu and W. Wang, Refined pointwise estimates for the Navier-Stokes-Poisson equations, Anal. Appl., 14 (2016), 739-762.  doi: 10.1142/S0219530515500153. Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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. Google Scholar

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##### References:
  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.  Google Scholar  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.  Google Scholar  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. Google Scholar  X. Hu and Y. Z. Wang, Asymptotic profiles of solutions to two dimensional degenerated wave equation with damping, submitted. Google Scholar  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.  Google Scholar  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.  Google Scholar  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. Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  A. M. Samsonov, Nonlinear strain waves in elastic waveguides, in: Nonlinear Waves in Solids, CISM Coyrses and Lecture, Springer, 341 (1994), 349-382. Google Scholar  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. Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  Y. -Z. Wang, S. Kawashima and J. Xu, Optimal decay estimates of solutions to the generalized double dispersion equation, preprint. Google Scholar  Z. Wu and W. Wang, Refined pointwise estimates for the Navier-Stokes-Poisson equations, Anal. Appl., 14 (2016), 739-762.  doi: 10.1142/S0219530515500153. Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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. Google Scholar
  Yacheng Liu, Runzhang Xu. Potential well method for initial boundary value problem of the generalized double dispersion equations. Communications on Pure & Applied Analysis, 2008, 7 (1) : 63-81. doi: 10.3934/cpaa.2008.7.63  Fei Guo, Bao-Feng Feng, Hongjun Gao, Yue Liu. On the initial-value problem to the Degasperis-Procesi equation with linear dispersion. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1269-1290. doi: 10.3934/dcds.2010.26.1269  Vladimir V. Varlamov. On the initial boundary value problem for the damped Boussinesq equation. Discrete & Continuous Dynamical Systems - A, 1998, 4 (3) : 431-444. doi: 10.3934/dcds.1998.4.431  Ivonne Rivas, Muhammad Usman, Bing-Yu Zhang. Global well-posedness and asymptotic behavior of a class of initial-boundary-value problem of the Korteweg-De Vries equation on a finite domain. Mathematical Control & Related Fields, 2011, 1 (1) : 61-81. doi: 10.3934/mcrf.2011.1.61  Changming Song, Hong Li, Jina Li. Initial boundary value problem for the singularly perturbed Boussinesq-type equation. Conference Publications, 2013, 2013 (special) : 709-717. doi: 10.3934/proc.2013.2013.709  Jun Zhou. Initial boundary value problem for a inhomogeneous pseudo-parabolic equation. Electronic Research Archive, 2020, 28 (1) : 67-90. doi: 10.3934/era.2020005  Shaoyong Lai, Yong Hong Wu, Xu Yang. The global solution of an initial boundary value problem for the damped Boussinesq equation. Communications on Pure & Applied Analysis, 2004, 3 (2) : 319-328. doi: 10.3934/cpaa.2004.3.319  Chengxia Lei, Yihong Du. Asymptotic profile of the solution to a free boundary problem arising in a shifting climate model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 895-911. doi: 10.3934/dcdsb.2017045  Yi Zhou, Jianli Liu. The initial-boundary value problem on a strip for the equation of time-like extremal surfaces. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 381-397. doi: 10.3934/dcds.2009.23.381  Martn P. Árciga Alejandre, Elena I. Kaikina. Mixed initial-boundary value problem for Ott-Sudan-Ostrovskiy equation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (2) : 381-409. doi: 10.3934/dcds.2012.32.381  Türker Özsarı, Nermin Yolcu. The initial-boundary value problem for the biharmonic Schrödinger equation on the half-line. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3285-3316. doi: 10.3934/cpaa.2019148  Ning-An Lai, Yi Zhou. Blow up for initial boundary value problem of critical semilinear wave equation in two space dimensions. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1499-1510. doi: 10.3934/cpaa.2018072  Masakazu Kato, Yu-Zhu Wang, Shuichi Kawashima. Asymptotic behavior of solutions to the generalized cubic double dispersion equation in one space dimension. Kinetic & Related Models, 2013, 6 (4) : 969-987. doi: 10.3934/krm.2013.6.969  Gilles Carbou, Bernard Hanouzet. Relaxation approximation of the Kerr model for the impedance initial-boundary value problem. Conference Publications, 2007, 2007 (Special) : 212-220. doi: 10.3934/proc.2007.2007.212  Xianpeng Hu, Dehua Wang. The initial-boundary value problem for the compressible viscoelastic flows. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 917-934. doi: 10.3934/dcds.2015.35.917  Jong-Shenq Guo, Masahiko Shimojo. Blowing up at zero points of potential for an initial boundary value problem. Communications on Pure & Applied Analysis, 2011, 10 (1) : 161-177. doi: 10.3934/cpaa.2011.10.161  Xu Liu, Jun Zhou. Initial-boundary value problem for a fourth-order plate equation with Hardy-Hénon potential and polynomial nonlinearity. Electronic Research Archive, 2020, 28 (2) : 599-625. doi: 10.3934/era.2020032  Linglong Du, Caixuan Ren. Pointwise wave behavior of the initial-boundary value problem for the nonlinear damped wave equation in $\mathbb{R}_{+}^{n}$. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3265-3280. doi: 10.3934/dcdsb.2018319  Roberto Camassa. Characteristics and the initial value problem of a completely integrable shallow water equation. Discrete & Continuous Dynamical Systems - B, 2003, 3 (1) : 115-139. doi: 10.3934/dcdsb.2003.3.115  Gen Nakamura, Michiyuki Watanabe. An inverse boundary value problem for a nonlinear wave equation. Inverse Problems & Imaging, 2008, 2 (1) : 121-131. doi: 10.3934/ipi.2008.2.121

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