December  2013, 6(4): 969-987. doi: 10.3934/krm.2013.6.969

Asymptotic behavior of solutions to the generalized cubic double dispersion equation in one space dimension

1. 

Muroran Institute of Technology, Muroran 050-8585, Japan

2. 

School of Mathematics and Information Sciences, North China University of Water Resources and Electric Power, Zhengzhou 450011

3. 

Faculty of Mathematics, Kyushu University, Fukuoka 819-0395

Received  July 2013 Revised  September 2013 Published  November 2013

We study the initial value problem for the generalized cubic double dispersion equation in one space dimension. We establish a nonlinear approximation result to our global solutions that was obtained in [6]. Moreover, we show that as time tends to infinity, the solution approaches the superposition of nonlinear diffusion waves which are given explicitly in terms of the self-similar solution of the viscous Burgers equation. The proof is based on the semigroup argument combined with the analysis of wave decomposition.
Citation: 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
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.  doi: 10.1016/j.jmaa.2004.05.044.  Google Scholar

[2]

M. Kato, Large time behavior of solutions to the generalized Burgers equations,, Osaka J. Math., 44 (2007), 923.   Google Scholar

[3]

S. Kawashima, Large-time behaviour of solutions to hyperbolic-parabolic systems of conservation laws and applications,, Proc. Roy. Soc. Edinburgh, 106 (1987), 169.  doi: 10.1017/S0308210500018308.  Google Scholar

[4]

S. Kawashima, Large-time behavior of solutions of the discrete Boltzmann equation,, Comm. Math. Phys., 109 (1987), 563.  doi: 10.1007/BF01208958.  Google Scholar

[5]

S. Kawashima, Y. Nikkuni and S. Nishibata, The initial value problem for hyperbolic-elliptic coupled systems and application to radiation hydrodynamics,, Analysis of Systems of Conservation Laws, (1999), 87.   Google Scholar

[6]

S. Kawashima and Y.-Z. Wang, Global Existence and Asymptotic Behavior of Solutions to the Generalized Cubic Double Dispersion Equation,, Analysis and Applications (accepted)., ().   Google Scholar

[7]

T. T. Li and Y. M. Chen, Nonlinear Evolution Equations,, Academic Press, (1989).   Google Scholar

[8]

T.-P. Liu, Hyperbolic and Viscous Conservation Laws,, CBMS-NSF Regional Conference Sereies in Applied Math., (2000).  doi: 10.1137/1.9780898719420.  Google Scholar

[9]

T.-P. Liu and Y. Zeng, Large time behavior of solutions to general quasilinear hyperbolic-parabolic systems of conservation laws,, Memoirs Amer. Math. Soc., 125 (1997).  doi: 10.1090/memo/0599.  Google Scholar

[10]

Y. Liu and S. Kawashima, Global existence and asymptotic behavior of solutions for quasi-linear dissipative plate equation,, Discrete Contin. Dyn. Syst., 29 (2011), 1113.  doi: 10.3934/dcds.2011.29.1113.  Google Scholar

[11]

Y. Liu and S. Kawashima, Global existence and decay of solutions for a quasi-linear dissipative plate equation,, J. Hyperbolic Differential Equations, 8 (2011), 591.  doi: 10.1142/S0219891611002500.  Google Scholar

[12]

Y. Liu and S. Kawashima, Decay property for a plate equation with memory-type dissipation,, Kinetic and Related Models, 4 (2011), 531.  doi: 10.3934/krm.2011.4.531.  Google Scholar

[13]

A. Matsumura, On the asymptotic behaviour of solutions of semi-linear wave equations,, Publ. Res. Inst. Math. Sci., 12 (1976), 169.  doi: 10.2977/prims/1195190962.  Google Scholar

[14]

M. Nakao and K. Ono, Existence of global solutions to the Cauchy problem for the semilinear dissipative wave equations,, Math. Z., 214 (1993), 325.  doi: 10.1007/BF02572407.  Google Scholar

[15]

K. Nishihara, $L^p$-$L^q$ estimates of solutions to the damped wave equation in 3-dimensional space and their applications,, Math. Z., 244 (2003), 631.  doi: 10.1007/s00209-003-0516-0.  Google Scholar

[16]

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.  doi: 10.1016/j.jmaa.2008.08.025.  Google Scholar

[17]

A. M. Samsonov, Nonlinear strain waves in elastic waveguides,, in Nonlinear Waves in Solids(Udine, 341 (1994), 349.   Google Scholar

[18]

A. M. Samsonov and E. V. Sokurinskaya, Energy exchange between nonlinear waves in elastic waveguides and external media,, in Nonlinear Waves in Active Media, (1989), 99.   Google Scholar

[19]

Y. Sugitani and S. Kawashima, Decay estimates of solution to a semi-linear dissipative plate equation,, J. Hyperbolic Differential Equations, 7 (2010), 471.  doi: 10.1142/S0219891610002207.  Google Scholar

[20]

H. Takeda and S. Yoshikawa, On the initial value problem of the semilinear beam equation with weak damping I: Smoothing effect,, J. Math. Anal. Appl., 401 (2013), 244.  doi: 10.1016/j.jmaa.2012.12.015.  Google Scholar

[21]

H. Takeda and S. Yoshikawa, On the initial value problem of the semilinear beam equation with weak damping II: Asymptotic profles,, J. Differential Equations, 253 (2012), 3061.  doi: 10.1016/j.jde.2012.07.014.  Google Scholar

[22]

G. Todorova and B. Yordanov, Critical exponent for a nonlinear wave equation with damping,, J. Differential Equations, 174 (2001), 464.  doi: 10.1006/jdeq.2000.3933.  Google Scholar

[23]

Y. Ueda and S. Kawashima, Large time behavior of solutions to a semilinear hyperbolic system with relaxation,, J. Hyperbolic Differential Equations, 4 (2007), 147.  doi: 10.1142/S0219891607001082.  Google Scholar

[24]

S. Wang and G. Chen, Cauchy problem of the generalized double dispersion equation,, Nonlinear Anal., 64 (2006), 159.  doi: 10.1016/j.na.2005.06.017.  Google Scholar

[25]

S. Wang and F. Da, On the asymptotic behavior of solution for the generalized double dipersion equation,, Appl. Anal., 92 (2013), 1179.  doi: 10.1080/00036811.2012.661044.  Google Scholar

[26]

S. Wang and H. Xu, On the asymptotic behavior of solution for the generalized IBq equation with hydrodynamical damped term,, J. Diff. Equations, 252 (2012), 4243.  doi: 10.1016/j.jde.2011.12.016.  Google Scholar

[27]

Y.-Z. Wang, Global existence and asymptotic behaviour of solutions for the generalized Boussinesq equation,, Nonlinear Anal., 70 (2009), 465.  doi: 10.1016/j.na.2007.12.018.  Google Scholar

[28]

Y.-Z. Wang, F. G. Liu and Y. Z. Zhang, Global existence and asymptotic of solutions for a semi-linear wave equation,, J. Math. Anal. Appl., 385 (2012), 836.  doi: 10.1016/j.jmaa.2011.07.010.  Google Scholar

[29]

Y.-Z. Wang and Y.-X. Wang, Global existence and asymptotic behavior of solutions to a nonlinear wave equation of fourth-order,, J. Math. Phys., 53 (2012).  doi: 10.1063/1.3677764.  Google Scholar

[30]

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.  doi: 10.1016/j.na.2009.03.069.  Google Scholar

[31]

S. M. Zheng, Nonlinear Evolution Equations,, Monographs and Surveys in Pure and Applied Mathematics, (2004).  doi: 10.1201/9780203492222.  Google Scholar

show all references

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.  doi: 10.1016/j.jmaa.2004.05.044.  Google Scholar

[2]

M. Kato, Large time behavior of solutions to the generalized Burgers equations,, Osaka J. Math., 44 (2007), 923.   Google Scholar

[3]

S. Kawashima, Large-time behaviour of solutions to hyperbolic-parabolic systems of conservation laws and applications,, Proc. Roy. Soc. Edinburgh, 106 (1987), 169.  doi: 10.1017/S0308210500018308.  Google Scholar

[4]

S. Kawashima, Large-time behavior of solutions of the discrete Boltzmann equation,, Comm. Math. Phys., 109 (1987), 563.  doi: 10.1007/BF01208958.  Google Scholar

[5]

S. Kawashima, Y. Nikkuni and S. Nishibata, The initial value problem for hyperbolic-elliptic coupled systems and application to radiation hydrodynamics,, Analysis of Systems of Conservation Laws, (1999), 87.   Google Scholar

[6]

S. Kawashima and Y.-Z. Wang, Global Existence and Asymptotic Behavior of Solutions to the Generalized Cubic Double Dispersion Equation,, Analysis and Applications (accepted)., ().   Google Scholar

[7]

T. T. Li and Y. M. Chen, Nonlinear Evolution Equations,, Academic Press, (1989).   Google Scholar

[8]

T.-P. Liu, Hyperbolic and Viscous Conservation Laws,, CBMS-NSF Regional Conference Sereies in Applied Math., (2000).  doi: 10.1137/1.9780898719420.  Google Scholar

[9]

T.-P. Liu and Y. Zeng, Large time behavior of solutions to general quasilinear hyperbolic-parabolic systems of conservation laws,, Memoirs Amer. Math. Soc., 125 (1997).  doi: 10.1090/memo/0599.  Google Scholar

[10]

Y. Liu and S. Kawashima, Global existence and asymptotic behavior of solutions for quasi-linear dissipative plate equation,, Discrete Contin. Dyn. Syst., 29 (2011), 1113.  doi: 10.3934/dcds.2011.29.1113.  Google Scholar

[11]

Y. Liu and S. Kawashima, Global existence and decay of solutions for a quasi-linear dissipative plate equation,, J. Hyperbolic Differential Equations, 8 (2011), 591.  doi: 10.1142/S0219891611002500.  Google Scholar

[12]

Y. Liu and S. Kawashima, Decay property for a plate equation with memory-type dissipation,, Kinetic and Related Models, 4 (2011), 531.  doi: 10.3934/krm.2011.4.531.  Google Scholar

[13]

A. Matsumura, On the asymptotic behaviour of solutions of semi-linear wave equations,, Publ. Res. Inst. Math. Sci., 12 (1976), 169.  doi: 10.2977/prims/1195190962.  Google Scholar

[14]

M. Nakao and K. Ono, Existence of global solutions to the Cauchy problem for the semilinear dissipative wave equations,, Math. Z., 214 (1993), 325.  doi: 10.1007/BF02572407.  Google Scholar

[15]

K. Nishihara, $L^p$-$L^q$ estimates of solutions to the damped wave equation in 3-dimensional space and their applications,, Math. Z., 244 (2003), 631.  doi: 10.1007/s00209-003-0516-0.  Google Scholar

[16]

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.  doi: 10.1016/j.jmaa.2008.08.025.  Google Scholar

[17]

A. M. Samsonov, Nonlinear strain waves in elastic waveguides,, in Nonlinear Waves in Solids(Udine, 341 (1994), 349.   Google Scholar

[18]

A. M. Samsonov and E. V. Sokurinskaya, Energy exchange between nonlinear waves in elastic waveguides and external media,, in Nonlinear Waves in Active Media, (1989), 99.   Google Scholar

[19]

Y. Sugitani and S. Kawashima, Decay estimates of solution to a semi-linear dissipative plate equation,, J. Hyperbolic Differential Equations, 7 (2010), 471.  doi: 10.1142/S0219891610002207.  Google Scholar

[20]

H. Takeda and S. Yoshikawa, On the initial value problem of the semilinear beam equation with weak damping I: Smoothing effect,, J. Math. Anal. Appl., 401 (2013), 244.  doi: 10.1016/j.jmaa.2012.12.015.  Google Scholar

[21]

H. Takeda and S. Yoshikawa, On the initial value problem of the semilinear beam equation with weak damping II: Asymptotic profles,, J. Differential Equations, 253 (2012), 3061.  doi: 10.1016/j.jde.2012.07.014.  Google Scholar

[22]

G. Todorova and B. Yordanov, Critical exponent for a nonlinear wave equation with damping,, J. Differential Equations, 174 (2001), 464.  doi: 10.1006/jdeq.2000.3933.  Google Scholar

[23]

Y. Ueda and S. Kawashima, Large time behavior of solutions to a semilinear hyperbolic system with relaxation,, J. Hyperbolic Differential Equations, 4 (2007), 147.  doi: 10.1142/S0219891607001082.  Google Scholar

[24]

S. Wang and G. Chen, Cauchy problem of the generalized double dispersion equation,, Nonlinear Anal., 64 (2006), 159.  doi: 10.1016/j.na.2005.06.017.  Google Scholar

[25]

S. Wang and F. Da, On the asymptotic behavior of solution for the generalized double dipersion equation,, Appl. Anal., 92 (2013), 1179.  doi: 10.1080/00036811.2012.661044.  Google Scholar

[26]

S. Wang and H. Xu, On the asymptotic behavior of solution for the generalized IBq equation with hydrodynamical damped term,, J. Diff. Equations, 252 (2012), 4243.  doi: 10.1016/j.jde.2011.12.016.  Google Scholar

[27]

Y.-Z. Wang, Global existence and asymptotic behaviour of solutions for the generalized Boussinesq equation,, Nonlinear Anal., 70 (2009), 465.  doi: 10.1016/j.na.2007.12.018.  Google Scholar

[28]

Y.-Z. Wang, F. G. Liu and Y. Z. Zhang, Global existence and asymptotic of solutions for a semi-linear wave equation,, J. Math. Anal. Appl., 385 (2012), 836.  doi: 10.1016/j.jmaa.2011.07.010.  Google Scholar

[29]

Y.-Z. Wang and Y.-X. Wang, Global existence and asymptotic behavior of solutions to a nonlinear wave equation of fourth-order,, J. Math. Phys., 53 (2012).  doi: 10.1063/1.3677764.  Google Scholar

[30]

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.  doi: 10.1016/j.na.2009.03.069.  Google Scholar

[31]

S. M. Zheng, Nonlinear Evolution Equations,, Monographs and Surveys in Pure and Applied Mathematics, (2004).  doi: 10.1201/9780203492222.  Google Scholar

[1]

Manil T. Mohan, Arbaz Khan. On the generalized Burgers-Huxley equation: Existence, uniqueness, regularity, global attractors and numerical studies. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020270

[2]

Weijiu Liu. Asymptotic behavior of solutions of time-delayed Burgers' equation. Discrete & Continuous Dynamical Systems - B, 2002, 2 (1) : 47-56. doi: 10.3934/dcdsb.2002.2.47

[3]

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

[4]

Carmen Cortázar, Manuel Elgueta, Fernando Quirós, Noemí Wolanski. Asymptotic behavior for a nonlocal diffusion equation on the half line. Discrete & Continuous Dynamical Systems - A, 2015, 35 (4) : 1391-1407. doi: 10.3934/dcds.2015.35.1391

[5]

Yongqin Liu, Shuichi Kawashima. Global existence and asymptotic behavior of solutions for quasi-linear dissipative plate equation. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 1113-1139. doi: 10.3934/dcds.2011.29.1113

[6]

Jean-Claude Saut, Jun-Ichi Segata. Asymptotic behavior in time of solution to the nonlinear Schrödinger equation with higher order anisotropic dispersion. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 219-239. doi: 10.3934/dcds.2019009

[7]

Ezzeddine Zahrouni. On the Lyapunov functions for the solutions of the generalized Burgers equation. Communications on Pure & Applied Analysis, 2003, 2 (3) : 391-410. doi: 10.3934/cpaa.2003.2.391

[8]

Jean-François Rault. A bifurcation for a generalized Burgers' equation in dimension one. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 683-706. doi: 10.3934/dcdss.2012.5.683

[9]

Engu Satynarayana, Manas R. Sahoo, Manasa M. Higher order asymptotic for Burgers equation and Adhesion model. Communications on Pure & Applied Analysis, 2017, 16 (1) : 253-272. doi: 10.3934/cpaa.2017012

[10]

Genni Fragnelli, A. Idrissi, L. Maniar. The asymptotic behavior of a population equation with diffusion and delayed birth process. Discrete & Continuous Dynamical Systems - B, 2007, 7 (4) : 735-754. doi: 10.3934/dcdsb.2007.7.735

[11]

Kin Ming Hui, Soojung Kim. Asymptotic large time behavior of singular solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5943-5977. doi: 10.3934/dcds.2017258

[12]

Bhargav Kumar Kakumani, Suman Kumar Tumuluri. Asymptotic behavior of the solution of a diffusion equation with nonlocal boundary conditions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 407-419. doi: 10.3934/dcdsb.2017019

[13]

Guenbo Hwang, Byungsoo Moon. Global existence and propagation speed for a Degasperis-Procesi equation with both dissipation and dispersion. Electronic Research Archive, 2020, 28 (1) : 15-25. doi: 10.3934/era.2020002

[14]

Oleg Yu. Imanuvilov, Jean Pierre Puel. On global controllability of 2-D Burgers equation. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 299-313. doi: 10.3934/dcds.2009.23.299

[15]

Long Wei, Zhijun Qiao, Yang Wang, Shouming Zhou. Conserved quantities, global existence and blow-up for a generalized CH equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1733-1748. doi: 10.3934/dcds.2017072

[16]

Rui Huang, Ming Mei, Yong Wang. Planar traveling waves for nonlocal dispersion equation with monostable nonlinearity. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3621-3649. doi: 10.3934/dcds.2012.32.3621

[17]

Marco Di Francesco, Alexander Lorz, Peter A. Markowich. Chemotaxis-fluid coupled model for swimming bacteria with nonlinear diffusion: Global existence and asymptotic behavior. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1437-1453. doi: 10.3934/dcds.2010.28.1437

[18]

Yuqian Zhou, Qian Liu. Reduction and bifurcation of traveling waves of the KdV-Burgers-Kuramoto equation. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 2057-2071. doi: 10.3934/dcdsb.2016036

[19]

Thierry Horsin, Peter I. Kogut. Optimal $L^2$-control problem in coefficients for a linear elliptic equation. I. Existence result. Mathematical Control & Related Fields, 2015, 5 (1) : 73-96. doi: 10.3934/mcrf.2015.5.73

[20]

Jerry L. Bona, Laihan Luo. Large-time asymptotics of the generalized Benjamin-Ono-Burgers equation. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 15-50. doi: 10.3934/dcdss.2011.4.15

2019 Impact Factor: 1.311

Metrics

  • PDF downloads (35)
  • HTML views (0)
  • Cited by (21)

Other articles
by authors

[Back to Top]