American Institute of Mathematical Sciences

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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

Masakazu Kato 1, , Yu-Zhu Wang 2, and  Shuichi Kawashima 3,

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.
Keywords: Burgers equation., Generalized cubic double dispersion equation, diffusion waves, global existence, asymptotic behavior.
Mathematics Subject Classification: Primary: 35L30, 35L76, 35B40; Secondary: 35C0.
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), 563-577. 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), 923-943.  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), 169-194. doi: 10.1017/S0308210500018308.  Google Scholar

[4]

S. Kawashima, Large-time behavior of solutions of the discrete Boltzmann equation, Comm. Math. Phys., 109 (1987), 563-589. 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,(Aachen, 1997), 87-127, Chapman & Hall/CRC Monogr. Surv. Pure Appl. Math., 99, Chapman & Hall/CRC,, Boca Raton, FL, (1999).  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, New York, 1989 (in Chinese).  Google Scholar

[8]

T.-P. Liu, Hyperbolic and Viscous Conservation Laws, CBMS-NSF Regional Conference Sereies in Applied Math., vol. 72, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 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), 120pp. 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), 1113-1139. 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), 591-614. 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), 531-547. 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), 169-189. doi: 10.2977/prims/1195190962.  Google Scholar

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M. Nakao and K. Ono, Existence of global solutions to the Cauchy problem for the semilinear dissipative wave equations, Math. Z., 214 (1993), 325-342. doi: 10.1007/BF02572407.  Google Scholar

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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), 631-649. doi: 10.1007/s00209-003-0516-0.  Google Scholar

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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

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A. M. Samsonov, Nonlinear strain waves in elastic waveguides, in Nonlinear Waves in Solids(Udine, 1993), CISM Courses and Lectures, Springer, Vienna, 341 (1994), 349-382.  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, Springer, Berlin, Heidelberg, (1989), 99-104.  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), 471-501. 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), 244-258. 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), 3061-3080. 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), 464-489. 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), 147-179. doi: 10.1142/S0219891607001082.  Google Scholar

[24]

S. Wang and G. Chen, Cauchy problem of the generalized double dispersion equation, Nonlinear Anal., 64 (2006), 159-173. 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), 1179-1193. 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), 4243-4258. 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), 465-482. 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), 836-853. 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), 013512, 13 pp. 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), 4977-4983. 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, vol. 133, Chapan & Hall/CRC, Boca Raton, FL, 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), 563-577. 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), 923-943.  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), 169-194. doi: 10.1017/S0308210500018308.  Google Scholar

[4]

S. Kawashima, Large-time behavior of solutions of the discrete Boltzmann equation, Comm. Math. Phys., 109 (1987), 563-589. 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,(Aachen, 1997), 87-127, Chapman & Hall/CRC Monogr. Surv. Pure Appl. Math., 99, Chapman & Hall/CRC,, Boca Raton, FL, (1999).  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, New York, 1989 (in Chinese).  Google Scholar

[8]

T.-P. Liu, Hyperbolic and Viscous Conservation Laws, CBMS-NSF Regional Conference Sereies in Applied Math., vol. 72, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 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), 120pp. 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), 1113-1139. 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), 591-614. 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), 531-547. 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), 169-189. 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), 325-342. 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), 631-649. 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), 10-20. 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, 1993), CISM Courses and Lectures, Springer, Vienna, 341 (1994), 349-382.  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, Springer, Berlin, Heidelberg, (1989), 99-104.  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), 471-501. 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), 244-258. 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), 3061-3080. 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), 464-489. 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), 147-179. doi: 10.1142/S0219891607001082.  Google Scholar

[24]

S. Wang and G. Chen, Cauchy problem of the generalized double dispersion equation, Nonlinear Anal., 64 (2006), 159-173. 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), 1179-1193. 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), 4243-4258. 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), 465-482. 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), 836-853. 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), 013512, 13 pp. 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), 4977-4983. 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, vol. 133, Chapan & Hall/CRC, Boca Raton, FL, 2004. doi: 10.1201/9780203492222.  Google Scholar

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