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May  2019, 39(5): 2877-2891. doi: 10.3934/dcds.2019119

Long-time existence of solutions to nonlocal nonlinear bidirectional wave equations

H. A. Erbay 1,, , S. Erbay 1, and  A. Erkip 2,

1. 

Department of Natural and Mathematical Sciences, Faculty of Engineering, Ozyegin University, Cekmekoy 34794, Istanbul, Turkey
2. 

Faculty of Engineering and Natural Sciences, Sabanci University, Tuzla 34956, Istanbul, Turkey

* Corresponding author: H. A. Erbay

Received  July 2018 Published  January 2019

We consider the Cauchy problem defined for a general class of nonlocal wave equations modeling bidirectional wave propagation in a nonlocally and nonlinearly elastic medium whose constitutive equation is given by a convolution integral. We prove a long-time existence result for the nonlocal wave equations with a power-type nonlinearity and a small parameter. As the energy estimates involve a loss of derivatives, we follow the Nash-Moser approach proposed by Alvarez-Samaniego and Lannes. As an application to the long-time existence theorem, we consider the limiting case in which the kernel function is the Dirac measure and the nonlocal equation reduces to the governing equation of one-dimensional classical elasticity theory. The present study also extends our earlier result concerning local well-posedness for smooth kernels to nonsmooth kernels.

Keywords: Long-time existence, nonlocal wave equation, Nash-Moser iteration, improved Boussinesq equation.
Mathematics Subject Classification: Primary: 35A01, 35L15; Secondary: 35L70, 35Q74, 74B20.
Citation: H. A. Erbay, S. Erbay, A. Erkip. Long-time existence of solutions to nonlocal nonlinear bidirectional wave equations. Discrete and Continuous Dynamical Systems, 2019, 39 (5) : 2877-2891. doi: 10.3934/dcds.2019119
References:
[1]

B. Alvarez-Samaniego and D. Lannes, A Nash-Moser theorem for singular evolution equations. Application to the Serre and Green-Naghdi equations, Indiana Univ. Math. Journal, 57 (2008), 97-131.  doi: 10.1512/iumj.2008.57.3200.

[2]

A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Rational Mech. Anal., 192 (2009), 165-186.  doi: 10.1007/s00205-008-0128-2.

[3]

N. Duruk, H. A. Erbay and A. Erkip, Global existence and blow-up for a class of nonlocal nonlinear Cauchy problems arising in elasticity, Nonlinearity, 23 (2010), 107-118.  doi: 10.1088/0951-7715/23/1/006.

[4]

M. Ehrnstrom, L. Pei and Y. Wang, A conditional well-posedness result for the bidirectional Whitham equation, preprint, arXiv: 1708.04551 [math.AP].

[5]

H. A. Erbay, S. Erbay and A. Erkip, The Camassa-Holm equation as the long-wave limit of the improved Boussinesq equation and of a class of nonlocal wave equations, Discrete Contin. Dyn. Syst., 36 (2016), 6101-6116.  doi: 10.3934/dcds.2016066.

[6]

T. J. R. Hughes, T. Kato and J. E. Marsden, Well-posed quasi-linear second-order hyperbolic systems with applications to nonlinear elastodynamics and general relativity, Arch. Ration. Mech. Anal., 63 (1977), 273-294.  doi: 10.1007/BF00251584.

[7]

S. Klainerman, Long time behaviour of solutions to nonlinear wave equations, in Proceedings of the International Congress of Mathematicians (Warsaw, 1983), PWN, Warsaw, (1984), 1209–1215.

[8]

M. Ming, J. C. Saut and P. Zhang, Long-time existence of solutions to Boussinesq systems, SIAM J. Math. Anal., 44 (2012), 4078-4100.  doi: 10.1137/110834214.

[9]

J. C. Saut and L. Xu, The Cauchy problem on large time for surface waves Boussinesq systems, J. Math. Pures. Appl., 97 (2012), 635-662.  doi: 10.1016/j.matpur.2011.09.012.

[10]

J. C. Saut, C. Wang and L. Xu, The Cauchy problem on large time for surface-waves-type Boussinesq systems Ⅱ, SIAM J. Math. Anal., 49 (2017), 2321-2386.  doi: 10.1137/15M1050203.

[11]

M. E. Taylor, Partial Differential Equations II. Qualitative Studies of Linear Equations, 2$^{nd}$ edition, Springer, New York, 2011. doi: 10.1007/978-1-4419-7052-7.

show all references

References:
[1]

B. Alvarez-Samaniego and D. Lannes, A Nash-Moser theorem for singular evolution equations. Application to the Serre and Green-Naghdi equations, Indiana Univ. Math. Journal, 57 (2008), 97-131.  doi: 10.1512/iumj.2008.57.3200.

[2]

A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Rational Mech. Anal., 192 (2009), 165-186.  doi: 10.1007/s00205-008-0128-2.

[3]

N. Duruk, H. A. Erbay and A. Erkip, Global existence and blow-up for a class of nonlocal nonlinear Cauchy problems arising in elasticity, Nonlinearity, 23 (2010), 107-118.  doi: 10.1088/0951-7715/23/1/006.

[4]

M. Ehrnstrom, L. Pei and Y. Wang, A conditional well-posedness result for the bidirectional Whitham equation, preprint, arXiv: 1708.04551 [math.AP].

[5]

H. A. Erbay, S. Erbay and A. Erkip, The Camassa-Holm equation as the long-wave limit of the improved Boussinesq equation and of a class of nonlocal wave equations, Discrete Contin. Dyn. Syst., 36 (2016), 6101-6116.  doi: 10.3934/dcds.2016066.

[6]

T. J. R. Hughes, T. Kato and J. E. Marsden, Well-posed quasi-linear second-order hyperbolic systems with applications to nonlinear elastodynamics and general relativity, Arch. Ration. Mech. Anal., 63 (1977), 273-294.  doi: 10.1007/BF00251584.

[7]

S. Klainerman, Long time behaviour of solutions to nonlinear wave equations, in Proceedings of the International Congress of Mathematicians (Warsaw, 1983), PWN, Warsaw, (1984), 1209–1215.

[8]

M. Ming, J. C. Saut and P. Zhang, Long-time existence of solutions to Boussinesq systems, SIAM J. Math. Anal., 44 (2012), 4078-4100.  doi: 10.1137/110834214.

[9]

J. C. Saut and L. Xu, The Cauchy problem on large time for surface waves Boussinesq systems, J. Math. Pures. Appl., 97 (2012), 635-662.  doi: 10.1016/j.matpur.2011.09.012.

[10]

J. C. Saut, C. Wang and L. Xu, The Cauchy problem on large time for surface-waves-type Boussinesq systems Ⅱ, SIAM J. Math. Anal., 49 (2017), 2321-2386.  doi: 10.1137/15M1050203.

[11]

M. E. Taylor, Partial Differential Equations II. Qualitative Studies of Linear Equations, 2$^{nd}$ edition, Springer, New York, 2011. doi: 10.1007/978-1-4419-7052-7.

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