# American Institute of Mathematical Sciences

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

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

 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.

Citation: H. A. Erbay, S. Erbay, A. Erkip. Long-time existence of solutions to nonlocal nonlinear bidirectional wave equations. Discrete & Continuous Dynamical Systems - A, 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.

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##### 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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