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: |
[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.![]() ![]() ![]() |