Long-time existence of solutions to nonlocal nonlinear bidirectional wave equations
Type :
Article
Publication Status :
Published
Access :
restrictedAccess
Abstract
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.
Source :
Discrete and Continuous Dynamical Systems- Series A
Date :
2019-05
Volume :
39
Issue :
5
Publisher :
American Institute of Mathematical Sciences
Collections
Share this page