Thresholds for global existence and blow-up in a general class of doubly dispersive nonlocal wave equations

In this article we study global existence and blow-up of solutions for a general class of nonlocal nonlinear wave equations with power-type nonlinearities, $u_{tt}-Lu_{xx}=B(- |u|^{p-1}u)_{xx}, ~(p>1)$, where the nonlocality enters through two pseudo-differential operators $L$ and $B$. We establish thresholds for global existence versus blow-up using the potential well method which relies essentially on the ideas suggested by Payne and Sattinger. Our results improve the global existence and blow-up results given in the literature for the present class of nonlocal nonlinear wave equations and cover those given for many well-known nonlinear dispersive wave equations such as the so-called double-dispersion equation and the traditional Boussinesq-type equations, as special cases.Comment: 17 pages. Accepted for publication in Nonlinear Analysis:Theory, Methods & Application

A higher-order Boussinesq equation in locally non-linear theory of one-dimensional non-local elasticity

In one space dimension, a non-local elastic model is based on a single integral law, giving the stress when the strain is known at all spatial points. In this study, we first derive a higher-order Boussinesq equation using locally non-linear theory of 1D non-local elasticity and then we are able to show that under certain conditions the Cauchy problem is globally well-posed

Global existence and blow-up for a class of nonlocal nonlinear Cauchy problems arising in elasticity

We study the initial-value problem for a general class of nonlinear nonlocal wave equations arising in one-dimensional nonlocal elasticity. The model involves a convolution integral operator with a general kernel function whose Fourier transform is nonnegative. We show that some well-known examples of nonlinear wave equations, such as Boussinesq-type equations, follow from the present model for suitable choices of the kernel function. We establish global existence of solutions of the model assuming enough smoothness on the initial data together with some positivity conditions on the nonlinear term. Furthermore, conditions for finite time blow-up are provided

Global existence and blow-up of solutions for a general class of doubly dispersive nonlocal nonlinear wave equations

This study deals with the analysis of the Cauchy problem of a general class of nonlocal nonlinear equations modeling the bi-directional propagation of dispersive waves in various contexts. The nonlocal nature of the problem is reflected by two different elliptic pseudodifferential operators acting on linear and nonlinear functions of the dependent variable, respectively. The well-known doubly dispersive nonlinear wave equation that incorporates two types of dispersive effects originated from two different dispersion operators falls into the category studied here. The class of nonlocal nonlinear wave equations also covers a variety of well-known wave equations such as various forms of the Boussinesq equation. Local existence of solutions of the Cauchy problem with initial data in suitable Sobolev spaces is proven and the conditions for global existence and finite-time blow-up of solutions are established.Comment: 17 page

Existence and stability of traveling waves for a class of nonlocal nonlinear equations

In this article we are concerned with the existence and orbital stability of traveling wave solutions of a general class of nonlocal wave equations: u_tt−Lu_xx=B(±|u|^(p−1)u)_xx, p>1. The main characteristic of this class of equations is the existence of two sources of dispersion, characterized by two coercive pseudo-differential operators L and B. Members of the class arise as mathematical models for the propagation of dispersive waves in a wide variety of situations. For instance, all Boussinesq-type equations and the so-called double-dispersion equation are members of the class. We first establish the existence of traveling wave solutions to the nonlocal wave equations considered. We then obtain results on the orbital stability or instability of traveling waves. For the case L=I, corresponding to a class of Klein-Gordon-type equations, we give an almost complete characterization of the values of the wave velocity for which the traveling waves are orbitally stable or unstable by blow-up

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

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

Cauchy problem for a higher-order boussinesq equation

In this thesis, we establish global well-posedness of the Cauchy problem for a particular higher-order Boussinesq equation. At the microscopic level this sixth order Boussinesq equation was derived in [11] for the longitudinal vibrations of a dense lattice, in which a unit length of the lattice contains a large number of lattice points. We take the initial data in the Sobolev space Hs with s > 1 2 . With smoothness assumptions on the nonlinear term, we establish local existence and uniqueness of the solution. Under further assumptions, we prove the global existence for s 1. Finally, we show continuous dependence of the solution on the initial data

Cauchy problem for a higher-order Boussinesq equation, Master’s Thesis,

In this study we establish global well-posedness of the Cauchy problem for a higher-order Boussinesq equation