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

Distortion-Memory Tradeoffs in Cache-Aided Wireless Video Delivery

Mobile network operators are considering caching as one of the strategies to keep up with the increasing demand for high-definition wireless video streaming. By prefetching popular content into memory at wireless access points or end user devices, requests can be served locally, relieving strain on expensive backhaul. In addition, using network coding allows the simultaneous serving of distinct cache misses via common coded multicast transmissions, resulting in significantly larger load reductions compared to those achieved with conventional delivery schemes. However, prior work does not exploit the properties of video and simply treats content as fixed-size files that users would like to fully download. Our work is motivated by the fact that video can be coded in a scalable fashion and that the decoded video quality depends on the number of layers a user is able to receive. Using a Gaussian source model, caching and coded delivery methods are designed to minimize the squared error distortion at end user devices. Our work is general enough to consider heterogeneous cache sizes and video popularity distributions.Comment: To appear in Allerton 2015 Proceedings of the 53rd annual Allerton conference on Communication, control, and computin

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

Instability and stability properties of traveling waves for the double dispersion equation

In this article we are concerned with the instability and stability properties of traveling wave solutions of the double dispersion equation $~u_{tt} -u_{xx}+a u_{xxxx}-bu_{xxtt} = - (|u|^{p-1}u)_{xx}~$ for $~p>1$, $~a\geq b>0$. The main characteristic of this equation is the existence of two sources of dispersion, characterized by the terms $u_{xxxx}$ and $u_{xxtt}$. We obtain an explicit condition in terms of $a$, $b$ and $p$ on wave velocities ensuring that traveling wave solutions of the double dispersion equation are strongly unstable by blow up. In the special case of the Boussinesq equation ($b=0$), our condition reduces to the one given in the literature. For the double dispersion equation, we also investigate orbital stability of traveling waves by considering the convexity of a scalar function. We provide both analytical and numerical results on the variation of the stability region of wave velocities with $a$, $b$ and $p$ and then state explicitly the conditions under which the traveling waves are orbitally stable.Comment: 16 pages, 4 figure

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 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

Analysis of simple inventory control systems with execution errors: Economic impact under correction opportunities

Cataloged from PDF version of article.Motivated by recent empirical evidence, we study the economic impact of inventory record inaccuracies that arise due to execution errors. We model a set of probable events regarding the erroneous registering of sales at each demand arrival. We define correction opportunities that can be used to (at least partially) correct inventory records. We analyze a simple inventory control model with execution errors and correction opportunities, and demonstrate that decisions that consider the existence of recording errors and the mechanisms with which they are corrected can be quite complicated and exhibit complex tradeoffs. To evaluate the economic impact of inventory record inaccuracies, we use a simulation model of a (Q,r) inventory control system and evaluate suboptimalities in cost and customer service that arise as a result of untimely triggering of orders due to inventory record inaccuracies. We show that the economic impact of inventory record inaccuracies can be significant, particularly in systems with small order sizes and low reorder levels. (C) 2010 Elsevier BM. All rights reserved