Cauchy problems for a class of nonlocal nonlinear bi-directional wave equations

Official URL: http://192.168.1.20/record=b1378234 (Table of Contents)

In this thesis study, two nonlocal models governing nonlinear wave motions in a continuous medium are proposed and the Cauchy problems corresponding to these nonlocal nonlinear wave equations are considered. Both of the models involve convolution integral operators with general kernel functions whose Fourier transforms are nonnegative. One of the models is based on a single equation governing the longitu­ dinal wave propagation, whereas the other model is based on two coupled equations governing the propagation of transverse waves. Some well-known examples of nonlinear wave equations, such as Boussinesq-type equations, follow from the proposed models for suitable choices of the kernel functions. The main aim of this thesis is to discuss well-posedness of the Cauchy problems . For this purpose, global existence of solutions of the models assuming enough smoothness ou the initial data together with some posit ivity conditions on the nonlinear term are established. Furthermore , sufficient conditions for finite time blow-up are provided.