Initial-boundary value problems for nonlinear wave equations.

摘要

In this dissertation, we consider two initial-boundary value problems for nonlinear wave equations. First, we consider an initial-boundary value problem for a wave equation in high space dimensions with a nonlinear damping term that is not Lipschitz in u_t. We establish the existence and uniqueness of a global weak solution using a Galerkin approximation scheme, utilizing a version of Aubin's compactness theorem and the monotonicity property of the nonlinearity to prove the convergence of the approximate solutions. We also consider an initial-boundary value problem for a wave equation in one space dimension with nonlinear damping and singular source terms. We establish the existence of local weak solutions using a standard fixed-point-theorem method. Moreover, the behavior of solutions is investigated. Under mild conditions, we prove several results concerning the quenching and non-quenching of solutions.