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Global regularity for nonlinear wave equations
Abstract
We consider two problems, the first of which is a nonlinear wave equation on the two-dimensional sphere with a blowing-up nonlinearity. The existence and uniqueness of a local regular solution to the initial value problem is established using a contraction mapping argument. Also, the behavior of solutions is examined. We show that a large class of solutions to the initial value problem quench in finite time. Second, we consider an initial-boundary value problem for a nonlinear wave equation in one space dimension. Here, the nonlinearity features the damping term :u: m−1ut and a competing source term of the form :u:p −1u, with m, p > 1. We show, using a compactness argument, that whenever m ≥ p, then local weak solutions are global. On the other hand, we prove that whenever p > m and the initial energy is negative, then local weak solutions cannot be global, regardless of the size of the initial data.
Subject Area
Mathematics
Recommended Citation
Strei, Theresa Anne, "Global regularity for nonlinear wave equations" (2000). ETD collection for University of Nebraska-Lincoln. AAI9992009.
https://digitalcommons.unl.edu/dissertations/AAI9992009