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Global regularity for nonlinear wave equations
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. ^
Strei, Theresa Anne, "Global regularity for nonlinear wave equations" (2000). ETD collection for University of Nebraska - Lincoln. AAI9992009.