Global Existence and Non-existence Theorems for Nonlinear Wave Equations

Authors

David R. Pitts, University of Nebraska-LincolnFollow
Mohammad A. Rammaha, University of Nebraska-LincolnFollow

Date of this Version

2002

Citation

Indiana University Mathematics Journal copyright , Vol. 51, No. 6 (2002)

Comments

2000 MATHEMATICS SUBJECT CLASSIFICATION: Primary: 35L05, 35L20; secondary: 58G16.

Abstract

In this article we focus on the global well-posedness of an initial-boundary value problem for a nonlinear wave equation in all space dimensions. The nonlinearity in the equation features the damping term |u|^k |u_t|^m sgn(u^t) and a source term of the form |u|^p-1u, where k, p ≥ 1 and 0 < m < 1. In addition, if the space dimension n ≥ 3, then the parameters k, m and p satisfy p, k/(1-m) ≤ n/(n - 2). We show that whenever k + m ≥ p, then local weak solutions are global. On the other hand, we prove that whenever p > k + m and the initial energy is negative, then local weak solutions blow-up in finite time, regardless of the size of the initial data.