The damped focusing cubic wave equation on a bounded domain
Abstract
For the focusing cubic wave (or Klein–Gordon) equation on a compact three-dimensional Riemannian manifold, the explicit decomposition of the set of initial data with energy below the ground state into blow-up and global-existence regimes has been known since the work of Payne and Sattinger. For the damped equation, we show that this decomposition remains valid. In particular, blow-up solutions cannot be stabilized. Additionally, we prove that any global solution is bounded in the energy space. Assuming that the damping satisfies the Geometric Control Condition, we establish that any global solution converges to a stationary solution along a time sequence, and that global solutions below the ground state energy can be stabilized.
Fields of Study & Subjects
Stability and Controllability of Differential Equations
Advanced Mathematical Physics Problems
Nonlinear Partial Differential Equations
Publication details
- Journal / source
- Mathematical Control and Related Fields
- Volume
- 20
- Issue
- 0
- Pages
- 186–216
- Publication date
- 27 January 2026
- Language
- EN