Morawetz estimates and stabilization for damped Klein-Gordon equation with small data

TL;DR

This paper proves that small initial data solutions to the damped Klein-Gordon equation in three dimensions decay exponentially, using Morawetz estimates and unique continuation principles.

Contribution

It introduces new decay results for the damped Klein-Gordon equation leveraging Morawetz estimates and unique continuation, extending understanding of stabilization mechanisms.

Findings

Solutions decay exponentially with small initial data

Morawetz estimates are effective for damping analysis

Unique continuation principles support decay proofs

Abstract

In the present paper, we show that the global solution to (partially) damped Klein-Gordon equation on the three dimensional Euclidean space with small data decays exponentially. The key ingredients in the proof are: Morawetz-type estimates for solutions with small data and Ruiz's unique continuation principle for wave equations.