# Decay of the local energy for the solutions of the critical Klein-Gordon equation

## TL;DR

This paper proves that solutions to the critical Klein-Gordon equation with localized nonlinearity exhibit exponential decay of local energy, using advanced mathematical tools like Strichartz estimates and Lax-Phillips semigroup.

## Contribution

It introduces a novel proof of exponential decay for the critical Klein-Gordon equation leveraging generalized Strichartz estimates and Lax-Phillips semigroup techniques.

## Key findings

- Exponential decay of local energy is established for the critical Klein-Gordon equation.
- The proof utilizes generalized Strichartz estimates and Lax-Phillips semigroup.
- Results contribute to understanding energy dissipation in nonlinear wave equations.

## Abstract

In this paper, we prove the exponential decay of local energy for the Klein-Gordon equation with localized critical nonlinearity. The proof relies on generalized Strichartz estimates, and semi-group of Lax-Phillips.