Global Dynamics Above The Ground State Energy For The Energy-Critical Klein-Gordon Equation

TL;DR

This paper analyzes the global behavior of solutions to the energy-critical Klein-Gordon equation in dimensions 3 to 5, especially those with energy slightly above the ground state, classifying their long-term dynamics.

Contribution

It provides a classification of solution flows near the ground states for the energy-critical Klein-Gordon equation in multiple dimensions.

Findings

Classified solution behaviors near ground states.

Described dynamics of solutions with energy slightly above ground state.

Identified solution trajectories ejected from neighborhoods of ground states.

Abstract

Consider the focusing energy-critical Klein-Gordon equation in dimension d=3,4,5. We describe the global dynamics of real-valued solutions of which the energy is slightly larger than that of the ground states'. We classify the flows of the solutions that are ejected from a small neighborhood of the ground states or that are away from them.