Global dynamics around 2-solitons for the nonlinear damped Klein-Gordon equations

TL;DR

This paper classifies the global behavior of solutions near 2-solitons in the damped nonlinear Klein-Gordon equation, identifying five types of long-term dynamics based on initial data proximity to ground states.

Contribution

It provides a complete classification of solution behaviors around 2-solitons, including the structure of manifolds leading to convergence, blow-up, or divergence, with new techniques for controlling unstable modes.

Findings

Solutions near 2-solitons are classified into five global behavior types.

Manifolds of solutions converging to ground states are characterized with codimension.

The interaction between unstable modes is shown to be uniformly integrable due to symmetry.

Abstract

Global behavior of solutions is studied for the nonlinear Klein-Gordon equation with a focusing power nonlinearity and a damping term in the energy space on the Euclidean space. We give a complete classification of solutions into 5 types of global behavior for all initial data in a small neighborhood of each superposition of two ground states (2-solitons) with the opposite signs and sufficient spatial distance. The neighborhood contains, for each sign of the ground state, the manifold with codimension one in the energy space, consisting of solutions that converge to the ground state at time infinity. The two manifolds are joined at their boundary by the manifold with codimension two of solutions that are asymptotic to 2-solitons moving away from each other. The connected union of these three manifolds separates the rest of the neighborhood into the open set of global decaying solutions and that of blow-up. The main ingredient in the proof is a difference estimate on two solutions starting near 2-solitons and asymptotic to 1-solitons. The main difficulty is in controlling the direction of the two unstable modes attached to 2-solitons, while the soliton interactions are not uniformly integrable in time. It is resolved by showing that the non-scalar part of the interaction between the unstable modes is uniformly integrable due to the symmetry of the equation and the 2-solitons.