Global Dynamics Around 2-Solitons for the Nonlinear Damped Klein-Gordon Equations

• Published in: Annals of PDE Vol. 9; no. 1; p. 2
• Main Authors: Ishizuka, Kenjiro, Nakanishi, Kenji
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.06.2023; Springer Nature B.V
• Subjects: Damping; Decomposition; Eigenvalues; Energy; Euclidean geometry; Ground state; Klein-Gordon equation; Manifolds (mathematics); Mathematical Methods in Physics; Nonlinearity; Partial Differential Equations; Physics; Physics and Astronomy; Solitary waves; Solitons; Secondary 37L25; 35B30; 35B40; Nonlinear Klein-Gordon equation; Invariant manifolds; Blow-up; Primary 35L71
• ISSN: 2524-5317; 2199-2576
• DOI: 10.1007/s40818-022-00128-3

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.