Long-time asymptotics of 3-solitary waves for the damped nonlinear Klein-Gordon equation

We consider the damped nonlinear Klein-Gordon equation: {align*} ∂ₜ^2u-Δ u+2α ∂ₜu+u-|u|ᵖ⁻¹u=0, \ & (t,x) ∈ R × R^d, {align*} where α>0, 1≤ d≤ 5 and energy sub-critical exponents $p>2$. In this paper, we prove that 3-solitary waves behave as if the three solitons are on a line. Furthermore, the solitary waves have alternative signs and their distances are of order logt.