Long-time asymptotics of the damped nonlinear Klein-Gordon equation with a delta potential

We consider the damped nonlinear Klein-Gordon equation with a delta potential {align*} ∂ₜ^2u-∂ₓ^2u+2α ∂ₜu+u-γ {δ}_0u-|u|ᵖ⁻¹u=0, \ & (t,x) ∈ R × R, {align*} where $p>2$, α>0,\ γ<2, and δ₀=δ₀ (x) denotes the Dirac delta with the mass at the origin. When γ=0, C\ᵒte, Martel and Yuan proved that any global solution either converges to 0 or to the sum of K≥ 1 decoupled solitary waves which have alternative signs. In this paper, we first prove that any global solution either converges to 0 or to the sum of K≥ 1 decoupled solitary waves. Next we construct a single solitary wave solution that moves away from the origin when γ<0 and construct an even 2-solitary wave solution when γ≤ -2. Last we give single solitary wave solutions and even 2-solitary wave solutions an upper bound for the distance between the origin and the solitary wave.