Asymptotics of solutions with a compactness property for the nonlinear damped Klein–Gordon equation. (May 2022) - Côte, Raphaël; Yuan, Xu
Abstract: We consider the nonlinear damped Klein–Gordon equation ∂ t t u + 2 α ∂ t u − Δ u + u − | u | p − 1 u = 0 on [ 0, ∞ ) × R N with α > 0, 2 ⩽ N ⩽ 5 and energy subcritical exponents p > 2 . We study the behavior of solutions for which it is supposed that only one nonlinear object appears asymptotically for large times, at least for a sequence of times. We first prove that the nonlinear object is necessarily a bound state. Next, we show that when the nonlinear object is a non-degenerate state or a degenerate excited state satisfying a simplicity condition, the convergence holds for all positive times, with an exponential or algebraic rate respectively. Last, we provide an example where the solution converges exactly at the rate t − 1 to the excited state.