2021/02/22 by Raphaël Côte, Xu Yuan, Côte, Raphaël +1
Engineering · Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Photonic Systems #Stability and Controllability of Differential Equations
paper · doi:10.48550/arxiv.2102.11178
openalex publication_date 2021/02/22 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
We consider the nonlinear damped Klein-Gordon equation ∂ttu+2α∂tu-Δu+u-|u|p-1u=0 on [0,∞)× ℝN 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.