Asymptotics of solutions with a compactness property for the nonlinear damped Klein-Gordon equation

We consider the nonlinear damped Klein-Gordon equation partial derivative(tt)u + 2 alpha partial derivative(t)u - Delta u + u - vertical bar u vertical bar(p-1)u = 0 on (0, infinity) x R-N with alpha > 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. (C) 2022 Elsevier Ltd. All rights reserved.

Automated summary

This paper considers the behavior of solutions to the nonlinear damped Klein-Gordon equation. It is shown that when only one nonlinear object appears asymptotically for large times, the nonlinear object necessarily takes the form of a bound state. Convergence of solutions is established for non-degenerate states or degenerate excited states satisfying a simplicity condition, with exponential or algebraic rates respectively. An example is provided where the solution converges exactly at a rate of t(-1) to the excited state.