Global Stability Dynamics of the Quasilinear Damped Klein–Gordon Equation with Variable Coefficients.

In this paper, we study global stability dynamics for quasilinear damped Klein–Gordon equation with variable coefficients in R n for dimension n ≥ 1 , without assuming that the quasilinear term satisfies the null condition. Due to the influence of damping term, we are able to prove that if it admits a global smooth solution (including time quasi-periodic solution), then this quasilinear system must be asymptotic stable in Sobolev space under some assumptions on the variable coefficients of it. This result also implies that a smooth large solution can be constructed for such quasilinear system, which includes time quasi-periodic stable dynamics of Klein–Gordon equation. We note the global stability for a class of semilinear wave equation satisfying null condition has been shown by Zha (J Funct Anal 282:109219, 2022). [ABSTRACT FROM AUTHOR]