Decay properties and asymptotic behaviors for a wave equation with general strong damping

In this paper, we study the Cauchy problem for a wave equation with general strong damping $-\mu(|D|)\Delta u_t$ motivated by [Tao, Anal. PDE (2009)] and [Ebert-Girardi-Reissig, Math. Ann. (2020)]. By employing energy methods in the Fourier space and WKB analysis, we derive decay estimates for solutions under a large class of $\mu(|D|)$. In particularly, a threshold $\lim\nolimits_{|\xi|\to\infty}\mu(|\xi|)=\infty$ is discovered for the regularity-loss phenomenon, where $\mu(|\xi|)$ denotes the symbol of $\mu(|D|)$. Furthermore, we investigate different asymptotic profiles of solution with additionally $L^1$ initial data, where some refined estimates in the sense of enhanced decay rate and reduced regularity are found. The derived results almost cover the known results with sufficiently small loss.