| Abstract: |
| We study the following Cauchy problem for the linear wave equation with both time-dependent frictional and viscoelastic damping terms:
\begin{equation*}
\begin{cases}
u_{tt}- \Delta u + b(t)u_t - g(t)\Delta u_t=0, &(t,x) \in [0,\infty) \times \mathbb{R}^n, \
u(0,x)= u_0(x),\quad u_t(0,x)= u_1(x), &x \in \mathbb{R}^n.
\end{cases}
\end{equation*}
Our goal is to derive decay estimates for higher-order energy norms of solutions to this problem. We focus on the interplay between the time-dependent coefficients in the frictional damping $b(t)u_t$ and viscoelastic damping $-g(t)\Delta u_t$, and their influence on the qualitative behavior of solutions. The analysis is based on the classification of the damping mechanisms and employs the WKB method in the extended phase space. |
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