A WAVE AND BEAM EQUATION WITH RIESZ DAMPING

Abstract

We introduce a mathematical model in $\mathbb{R}^{n}$ for linear classical damped wave and beam equations with additional nonlocal (in space) damping mechanism:
\begin{equation*}
u_{tt}(t,x)+(-\Delta)^{\sigma}u(t,x)+u_{t}(t,x)+(-\Delta)^{-\delta}u_{t}(t,x)=0,
\end{equation*}
$$u(0,x)=u_{0}(x), \ \ u_{t}(0,x)=u_{1}(x),$$
where $t\geq 0$, $x\in \mathbb{R}^{n}$, $\sigma\geq 1$, $2\delta \in \left(0,n\right)$. Our main tool is the Fourier analysis and the main goal is to derive the $(L^1\cap L^2)-L^2$ decay estimates for the solution $u$ as follows

$$
\left\|u(t,\cdot)\right\|_{L^{2}} \lesssim(1+t)^{-\frac{n}{4(\sigma+\delta)}}\|u_{0}\|_{L^{1}\cap L^{2}}+(1+t)^{-\frac{n}{4(\sigma+\delta)}-\frac{\delta}{\sigma+\delta}}\|u_{1}\|_{L^{1}\cap L^{2}}.
$$
This energy estimates shows the fact that the decay rate of the solution is influenced by the vanishing property of the initial displacement $u_{0}$ as opposed to the well-known classical damped wave and beam models, in which the decay rate is always the one corresponding to $u_{1}$ even if $u_{0}\neq0$. More precisely, if $u_{0}=0$ and $n<4\sigma$, then the inverse structural damping or Riesz damping $(-\Delta)^{-\delta}u_{t}$ will enhance the damping effect in the low-frequency zone compared to the frictional damping $u_{t}$ without imposing any higher regularity on the data.

As a result, we will introduce a new classification of the damping mechanism $(-\Delta)^{\gamma}u_{t}$ depending on the fractional power $\gamma\in \mathbb{R}$, which clarifies the comparison between decay rate corresponding to the initial displacement $u_{0}$ and the initial velocity $u_{1}$.

Finally, the regularity-loss class models include classical wave and beam equations with only Riesz damping.