Asymptotic estimates of solution to damped fractional wave equation

It is known that the damped fractional wave equation has the diffusive structure as $t\to \mathrm{\infty }$ $t\rightarrow \infty $ . Let $u(t,x)=e^{-t}cosh(t\sqrt{L})f(x)+e^{-t}\frac{sinh(t\sqrt{L})}{\sqrt{L}}(f(x)+g(x))$ $u(t,x)=e^{-t}\cosh (t\sqrt{L})f(x)+e^{-t} \frac{\sinh (t\sqrt{L})}{\sqrt{L}}(f(x)+g(x))$ be the solution of the Cauchy problem for the damped fractional wave equation, where $\sqrt{L}$ $\sqrt{L}$ involves the fractional Laplacian ${(-\mathrm{△})}^{\alpha }$ $(-\triangle )^{\alpha}$ on the space variable. We can study the decay estimate of the solution $u(t,x)$ $u(t,x)$ over the time t by means of the Cauchy problem for the parabolic equation. In this paper, we consider, for $0<\alpha <1$ $0<\alpha <1$ , the Cauchy problem in the two- and three-dimensional spaces for the damped fractional wave equation and the corresponding parabolic equation and obtain the Triebel–Lizorkin space estimate of the difference of solutions. At the same time, we also consider, for $\alpha =1$ $\alpha =1$ , the case of the Cauchy problem in the four-dimensional space and obtain a Triebel–Lizorkin space estimate.