Sharp time decay rates on a hyperbolic plate model under effects of an intermediate damping with a time-dependent coefficient

Abstract

In this work we study decay rates for a hyperbolic plate equationunder effects of an intermediate damping term represented by theaction of a fractional Laplacian operator and a time-dependentcoefficient. We obtain decay rates with very general conditions onthe time-dependent coefficient (Theorem 2.1,Section 2), for the power fractional exponent of theLaplace operator $(-\Delta)^\theta$, in the damping term, $\theta \in[0,1]$. For the special time-dependent coefficient $b(t)=\mu(1+t)^{\alpha}$, $\alpha \in (0,1]$, we get optimal decay rates(Theorem 3.1, Section 3).