Optimal decay for plates with rotational inertia and memory

We study the asymptotic behavior of a linear plate equation with effects of rotational inertia and a fractional damping in the memory term:u t t − γ Δ u t t + β Δ 2 u − ∫ 0 ∞ g ( s ) Δ 2 θ u ( t − s )d s = 0 , where θ ≤ 1 and the kernel g is exponentially decreasing. The main result of this work is the polynomial decay of their solutions when θ < 1 . We prove that the solutions decay with the rate t − 1 / ( 4 − 4 θ )and also that the decay rate is optimal. Furthermore, when θ = 1 , we obtain the exponential decay of the solutions.