Optimal decay for plates with rotational inertia and memory

Abstract

We study the asymptotic behavior of a linear plate equation with effects of rotational inertia and a fractional damping in the memory term: utt−γΔutt+βnormalΔ2u−∫0∞gfalse(sfalse)normalΔ2θufalse(t−sfalse)ds=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/false(4−4θfalse) and also that the decay rate is optimal. Furthermore, when θ=1, we obtain the exponential decay of the solutions.