On exponential decay rate of semigroup associated with second order linear differential equation in Hilbert space with strong damping operator

TL;DR

This paper derives estimates for the exponential decay rate of semigroups generated by second order linear differential equations in Hilbert spaces, considering operators with specific properties, and analyzes the spectrum of the associated operator pencil.

Contribution

It provides new bounds on the decay rate and spectral location for semigroups linked to second order differential equations with strong damping in Hilbert spaces.

Findings

Established decay rate estimates for the semigroup.

Determined the spectrum location of the associated operator pencil.

Extended understanding of damping effects in infinite-dimensional systems.

Abstract

We obtain estimate of the exponential decay rate of semigroup associated with second order linear differential equation $u"+Du'+Au=0$ in Hilbert space. We assume that $A$ is a selfadjoint positive definite operator, $D$ is an accretive sectorial operator and $\Ree D\geq\delta A$, $\delta>0$. We obtain a location of the spectrum of a pencil associated with linear differential equation.