Polynomial decay rate of C₀ semigroup in Hilbert spaces

Abstract

In the paper, we characterize the polynomial stability of $C_0$ semigroup $T\left(t\right)$ with generator A on Hilbert space $\mathcal{H}$. Let A have compact resolvent and there be a sequence of the eigenvectors of A that forms a Riesz basis for $\mathcal{H}$. By the asymptotic relation of the real part and imaginary part of eigenvalues of A, we give the optimal decay rate of polynomial stability of $T\left(t\right)$. Moreover, we give the zeros distribution of certain equations. As an application, we illustrate our general results by an acoustical system.

COMMUNICATED BY:

• R. Triggiani

Keywords:

• $C_0$ semigroup
• polynomial stability
• Riesz basis
• spectrum

AMS Subject Classifications:

• 47D06
• 35B40
• 35L20

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This research is supported by the National Natural Science Foundation of China [grant number NSFC-61773277] and the Doctoral Scientific Research Foundation of Henan Normal University qd18088.