# American Institute of Mathematical Sciences

June  2008, 1(2): 207-218. doi: 10.3934/dcdss.2008.1.207

## Strong stability of PDE semigroups via a generator resolvent criterion

 1 Department of Mathematics, University of Nebraska-Lincoln, Lincoln, Nebraska 68588, United States

Received  September 2006 Revised  November 2007 Published  March 2008

In the context of a coupled partial differential equation (PDE) model, we provide a rather general procedure by which one may invoke a recently derived operator theoretic result in [21], so as to obtain strong stability of those dissipative $C_{0}$-semigroups which model PDEs in Hilbert space. In particular, the procedure is applied here to a PDE which models structural acoustic interactions; it is wellknown that for this interactive PDE the classical stability tools--i.e., the Nagy-Foias decomposition--is inapplicable. The novelty of adopting the present strong stability technique is that one does not need to have an explicit representation of the resolvent.
Citation: George Avalos. Strong stability of PDE semigroups via a generator resolvent criterion. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 207-218. doi: 10.3934/dcdss.2008.1.207
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