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The lack of exponential stability for a class of second-order systems with memory

Part of: Qualitative behavior Partial differential equations

Published online by Cambridge University Press:  31 May 2017

Valeria Danese  and
Filippo Dell'Oro
Valeria Danese
Affiliation:
Dipartimento di Matematica, Politecnico di Milano, Via Bonardi 9, 20133 Milano, Italy (valeria.danese@polimi.it; filippo.delloro@polimi.it)
Filippo Dell'Oro
Affiliation:
Dipartimento di Matematica, Politecnico di Milano, Via Bonardi 9, 20133 Milano, Italy (valeria.danese@polimi.it; filippo.delloro@polimi.it)
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We analyse the decay properties of the solution semigroup S(t) generated by the linear integrodifferential equation

where the operator A is strictly positive self-adjoint with A –1 not necessarily compact. The asymptotic stability of S(t) is investigated in terms of the dependence of the parameter γ ∈ ℝ. In particular, we show that S(t) is not exponentially stable when γ ≠ 1.

Keywords

memory kernelcontraction semigroupstabilityexponential stability

MSC classification

Secondary: 35B40: Asymptotic behavior of solutions 45K05: Integro-partial differential equations 45M10: Stability theory
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 147 , Issue 4 , August 2017 , pp. 683 - 702
Copyright
Copyright © Royal Society of Edinburgh 2017