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Perturbation of analytic semigroups and applications to partial differential equations

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Abstract

In a recent paper we presented a general perturbation result for generators of \(C_0\)-semigroups, c.f. Theorem 2.1 below. The aim of the present work is to replace, in case the unperturbed semigroup is analytic, the various admissibility conditions appearing in this result by simpler inclusion assumptions on the domain and the range of the perturbation. This is done in Theorem 2.4 and allows to apply our results also in situations which are only in part governed by analytic semigroups. The power of our approach to treat in a unified and systematic way wide classes of PDE’s is illustrated by a generic example, a degenerate differential operator with generalized Wentzell boundary conditions, a reaction diffusion equation with unbounded delay and a perturbed Laplacian.

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Authors and Affiliations

  1. Arbeitsbereich Funktionalanalysis, Mathematisches Institut, Auf der Morgenstelle 10, 72076, Tübingen, Germany

    Martin Adler & Miriam Bombieri

  2. Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica (DISIM), Università degli Studi dell’Aquila, Via Vetoio, 67100, L’Aquila (AQ), Italy

    Klaus-Jochen Engel

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  1. Martin Adler
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  2. Miriam Bombieri
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Correspondence to Klaus-Jochen Engel.

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Dedicated to Rainer Nagel on the occasion of his 75th birthday

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Adler, M., Bombieri, M. & Engel, KJ. Perturbation of analytic semigroups and applications to partial differential equations. J. Evol. Equ. 17, 1183–1208 (2017). https://doi.org/10.1007/s00028-016-0377-8

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