Regularity properties of some perturbations of non-densely defined operators with applications

Abstract

This paper is to study some conditions on semigroups, generated by some class of non-densely defined operators in the closure of its domain, in order that certain bounded perturbations preserve some regularity properties of the semigroup such as norm continuity, compactness, differentiability and analyticity. Furthermore, we study the critical and essential growth bound of the semigroup under bounded perturbations. The main results generalize the corresponding results in the case of Hille–Yosida operators. As an illustration, we apply the main results to study the asymptotic behaviors of a class of age-structured population models in \( L^p \) spaces (\( 1 \le p < \infty \)).

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  • 24 May 2020

    The original publication of the article contains errors which need to be amended as mentioned below.

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Correspondence to Deliang Chen.

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Part of this work was done at East China Normal University. The author would like to thank Shigui Ruan, Ping Bi and Dongmei Xiao for their useful discussions and encouragement. The author is grateful to the referee(s) for useful comments and suggestions and particularly pointing out Lemma 6.9 and a mistake in Theorem 7.4, which improved significantly the presentation of the original manuscript.

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Chen, D. Regularity properties of some perturbations of non-densely defined operators with applications. J. Evol. Equ. 20, 659–702 (2020). https://doi.org/10.1007/s00028-019-00510-y

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Keywords

  • Regularity
  • Perturbation
  • Non-densely defined operators
  • Critical growth bound
  • Essential growth bound
  • Integrated semigroup
  • Age-structured population model

Mathematics Subject Classification

  • Primary 47A55
  • 34D10
  • Secondary 34K12
  • 47N20
  • 47D62