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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Local and global smoothing effects for some linear hyperbolic equations with a strong dissipation
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by Marina Ghisi, Massimo Gobbino and Alain Haraux PDF
Trans. Amer. Math. Soc. 368 (2016), 2039-2079 Request permission

Abstract:

We consider an abstract second order linear equation with a strong dissipation, namely a friction term which depends on a power of the “elastic” operator.

In the homogeneous case, we investigate the phase spaces in which the initial value problem gives rise to a semigroup and the further regularity of solutions. In the non-homogeneous case, we study how the regularity of solutions depends on the regularity of forcing terms, and we characterize the spaces where a bounded forcing term yields a bounded solution.

What we discover is a variety of different regimes, with completely different behaviors, depending on the exponent in the friction term.

We also provide counterexamples in order to show the optimality of our results.

References
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Additional Information
  • Marina Ghisi
  • Affiliation: Dipartimento di Matematica, Università degli Studi di Pisa, Pisa, Italy
  • Email: ghisi@dm.unipi.it
  • Massimo Gobbino
  • Affiliation: Dipartimento di Matematica, Università degli Studi di Pisa, Pisa, Italy
  • Email: massimo.gobbino@unipi.it
  • Alain Haraux
  • Affiliation: Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Paris, France
  • Email: haraux@ann.jussieu.fr
  • Received by editor(s): February 26, 2014
  • Published electronically: April 3, 2015
  • © Copyright 2015 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 368 (2016), 2039-2079
  • MSC (2010): Primary 35L10, 35L15, 35L20
  • DOI: https://doi.org/10.1090/tran/6520
  • MathSciNet review: 3449233