## 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.

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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