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Local and global smoothing effects for some linear hyperbolic equations with a strong dissipation


Authors: Marina Ghisi, Massimo Gobbino and Alain Haraux
Journal: Trans. Amer. Math. Soc. 368 (2016), 2039-2079
MSC (2010): Primary 35L10, 35L15, 35L20
DOI: https://doi.org/10.1090/tran/6520
Published electronically: April 3, 2015
MathSciNet review: 3449233
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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

DOI: https://doi.org/10.1090/tran/6520
Keywords: Linear hyperbolic equations, dissipative hyperbolic equations, strong dissipation, fractional damping, bounded solutions
Received by editor(s): February 26, 2014
Published electronically: April 3, 2015
Article copyright: © Copyright 2015 American Mathematical Society