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Strong convergence of solutions to nonautonomous Kolmogorov equations


Authors: Luca Lorenzi, Alessandra Lunardi and Roland Schnaubelt
Journal: Proc. Amer. Math. Soc. 144 (2016), 3903-3917
MSC (2010): Primary 35K10; Secondary 35K15, 35B40
DOI: https://doi.org/10.1090/proc/13031
Published electronically: April 28, 2016
MathSciNet review: 3513547
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Abstract: We study a class of nonautonomous, linear, parabolic equations with unbounded coefficients on $ \mathbb{R}^{d}$ which admit an evolution system of measures. It is shown that the solutions of these equations converge to constant functions as $ t\to +\infty $. We further establish the uniqueness of the tight evolution system of measures and treat the case of converging coefficients.


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Additional Information

Luca Lorenzi
Affiliation: Dipartimento di Matematica e Informatica, Università degli Studi di Parma, Parco Area delle Scienze 53/A, I-43124 Parma, Italy
Email: luca.lorenzi@unipr.it

Alessandra Lunardi
Affiliation: Dipartimento di Matematica e Informatica, Università degli Studi di Parma, Parco Area delle Scienze 53/A, I-43124 Parma, Italy
Email: alessandra.lunardi@unipr.it

Roland Schnaubelt
Affiliation: Department of Mathematics, Karlsruhe Institute of Technology, D-76128 Karlsruhe, Germany
Email: schnaubelt@kit.edu

DOI: https://doi.org/10.1090/proc/13031
Keywords: Nonautonomous parabolic problem, unbounded coefficients, invariant measures, uniqueness, convergence, evolution semigroup, Green's function, gradient estimates.
Received by editor(s): August 17, 2015
Received by editor(s) in revised form: November 6, 2015
Published electronically: April 28, 2016
Additional Notes: The first author wishes to thank the Department of Mathematics of the Karlsruhe Institute of Technology for the kind hospitality during his visit.
This work was supported by the M.I.U.R. Research Project PRIN 2010-2011 “Problemi differenziali di evoluzione: approcci deterministici e stocastici e loro interazioni”
Communicated by: Joachim Krieger
Article copyright: © Copyright 2016 American Mathematical Society

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