Nonautonomous Kolmogorov parabolic equations with unbounded coefficients
Authors:
Markus Kunze, Luca Lorenzi and Alessandra Lunardi
Journal:
Trans. Amer. Math. Soc. 362 (2010), 169-198
MSC (2000):
Primary 35K10, 35K15, 37L40
DOI:
https://doi.org/10.1090/S0002-9947-09-04738-2
Published electronically:
August 3, 2009
MathSciNet review:
2550148
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: We study a class of elliptic operators with unbounded coefficients defined in
for some unbounded interval
. We prove that, for any
, the Cauchy problem
for the parabolic equation
admits a unique bounded classical solution
. This allows to associate an evolution family
with
, in a natural way. We study the main properties of this evolution family and prove gradient estimates for the function
. Under suitable assumptions, we show that there exists an evolution system of measures for
and we study the first properties of the extension of
to the
-spaces with respect to such measures.
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Additional Information
Markus Kunze
Affiliation:
Graduiertenkolleg 1100, Ulm University, Helmholtzstrasse 18, 89069 Ulm, Germany
Address at time of publication:
Delft Institute of Applied Mathematics, Delft University of Technology, P.O. Box 5031, 2600 GA Delft, The Netherlands
Email:
markus.kunze@uni-ulm.de, M.C.Kunze@tudelft.nl
Luca Lorenzi
Affiliation:
Dipartimento di Matematica, Università degli Studi di Parma, Viale G.P. Usberti, 53/A, I-43100 Parma, Italy
Email:
luca.lorenzi@unipr.it
Alessandra Lunardi
Affiliation:
Dipartimento di Matematica, Università degli Studi di Parma, Viale G.P. Usberti, 53/A, I-43100 Parma, Italy
Email:
alessandra.lunardi@unipr.it
DOI:
https://doi.org/10.1090/S0002-9947-09-04738-2
Keywords:
Nonautonomous parabolic equations,
evolution operators,
gradient estimates,
evolution systems of measures,
evolution semigroups,
invariant measures
Received by editor(s):
July 5, 2007
Published electronically:
August 3, 2009
Additional Notes:
This work was supported by the M.I.U.R. research projects Prin 2004 and 2006 “Kolmogorov equations”
Article copyright:
© Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.