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Transactions of the American Mathematical Society

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

 

 

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
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Abstract: We study a class of elliptic operators $ A$ with unbounded coefficients defined in $ I\times\mathbb{R}^d$ for some unbounded interval $ I\subset\mathbb{R}$. We prove that, for any $ s\in I$, the Cauchy problem $ u(s,\cdot)=f\in C_b(\mathbb{R}^d)$ for the parabolic equation $ D_tu=Au$ admits a unique bounded classical solution $ u$. This allows to associate an evolution family $ \{G(t,s)\}$ with $ A$, in a natural way. We study the main properties of this evolution family and prove gradient estimates for the function $ G(t,s)f$. Under suitable assumptions, we show that there exists an evolution system of measures for $ \{G(t,s)\}$ and we study the first properties of the extension of $ G(t,s)$ to the $ L^p$-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.