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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Exponential separation and principal Floquet bundles for linear parabolic equations on general bounded domains: Nondivergence case


Author: Juraj Húska
Journal: Trans. Amer. Math. Soc. 360 (2008), 4639-4679
MSC (2000): Primary 35K10; Secondary 35B05
Published electronically: April 7, 2008
MathSciNet review: 2403700
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Abstract: We consider the Dirichlet problem for linear nonautonomous second order parabolic equations of nondivergence type on general bounded domains with bounded measurable coefficients. Under such minimal regularity assumptions, we establish the existence of a principal Floquet bundle exponentially separated from a complementary invariant bundle. As a special case of our main theorem, assuming the coefficients are time-periodic, we obtain a new result on the existence of a principal eigenvalue of an associated (time-periodic) parabolic eigenvalue problem. We also show the existence of a uniform spectral gap between the principal eigenvalue and the rest of the spectrum for a class of time-periodic uniformly parabolic operators. Finally, we prove the uniqueness of positive entire solutions in the class of solutions whose supremum norms do not grow superexponentially as time goes to negative infinity.


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Additional Information

Juraj Húska
Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
Email: huska@math.umn.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-08-04413-9
PII: S 0002-9947(08)04413-9
Keywords: Exponential separation, nonsmooth domains, positive entire solutions, principal Floquet bundle, spectral gap
Received by editor(s): April 26, 2006
Published electronically: April 7, 2008
Additional Notes: The author was supported by the Doctoral Dissertation Fellowship of the University of Minnesota
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.