Remote Access Transactions of the American Mathematical Society
Green Open Access

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • 1. J.M. Arrieta.
    Elliptic equations, principal eigenvalue and dependence on the domain.
    Comm. Partial Differential Equations , 21:971-991, 1996. MR 1391529 (97h:35037)
  • 2. H. Berestycki and L. Nirenberg.
    On the method of moving planes and the sliding method.
    Bol. Soc. Brasil. Mat. (N.S.), 22:1-37, 1991. MR 1159383 (93a:35048)
  • 3. H. Berestycki, L. Nirenberg, and S. R. S. Varadhan.
    The principal eigenvalue and maximum principle for second-order elliptic operators in general domains.
    Comm. Pure Appl. Math., 47:47-92, 1994. MR 1258192 (95h:35053)
  • 4. I. Birindelli.
    Hopf's lemma and anti-maximum principle in general domains.
    J. Differential Equations, 119(2):450-472, 1995. MR 1340547 (96g:35047)
  • 5. S.-N. Chow, K. Lu, and J. Mallet-Paret.
    Floquet theory for parabolic differential equations.
    J. Differential Equations, 109:147-200, 1994. MR 1272403 (95c:35116)
  • 6. S.-N. Chow, K. Lu, and J. Mallet-Paret.
    Floquet bundles for scalar parabolic equations.
    Arch. Rational Mech. Anal., 129:245-304, 1995. MR 1328478 (96c:35070)
  • 7. D. Daners.
    Existence and perturbation of principal eigenvalues for a periodic-parabolic problem.
    Electron. J. Diff. Eqns., Conf. 05:51-67, 2000. MR 1799044 (2001j:35125)
  • 8. D.-E. Edmunds, L.-A. Peletier.
    Removable singularities of solutions of quasi-linear parabolic equations.
    J. London Math. Soc., (2) 2:273-283, 1970. MR 0259365 (41:4003)
  • 9. T. Godoy and U. Kaufmann.
    On principal eigenvalues for periodic parabolic problems with optimal condition on the weight function.
    J. Math. Anal. Appl., 262(1):208-220, 2001. MR 1857223 (2002g:35162)
  • 10. M. Gruber.
    Harnack inequalities for solutions of general second order parabolic equations and estimates of their Hölder constants.
    Math. Z., 185(1):23-43, 1984. MR 724044 (86b:35089)
  • 11. P. Hess.
    Periodic-parabolic boundary value problems and positivity.
    Longman Scientific & Technical, Harlow, 1991. MR 1100011 (92h:35001)
  • 12. P. Hess and P. Poláčik.
    Boundedness of prime periods of stable cycles and convergence to fixed points in discrete monotone dynamical systems.
    SIAM J. Math. Anal., 24:1312-1330, 1993. MR 1234018 (94i:47087)
  • 13. J. Húska.
    Exponential separation and principal Floquet bundles for linear parabolic equations on general bounded domains: The divergence case.
    Indiana Univ. Math. J., 55(3):1015-1044, 2006. MR 2244596
  • 14. J. Húska.
    Harnack inequality and exponential separation for oblique derivative problems on Lipschitz domains.
    J. Differential Equations, 226:541-557, 2006. MR 2237690
  • 15. J. Húska and P. Poláčik.
    The principal Floquet bundle and exponential separation for linear parabolic equations.
    J. Dynam. Differential Equations, 24:1312-1330, 2004. MR 2105779 (2006e:35147)
  • 16. J. Húska, P. Poláčik and M.V. Safonov.
    Principal eigenvalues, spectral gaps and exponential separation between positive and sign-changing solutions of parabolic equations.
    Proceedings of the 5th International Conference on Dynamical Systems and Differential Equations, Pomona 2004, Disc. Cont. Dynamical Systems, Supplement 2005, pp. 427-435. MR 2192700 (2006m:35131)
  • 17. J. Húska, P. Poláčik, and M. Safonov.
    Harnack inequalities, exponential separation, and perturbations of principal Floquet bundles for linear parabolic equations. Ann. Inst. H. Poincaré Anal. Non Linéaire, 24:711-739, 2007. MR 2348049
  • 18. V. Hutson, W. Shen, and G. T. Vickers.
    Estimates for the principal spectrum point for certain time-dependent parabolic operators.
    Proc. Amer. Math. Soc., 129(6):1669-1679 (electronic), 2001. MR 1814096 (2001m:35243)
  • 19. N.V. Krylov and M.V. Safonov.
    A property of the solutions of parabolic equations with measurable coefficients.
    Izv. Akad. Nauk SSSR Ser. Mat., 44(1):161-175, 1980. MR 563790 (83c:35059)
  • 20. N.V. Krylov.
    Nonlinear Elliptic and Parabolic Equations of the Second Order.
    Mathematics and its Applications, Reidel, Dordrecht, 1987. MR 0901759 (88d:35005)
  • 21. G.M. Lieberman.
    Second order parabolic differential equations.
    World Scientific Publishing Co. Inc., River Edge, NJ, 1996. MR 1465184 (98k:35003)
  • 22. J. Mierczyński.
    Flows on order bundles,
  • 23. J. Mierczyński.
    $ p$-arcs in strongly monotone discrete-time dynamical systems.
    Differential Integral Equations, 7:1473-1494, 1994. MR 1269666 (95c:58121)
  • 24. J. Mierczyński.
    Globally positive solutions of linear PDEs of second order with Robin boundary conditions.
    J. Math. Anal. Appl., 209:47-59, 1997. MR 1444510 (98c:35071)
  • 25. J. Mierczyński.
    Globally positive solutions of linear parabolic partial differential equations of second order with Dirichlet boundary conditions.
    J. Math. Anal. Appl., 226:326-347, 1998. MR 1650236 (99m:35096)
  • 26. J. Mierczyński.
    The principal spectrum for linear nonautonomous parabolic pdes of second order: Basic properties.
    J. Differential Equations, 168:453-476, 2000. MR 1808456 (2001m:35147)
  • 27. J. Mierczyński and W. Shen.
    Exponential separation and principal Lyapunov exponent/spectrum for random/nonautonomous parabolic equations.
    J. Differential Equations, 191:175-205, 2003. MR 1973287 (2004h:35232)
  • 28. K. Miller. Barriers on Cones for Uniformly Elliptic Operators,
    Ann. Mat. Pura Appl., 76:93-106, 1967. MR 0221087 (36:4139)
  • 29. M. Nishio.
    The uniqueness of positive solutions of parabolic equations of divergence form on an unbounded domain.
    Nagoya Math. J., 130:111-121, 1993. MR 1223732 (94f:35058)
  • 30. P. Poláčik.
    Estimates of solutions and asymptotic symmetry for parabolic equations on bounded domains.
    Arch. Rational Mech. Anal., 183:59-91, 2007. MR 2259340
  • 31. P. Poláčik.
    On uniqueness of positive entire solutions and other properties of linear parabolic equations.
    Discrete Contin. Dynamical Systems, 12:13-26, 2005. MR 2121246 (2005k:35170)
  • 32. P. Poláčik and I. Tereščák.
    Convergence to cycles as a typical asymptotic behavior in smooth strongly monotone discrete-time dynamical systems.
    Arch. Rational Mech. Anal., 116:339-360, 1992. MR 1132766 (93b:58088)
  • 33. P. Poláčik and I. Tereščák.
    Exponential separation and invariant bundles for maps in ordered Banach spaces with applications to parabolic equations.
    J. Dynamics Differential Equations, 5:279-303, 1993.
    Erratum: 6(1):245-246 1994.MR 1223450 (94d:47064); MR 1262730
  • 34. W. Shen and Y. Yi.
    Almost automorphic and almost periodic dynamics in skew-product semiflows.
    Mem. Amer. Math. Soc., 647:93 p., 1998. MR 1445493 (99d:34088)
  • 35. I. Tereščák.
    Dynamics of $ {C}^1$ smooth strongly monotone discrete-time dynamical systems.
  • 36. I. Tereščák.
    Dynamical systems with discrete Lyapunov functionals.
    Ph.D. thesis, Comenius University, 1994.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 35K10, 35B05

Retrieve articles in all journals with MSC (2000): 35K10, 35B05

Additional Information

Juraj Húska
Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455

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.

American Mathematical Society