The Principal Floquet Bundle and Exponential Separation for Linear Parabolic Equations

  • Juraj Húska
  • Peter Poláčik


We consider linear nonautonomous second order parabolic equations on bounded domains subject to Dirichlet boundary condition. Under mild regularity assumptions on the coefficients and the domain, we establish the existence of a principal Floquet bundle exponentially separated from a complementary invariant bundle. Our main theorem extends in a natural way standard results on principal eigenvalues and eigenfunctions of elliptic and time-periodic parabolic equations. Similar theorems were earlier available only for smooth domains and coefficients. As a corollary of our main result, we obtain the uniqueness of positive entire solutions of the equations in

Nonautonomous parabolic equations principal Floquet bundle exponential separation positive entire solutions 


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  1. 1.
    Aronson, D.G. (1968). Non-negative solutions of linear parabolic equations. Ann. Scuola Norm. Sup. Pisa 22, 607–694.Google Scholar
  2. 2.
    Berestycki, H., Nirenberg, L., and Varadhan, S.R.S. (1994). The principal eigenvalue and maximum principle for second-order elliptic operators in general domains. Comm. Pure Appl. Math. 47, 47–92.Google Scholar
  3. 3.
    Birindelli, I. (1995). Hopf's lemma and anti-maximum principle in general domains. J. Diff. Eqns 119 (2), 450–472.Google Scholar
  4. 4.
    Chow, S.-N., Lu, K., and Mallet-Paret, J. (1994). Floquet theory for parabolic differential equations. J. Diff. Eqns 109, 147–200.Google Scholar
  5. 5.
    Chow, S.-N., Lu, K., and Mallet-Paret, J. (1995). Floquet bundles for scalar parabolic equations. Arch. Rational Mech. Anal. 129, 245–304.Google Scholar
  6. 6.
    Daners, D. (1996). Domain perturbation for linear and nonlinear parabolic equations. J. Diff. Eqns 129, 358–402.Google Scholar
  7. 7.
    Daners, D. (2000). Existence and perturbation of principal eigenvalues for a periodic-parabolic problem. Electron. J. Diff. Eqns Conf. 5, 51–67.Google Scholar
  8. 8.
    Daners, D. (2000). Heat kernel estimates for operators with boundary conditions. Math. Nachr. 217, 13–41.Google Scholar
  9. 9.
    Daners, D., and Koch Medina, P. (1992). Abstract evolution equations, periodic problems and applications, Longman Scientific & Technical, Harlow.Google Scholar
  10. 10.
    Escauriaza, L., and Fernandez, J. (2003). Unique continuation for parabolic operators. Ark. Mat. 41, 35–60.Google Scholar
  11. 11.
    Ferretti, E., and Safonov, M.V. (2001). Harmonic analysis and boundary value problems, Contemp. Math. 277, American Mathematical Society Providence, RI, pp. 87–112.Google Scholar
  12. 12.
    Fabes, E.B., and Safonov, M.V. (1997). Behavior near the boundary of positive solutions of second order parabolic equations. J. Fourier Anal. Appl. 3, 871–882.Google Scholar
  13. 13.
    Gilbarg, D., and Trudinger, N. (1977). Elliptic Partial Differential Equations of Second Order, Springer, Berlin Heidelberg.Google Scholar
  14. 14.
    Hess, P. (1991). Periodic-parabolic boundary value problems and positivity, Longman Scientific & Technical, Harlow.Google Scholar
  15. 15.
    Hess, P., and Poláčik, P. (1993). Boundedness of prime periods of stable cycles and convergence to xed points in discrete monotone dynamical systems. SIAM J. Math. Anal. 24, 1312–1330.Google Scholar
  16. 16.
    Hutson, V., Shen, W., and Vickers, G.T. (2001). Estimates for the principal spectrum point for certain time-dependent parabolic operators. Proc. Amer. Math. Soc. 129 (6), 1669–1679 (electronic).Google Scholar
  17. 17.
    Ladyzhenskaya, O.A., Solonnikov, V.A., and Uralceva, N.N. (1968). Linear and Quasilinear Equations of Parabolic Type. Providence, Translation of Mathematical Monographs, Rhode Island.Google Scholar
  18. 18.
    Mierczyński, J. Flows on order bundles. unpublished.Google Scholar
  19. 19.
    Mierczyński, J. (1994). p-arcs in strongly monotone discrete-time dynamical systems. Diff. Integ. Eqns 7, 1473–1494.Google Scholar
  20. 20.
    Mierczyński, J. (1997). Globally positive solutions of linear PDEs of second order with Robin boundary conditions. J. Math. Anal. Appl. 209, 47–59.Google Scholar
  21. 21.
    Mierczyński, J. (1998). Globally positive solutions of linear parabolic partial differential equations of second order with Dirichlet boundary conditions. J. Math. Anal. Appl. 226, 326–347.Google Scholar
  22. 22.
    Mierczyński, J. (2000). The principal spectrum for linear nonautonomous parabolic pdes of second order: Basic properties. J. Diff. Eqns 168, 453–476.Google Scholar
  23. 23.
    Mierczyński, J., and Shen, W. (2003). Exponential separation and principal Lyapunov exponent/spectrum for random/nonautonomous parabolic equations. J. Diff. Eqns 191, 175–205.Google Scholar
  24. 24.
    Miller, K. (1973). Non-unique continuation for certain ODE's in Hilbert space and for uniformly parabolic and elliptic equations in self-adjoint divergence form. Symposium on Non-Well-Posed Problems and Logarithmic Convexity (Heriot-Watt Univ., Edinburgh, 1972) Lecture Notes in Mathematics, Vol. 316, Springer, Berlin, pp. 85–101.Google Scholar
  25. 25.
    Moser, J. (1964). A Harnack inequality for parabolic differential equations. Comm. Pure Appl. Math. 17, 101–134, Correction in Comm. Pure Appl. Math. 20, 231–236, 1967.Google Scholar
  26. 26.
    Nishio, M. (1933). The uniqueness of positive solutions of parabolic equations of divergence form on an unbounded domain. Nagoya Math J. 130, 111–121.Google Scholar
  27. 27.
    Poláčik, P. (2003). On Uniqueness of Positive Entire Solutions and Other Properties of Linear Parabolic Equations. Preprint.Google Scholar
  28. 28.
    Poláčik, P., and Tereščák, I. (1992). Convergence to cycles as a typical asymptotic behavior in smooth strongly monotone discrete-time dynamical systems. Arch. Rational Mech. Anal. 116, 339–360.Google Scholar
  29. 29.
    Poláčik, P., and Tereščák, I. (1993). Exponential separation and invariant bundles for maps in ordered Banach spaces with applications to parabolic equations. J. Dyn. Diff. Eqns 5, 279–303; Erratum 6 (1), 245–246, 1994.Google Scholar
  30. 30.
    Ruelle, D. (1979). Analycity properties of the characteristic exponents of random matrix products. Adv. Math. 32, 68–80.Google Scholar
  31. 31.
    Shen, W., and Yi, Y. (1998). Almost automorphic and almost periodic dynamics in skew-product semiflows. Mem. Amer. Math. Soc. 647, 93.Google Scholar
  32. 32.
    Tereščák, I. (1994). Dynamical Systems with Discrete Lyapunov Functionals. PhD Thesis, Comenius University.Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2004

Authors and Affiliations

  • Juraj Húska
    • 1
  • Peter Poláčik
    • 1
  1. 1.School of MathematicsUniversity of MinnesotaMinneapolisUSA

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