Harnack inequalities, exponential separation, and perturbations of principal Floquet bundles for linear parabolic equations
Annales de l'I.H.P. Analyse non linéaire, Volume 24 (2007) no. 5, pp. 711-739.
@article{AIHPC_2007__24_5_711_0,
     author = {H\'uska, Juraj and Pol\'a\v{c}ik, Peter and Safonov, Mikhail V.},
     title = {Harnack inequalities, exponential separation, and perturbations of principal {Floquet} bundles for linear parabolic equations},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {711--739},
     publisher = {Elsevier},
     volume = {24},
     number = {5},
     year = {2007},
     doi = {10.1016/j.anihpc.2006.04.006},
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     language = {en},
     url = {http://archive.numdam.org/articles/10.1016/j.anihpc.2006.04.006/}
}
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Húska, Juraj; Poláčik, Peter; Safonov, Mikhail V. Harnack inequalities, exponential separation, and perturbations of principal Floquet bundles for linear parabolic equations. Annales de l'I.H.P. Analyse non linéaire, Volume 24 (2007) no. 5, pp. 711-739. doi : 10.1016/j.anihpc.2006.04.006. http://archive.numdam.org/articles/10.1016/j.anihpc.2006.04.006/

[1] Aronson D.G., Non-negative solutions of linear parabolic equations, Ann. Scuola Norm. Sup. Pisa 22 (1968) 607-694. | Numdam | MR | Zbl

[2] Arrieta J.M., Elliptic equations, principal eigenvalue and dependence on the domain, Comm. Partial Differential Equations 21 (1996) 971-991. | MR | Zbl

[3] Berestycki H., Nirenberg L., Varadhan S.R.S., The principal eigenvalue and maximum principle for second-order elliptic operators in general domains, Comm. Pure Appl. Math. 47 (1994) 47-92. | MR | Zbl

[4] Birindelli I., Hopf's lemma and anti-maximum principle in general domains, J. Differential Equations 119 (2) (1995) 450-472. | MR | Zbl

[5] Chow S.-N., Lu K., Mallet-Paret J., Floquet bundles for scalar parabolic equations, Arch. Rational Mech. Anal. 129 (1995) 245-304. | MR | Zbl

[6] Daners D., Dirichlet problems on varying domains, J. Differential Equations 188 (2003) 591-624. | MR | Zbl

[7] Daners D., Domain perturbation for linear and nonlinear parabolic equations, J. Differential Equations 129 (1996) 358-402. | MR | Zbl

[8] Daners D., Existence and perturbation of principal eigenvalues for a periodic-parabolic problem, Electron. J. Differential Equations, Conf. 05 (2000) 51-67. | MR | Zbl

[9] Daners D., Heat kernel estimates for operators with boundary conditions, Math. Nachr. 217 (2000) 13-41. | MR | Zbl

[10] Evans L.C., Partial Differential Equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998. | MR | Zbl

[11] Fabes E.B., Garofalo N., Salsa S., A backward Harnack inequality and Fatou theorem for nonnegative solutions of parabolic equations, Illinois J. Math. 30 (1986) 536-565. | MR | Zbl

[12] Fabes E.B., Safonov M.V., Behavior near the boundary of positive solutions of second order parabolic equations, J. Fourier Anal. Appl. 3 (1997) 871-882. | MR | Zbl

[13] Fabes E.B., Safonov M.V., Yuan Y., Behavior near the boundary of positive solutions of second order parabolic equations II, Trans. Amer. Math. Soc. 12 (1999) 4947-4961. | MR | Zbl

[14] Ferretti E., Safonov M.V., Growth theorems and Harnack inequality for second order parabolic equations, in: Harmonic Analysis and Boundary Value Problems, Contemp. Math., vol. 277, Amer. Math. Soc., Providence, RI, 2001, pp. 87-112. | MR | Zbl

[15] Garofalo N., Second order parabolic equations in nonvariational form: boundary Harnack principle and comparison theorems for nonnegative solutions, Ann. Mat. Pura Appl. 138 (1984) 267-296. | MR | Zbl

[16] Henry P., Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, vol. 840, Springer, New York, 1981. | MR | Zbl

[17] Hess P., Periodic-Parabolic Boundary Value Problems and Positivity, Longman Scientific & Technical, Harlow, 1991. | MR | Zbl

[18] Hess P., Poláčik P., Boundedness of prime periods of stable cycles and convergence to fixed points in discrete monotone dynamical systems, SIAM J. Math. Anal. 24 (1993) 1312-1330. | MR | Zbl

[19] Húska J., Harnack inequality and exponential separation for oblique derivative problems on Lipschitz domains, J. Differential Equations 226 (2006) 541-557. | MR | Zbl

[20] Húska J., Poláčik P., The principal Floquet bundle and exponential separation for linear parabolic equations, J. Dynam. Differential Equations 24 (2004) 1312-1330. | MR | Zbl

[21] J. Húska, P. Poláčik, M.V. Safonov, Principal eigenvalues, spectral gaps and exponential separation between positive and sign-changing solutions of parabolic equations, in: Disc. Cont. Dynamical Systems, Supplement, Proceedings of the 5th International Conference on Dynamical Systems and Differential Equations, Pomona 2004, 2005, pp. 427-435.

[22] Hutson V., Shen W., Vickers G.T., Estimates for the principal spectrum point for certain time-dependent parabolic operators, Proc. Amer. Math. Soc. 129 (6) (2001) 1669-1679, (electronic). | MR | Zbl

[23] Krylov N.V., Safonov M.V., A property of the solutions of parabolic equations with measurable coefficients, Izv. Akad. Nauk SSSR Ser. Mat. 44 (1) (1980) 161-175. | MR | Zbl

[24] Ladyzhenskaya O.A., Solonnikov V.A., Uralceva N.N., Linear and Quasilinear Equations of Parabolic Type, Translation of Mathematical Monographs, American Mathematical Society, Providence, RI, 1968. | Zbl

[25] Landis E.M., Second Order Equations of Elliptic and Parabolic Type, Translation of Mathematical Monographs, American Mathematical Society, Providence, RI, 1998. | MR | Zbl

[26] Lieberman G.M., Second Order Parabolic Differential Equations, World Scientific Publishing Co. Inc., River Edge, NJ, 1996. | MR | Zbl

[27] J. Mierczyński, Flows on order bundles, unpublished.

[28] Mierczyński J., p-arcs in strongly monotone discrete-time dynamical systems, Differential Integral Equations 7 (1994) 1473-1494. | MR | Zbl

[29] Mierczyński J., Globally positive solutions of linear PDEs of second order with Robin boundary conditions, J. Math. Anal. Appl. 209 (1997) 47-59. | MR | Zbl

[30] Mierczyński J., Globally positive solutions of linear parabolic partial differential equations of second order with Dirichlet boundary conditions, J. Math. Anal. Appl. 226 (1998) 326-347. | MR | Zbl

[31] Mierczyński J., The principal spectrum for linear nonautonomous parabolic pdes of second order: Basic properties, J. Differential Equations 168 (2000) 453-476. | MR | Zbl

[32] Mierczyński J., Shen W., Exponential separation and principal Lyapunov exponent/spectrum for random/nonautonomous parabolic equations, J. Differential Equations 191 (2003) 175-205. | MR | Zbl

[33] Moser J., A Harnack inequality for parabolic differential equations, Comm. Pure Appl. Math. 17 (1964) 101-134, Correction in, Comm. Pure Appl. Math. 20 (1967) 231-236. | MR | Zbl

[34] Nishio M., The uniqueness of positive solutions of parabolic equations of divergence form on an unbounded domain, Nagoya Math. J. 130 (1993) 111-121. | MR | Zbl

[35] Poláčik P., Parabolic equations: asymptotic behavior and dynamics on invariant manifolds, in: Fiedler B. (Ed.), Handbook on Dynamical Systems, vol. 2, Elsevier, Amsterdam, 2002, pp. 835-883. | MR | Zbl

[36] Poláčik P., On uniqueness of positive entire solutions and other properties of linear parabolic equations, Discrete Contin. Dynamical Systems 12 (2005) 13-26. | MR | Zbl

[37] Poláčik P., Tereščák I., Convergence to cycles as a typical asymptotic behavior in smooth strongly monotone discrete-time dynamical systems, Arch. Rational Mech. Anal. 116 (1992) 339-360. | MR | Zbl

[38] Poláčik P., Tereščák I., Exponential separation and invariant bundles for maps in ordered Banach spaces with applications to parabolic equations, J. Dynamics Differential Equations 5 (1993) 279-303, Erratum, J. Dynamics Differential Equations 6 (1) (1994) 245-246. | MR | Zbl

[39] Ruelle D., Analycity properties of the characteristic exponents of random matrix products, Adv. in Math. 32 (1979) 68-80. | MR | Zbl

[40] Shen W., Yi Y., Almost automorphic and almost periodic dynamics in skew-product semiflows, Mem. Amer. Math. Soc. 647 (1998) 93. | MR | Zbl

[41] I. Tereščák, Dynamics of C 1 smooth strongly monotone discrete-time dynamical systems, Preprint.

[42] I. Tereščák, Dynamical systems with discrete Lyapunov functionals, PhD thesis, Comenius University, 1994.

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