Harnack inequalities, exponential separation, and perturbations of principal Floquet bundles for linear parabolic equations

Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire

Volumn 24, Issue 5, 2007, Pages 711-739

  Húska, Juraj ^a   Poláčik, Peter ^a   Safonov, Mikhail V ^a  

^a UNIVERSITY OF MINNESOTA   (United States)

Author keywords

Exponential separation; Harnack inequalities; Perturbations; Positive entire solutions; Principal Floquet bundle; Robustness

Indexed keywords

PERTURBATION TECHNIQUES; PROBLEM SOLVING; ROBUSTNESS (CONTROL SYSTEMS);

EXPONENTIAL SEPARATION; HARNACK INEQUALITIES; POSITIVE ENTIRE SOLUTIONS; PRINCIPAL FLOQUET BUNDLE;

LINEAR EQUATIONS;

EID: 34547575964     PISSN: 02941449     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.anihpc.2006.04.006     Document Type: Article

References (44)

1. Aronson D.G. Non-negative solutions of linear parabolic equations. Ann. Scuola Norm. Sup. Pisa 22 (1968) 607-694
2. Arrieta J.M. Elliptic equations, principal eigenvalue and dependence on the domain. Comm. Partial Differential Equations 21 (1996) 971-991
3. Berestycki H., Nirenberg L., and 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
4. Birindelli I. Hopf's lemma and anti-maximum principle in general domains. J. Differential Equations 119 2 (1995) 450-472
5. Chow S.-N., Lu K., and Mallet-Paret J. Floquet bundles for scalar parabolic equations. Arch. Rational Mech. Anal. 129 (1995) 245-304
6. Daners D. Dirichlet problems on varying domains. J. Differential Equations 188 (2003) 591-624
7. Daners D. Domain perturbation for linear and nonlinear parabolic equations. J. Differential Equations 129 (1996) 358-402
8. Daners D. Existence and perturbation of principal eigenvalues for a periodic-parabolic problem. Electron. J. Differential Equations, Conf. 05 (2000) 51-67
9. Daners D. Heat kernel estimates for operators with boundary conditions. Math. Nachr. 217 (2000) 13-41
10. Evans L.C. Partial Differential Equations. Graduate Studies in Mathematics vol. 19 (1998), American Mathematical Society, Providence, RI
11. Fabes E.B., Garofalo N., and Salsa S. A backward Harnack inequality and Fatou theorem for nonnegative solutions of parabolic equations. Illinois J. Math. 30 (1986) 536-565
12. Fabes E.B., and Safonov M.V. Behavior near the boundary of positive solutions of second order parabolic equations. J. Fourier Anal. Appl. 3 (1997) 871-882
13. Fabes E.B., Safonov M.V., and Yuan Y. Behavior near the boundary of positive solutions of second order parabolic equations II. Trans. Amer. Math. Soc. 12 (1999) 4947-4961
14. Ferretti E., and Safonov M.V. Growth theorems and Harnack inequality for second order parabolic equations. Harmonic Analysis and Boundary Value Problems. Contemp. Math. vol. 277 (2001), Amer. Math. Soc., Providence, RI 87-112
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
16. Henry P. Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Mathematics vol. 840 (1981), Springer, New York
17. Hess P. Periodic-Parabolic Boundary Value Problems and Positivity (1991), Longman Scientific & Technical, Harlow
18. Hess P., and 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
19. Húska J. Harnack inequality and exponential separation for oblique derivative problems on Lipschitz domains. J. Differential Equations 226 (2006) 541-557
20. Húska J., and Poláčik P. The principal Floquet bundle and exponential separation for linear parabolic equations. J. Dynam. Differential Equations 24 (2004) 1312-1330
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., and 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)
23. Krylov N.V., and 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
24. Ladyzhenskaya O.A., Solonnikov V.A., and Uralceva N.N. Linear and Quasilinear Equations of Parabolic Type. Translation of Mathematical Monographs (1968), American Mathematical Society, Providence, RI
25. Landis E.M. Second Order Equations of Elliptic and Parabolic Type. Translation of Mathematical Monographs (1998), American Mathematical Society, Providence, RI
26. Lieberman G.M. Second Order Parabolic Differential Equations (1996), World Scientific Publishing Co. Inc., River Edge, NJ
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
29. Mierczyński J. Globally positive solutions of linear PDEs of second order with Robin boundary conditions. J. Math. Anal. Appl. 209 (1997) 47-59
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
31. Mierczyński J. The principal spectrum for linear nonautonomous parabolic pdes of second order: Basic properties. J. Differential Equations 168 (2000) 453-476
32. Mierczyński J., and Shen W. Exponential separation and principal Lyapunov exponent/spectrum for random/nonautonomous parabolic equations. J. Differential Equations 191 (2003) 175-205
33. Moser J. A Harnack inequality for parabolic differential equations. Comm. Pure Appl. Math. 17 (1964) 101-134 Correction in
34. Moser J. A Harnack inequality for parabolic differential equations. Comm. Pure Appl. Math. 20 (1967) 231-236
35. Nishio M. The uniqueness of positive solutions of parabolic equations of divergence form on an unbounded domain. Nagoya Math. J. 130 (1993) 111-121
36. Poláčik P. Parabolic equations: asymptotic behavior and dynamics on invariant manifolds. In: Fiedler B. (Ed). Handbook on Dynamical Systems, vol. 2 (2002), Elsevier, Amsterdam 835-883
37. Poláčik P. On uniqueness of positive entire solutions and other properties of linear parabolic equations. Discrete Contin. Dynamical Systems 12 (2005) 13-26
38. Poláčik P., and 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
39. Poláčik P., and 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
40. Poláčik P., and Tereščák I. Exponential separation and invariant bundles for maps in ordered Banach spaces with applications to parabolic equations. J. Dynamics Differential Equations 6 1 (1994) 245-246
41. Ruelle D. Analycity properties of the characteristic exponents of random matrix products. Adv. in Math. 32 (1979) 68-80
42. Shen W., and Yi Y. Almost automorphic and almost periodic dynamics in skew-product semiflows. Mem. Amer. Math. Soc. 647 (1998) 93
43. 1 smooth strongly monotone discrete-time dynamical systems, Preprint
44. I. Tereščák, Dynamical systems with discrete Lyapunov functionals, PhD thesis, Comenius University, 1994