Time averaging for nonautonomous/random linear parabolic equations

Discrete and Continuous Dynamical Systems - Series B

Volumn 9, Issue 3-4, 2008, Pages 661-669

  Mierczyński, Janusz ^a   Shen, Wenxian ^b  

^a WROCLAW UNIVERSITY OF TECHNOLOGY   (Poland)

^b AUBURN UNIVERSITY   (United States)

Author keywords

Averaging; Nonautonomous linear partial differential equation of parabolic type; Principal Lya punov exponent; Principal spectrum; Random linear partial differential equation of parabolic type

Indexed keywords

EID: 73149089428     PISSN: 15313492     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (41)

1. R. A. Adams, "Sobolev Spaces," Pure and Applied Mathematics, Vol.65, Academic Press, New York-London, 1975.
2. H. Amann, Existence and regularity for semilinear parabolic evolution equations, Ann. Scuola Norm. Sup. Pisa C1. Sci. (4), 11 (1984), 593-676.
3. H. Amann, "Linear and Quasilinear Parabolic Problems, Vol. I: Abstract Linear Theory," Monogr. Math., 89, Birkhäuser Boston, Inc., Boston, MA, 1995.
4. H. Amann, Quasilinear evolution equations and parabolic systems, Trans. Amer. Math. Soc, 292 (1986), 191-227.
5. H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, in "Function spaces, differential operators and nonlinear analysis" (Friedrichroda, 1992), 9-126, Teubner-Texte Math., 133, Teubner, Stuttgart, 1993.
6. L. Arnold, "Random Dynamical Systems," Springer Monogr. Math., Springer, Berlin, 1998.
7. D. G. Aronson, Non-negative solutions of linear parabolic equations, Ann. Scuola Norm. Sup. Pisa (3), 22 (1968), 607-694
8. Ann. Scuola Norm. Sup. Pisa (3), 25 (1971), 221-228.
9. C. Bandle, "Isoperimetric Inequalities and Applications," Monographs and Studies in Mathematics, 7, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1980.
10. J. Bergh and J. Löfström, "Interpolation Spaces. An Introduction," Grundlehren Math. Wiss., No. 223, Springer, Berlin-New York, 1976.
11. R. S. Cantrell and C. Cosner, "Spatial Ecology via Reaction-Diffusion Equations," Wiley Series in Mathematical and Computational Biology, Wiley, Chichester, 2003.
12. D. Daners, Heat kernel estimates for operators with boundary conditions, Math. Nachr., 217 (2000), 13-41.
13. R. Dautray and J.-L. Lions, "Mathematical Analysis and Numerical Methods for Science and Technology. Vol. 5: Evolution problems. I," with the collaboration of M. Artola, M. Cessenat and H. Lanchon, translated from the French by A. Craig, Springer, Berlin, 1992.
14. G. B. Folland, "Real Analysis, Modern Techniques and Their Applications," Second Edition, Pure Appl. Math. (N. Y.), John Wiley & Sons, Inc., New York, 1999.
15. A. Friedman, "Partial Differential Equations," Holt, Rinehart and Winston, New York-Montreal, Que-London, 1969.
16. A. Friedman, "Partial Differential Equations of Parabolic Type," Prentice-Hall, Inc., Engle-wood Cliffs, N.J., 1964.
17. P. Hess, "Periodic-Parabolic Boundary Value Problems and Positivity," Pitman Res. Notes Math. Ser., 247, Longman/Wiley, Harlow/New York, 1991.
18. J. Húska, Harnack inequality and exponential separation for oblique derivative problems on Lipschitz domains, J. Differential Equations, 226 (2006), 541-557.
19. J. Húska, Exponential separation and principal Floquet bundles for linear parabolic equations on general bounded domains: the divergence case, Indiana Univ. Math. J., 55 (2006), 1015-1043.
20. J. Húska and P. Poláčlk, The principal Floquet bundle and exponential separation for linear parabolic equations, J. Dynam. Differential Equations, 16 (2004), 347-375.
21. J. Huska, P. Poláčik and M. V. Safonov, Principal eigenvalues, spectral gaps and exponential separation between positive and sign-changing solutions of parabolic equations, Discrete Contin. Dyn. Syst., (2005), suppl., 427-435.
22. J. Húska, P. Poláčik and M. V. Safonov, Harnack inequality, exponential separation, and perturbations of principal Floquet bundles for linear parabolic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, Ann. Inst: H. Poincaré Anal. Non Linéaire, 24 (2007), 711-739.
23. V. Hutson, W. Shen and G. Vickers, Estimates for the principal spectrum point for certain time-dependent parabolic operators, Proc. Amer. Math. Soc, 129 (2000), 1669-1679. (Pubitemid 33766226)
24. R. A. Johnson, On almost-periodic linear differential systems of Millionshchikov and Vino-grad, J. Math. Anal. Appl., 85 (1982), 452-460.
25. R. A. Johnson, K. J. Palmer and G. R. Sell, Ergodic properties of linear dynamical systems, SIAM J. Math. Anal., 18 (1987), 1-33.
26. U. Krengel, "Ergodic Theorems," with a supplement by A. Brunei, de Gruyter Stud. Math. 6, Walter de Gruyter, Berlin, 1985.
27. O. A. Ladyzhenskaya [O. A. Ladyženskaja], V. A. Solonnikov and N. N. Uraĺtseva [N. N. Ural'ceva], "Linear and Quasilinear Equations of Parabolic Type," translated from the Russian by S. Smith, Transl. Math. Monogr., Vol. 23, Amer. Math. Soc, Providence, RI, 1967.
28. G. M. Lieberman, "Second Order Parabolic Differential Equations," World Scientific, River Edge, NJ, 1996.
29. J. Mierczyński, Globally positive solutions of linear parabolic PDEs of second order with Robin boundary conditions, J. Math. Anal. Appl., 209 (1997), 47-59.
30. J. Mierczyński, 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. J. Mierczyński, The principal spectrum for linear nonautonomous parabolic PDEs of second order: Basic properties, in: Special issue in celebration of Jack K. Hale's 70th birthday, Part 2 (Atlanta, GA/Lisbon, 1998), J. Differential Equations, 168 (2000), 453-476.
32. J. Mierczyński and W. Shen, Exponential separation and principal Lyapunov exponent/spectrum for random/nonautonomous parabolic equations, J. Differential Equations, 191 (2003), 175-205.
33. J. Mierczyński and W. Shen, "Spectral Theory for Random and Nonautonomous Parabolic Equations and Applications," Chapman & Hall/CRC Monogr. Surv. Pure Appl. Math., Chapman & Hall/CRC, Boca Raton, FL, in press.
34. J. Oxtoby, Ergodic sets, Bull. Amer. Math. Soc, 58 (1952), 116-136.
35. F. D. Penning, Continuous dependence results for parabolic problems with time-dependent coefficients, Quaestiones Math., 14 (1991), 33-49.
36. P. Poláčik, On uniqueness of positive entire solutions and other properties of linear parabolic equations, Discrete Contin. Dyn. Syst., 12 (2005), 13-26.
37. P. Poláčik and I. Tereśčák, Exponential separation and invariant bundles for maps in ordered Banach spaces with applications to parabolic equations, J. Dynam. Differential Equations, 5 (1993), 279-303;
38. Erratum, J. Dynam. Differential Equations, 6 (1994), 245-246.
39. W. Shen and G. Vickers, Spectral theory for general nonautonomous/random dispersal evolution operators, J. Differential Equations, 235 (2007), 262-297.
40. H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators," North-Holland Math. Library, 18, North-Holland, Amsterdam-New York, 1978.
41. A. Yagi, On the abstract linear evolution equations in Banach spaces, J. Math. Soc. Japan, 28 (1976), 290-303.