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Parabolic evolution equations with asymptotically autonomous delay

Author: Roland Schnaubelt
Journal: Trans. Amer. Math. Soc. 356 (2004), 3517-3543
MSC (2000): Primary 35R10; Secondary 34K30, 47D06
Published electronically: November 25, 2003
MathSciNet review: 2055745
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Abstract: We study retarded parabolic non-autonomous evolution equations whose coefficients converge as $t\to\infty$, such that the autonomous problem in the limit has an exponential dichotomy. Then the non-autonomous problem inherits the exponential dichotomy, and the solution of the inhomogeneous equation tends to the stationary solution at infinity. We use a generalized characteristic equation to deduce the exponential dichotomy and new representation formulas for the solution of the inhomogeneous equation.

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  • 1. P. Acquistapace, Evolution operators and strong solutions of abstract linear parabolic equations, Differential Integral Equations 1 (1988), 433-457. MR 90b:34094
  • 2. P. Acquistapace, B. Terreni, A unified approach to abstract linear nonautonomous parabolic equations, Rend. Sem. Mat. Univ. Padova 78 (1987), 47-107. MR 89e:34099
  • 3. H. Amann, Linear and Quasilinear Parabolic Problems. Volume 1: Abstract Linear Theory, Birkhäuser, 1995. MR 96g:34088
  • 4. A. Batkái, Hyperbolicity of linear partial differential equations with delay, Integral Equations Operator Theory 44 (2002), 383-396. MR 2003j:34148
  • 5. A. Batkái, R. Schnaubelt, Asymptotic behaviour of parabolic problems with delays in the highest order derivatives, submitted.
  • 6. C.J.K. Batty, R. Chill, Approximation and asymptotic behaviour of evolution families, Differential Integral Equations 15 (2002), 477-512. MR 2002i:34100
  • 7. C. Chicone, Y. Latushkin, Evolution Semigroups in Dynamical Systems and Differential Equations, Amer. Math. Soc., 1999. MR 2001e:47068
  • 8. K.L. Cooke, Linear functional differential equations of asymptotically autonomous type, J. Differential Equations 7 (1970), 154-174. MR 41:604
  • 9. R. Datko, Not all feedback stabilized systems are robust with respect to small time delays, SIAM J. Control Optim. 26 (1988), 697-713. MR 89c:93057
  • 10. W. Desch, I. Gyori, G. Gühring, Stability of nonautonomous delay equations with a positive fundamental solution, as a preprint in: Tübinger Berichte zur Funktionalanalysis 9 (2000), 125-139.
  • 11. K.J. Engel, R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer, 2000. MR 2000i:47075
  • 12. G. Gühring, Asymptotic properties of nonautonomous evolution equations and nonautonomous retarded equations, Ph.D. thesis, Tübingen, 1999.
  • 13. G. Gühring, F. Räbiger, Asymptotic properties of mild solutions of nonautonomous evolution equations with applications to retarded differential equations, Abstr. Appl. Anal. 4 (1999), 169-194. MR 2001m:34130
  • 14. G. Gühring, F. Räbiger, W. Ruess, Principle of linearized stability for semilinear non-autonomous evolution equations with applications to retarded differential equations, Differential Integral Equations 13 (2000), 503-528. MR 2001c:34124
  • 15. G. Gühring, F. Räbiger, R. Schnaubelt, A characteristic equation for non-autonomous partial functional differential equations, J. Differential Equations 181 (2002), 439-462. MR 2003d:34168
  • 16. D. Guidetti, On the asymptotic behavior of solutions of linear nonautonomous parabolic equations, Boll. Un. Mat. Ital. B (7) 1 (1987), 1055-1076. MR 89d:35077
  • 17. I. Gyori, F. Hartung, J. Turi, Preservation of stability in delay equations under delay perturbations, J. Math. Anal. Appl. 220 (1998), 290-312. MR 99a:34212
  • 18. J.K. Hale, Theory of Functional Differential Equations, Springer, 1977. MR 58:22904
  • 19. J.K. Hale, S.M. Verduyn Lunel, Effects of small delays on stability and control, in: Bart, Gohberg, Ran (Eds.): Operator Theory and Analysis, The M.A. Kaashoek Anniversary Volume, Birkhäuser, 2001, pp. 275-301. MR 2002e:93043
  • 20. T. Kato, Perturbation Theory for Linear Operators, Corrected Printing of 2nd Edition, Springer, 1980. MR 53:11389
  • 21. X.-B. Lin, Exponential dichotomies and homoclinic orbits in functional differential equations, J. Differential Equations 63 (1986), 227-254. MR 87j:34138
  • 22. X.-B. Lin, Exponential dichotomies in intermediate spaces with applications to a diffusively perturbed predator-prey model, J. Differential Equations 108 (1994), 36-63. MR 95c:35139
  • 23. A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser, 1995. MR 96e:47039
  • 24. J. Mallet-Paret, The Fredholm alternative for functional differential equations of mixed type, J. Dynam. Differential Equations 11 (1999), 1-48. MR 2000j:34107
  • 25. J. Prüss, On resolvent operators for linear integrodifferential equations of Volterra type, J. Integral Equations 5 (1983), 211-236. MR 85d:45026
  • 26. A. Rhandi, Extrapolation methods to solve nonautonomous retarded partial differential equations, Studia Math. 126 (1997), 219-233. MR 99c:47058
  • 27. W.M. Ruess, Existence of solutions to partial functional differential equations with delay, in: A.G. Kartsatos (ed.), Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, Marcel Dekker, 1996, 259-288. MR 97b:34085
  • 28. R. Schnaubelt, Exponential bounds and hyperbolicity of evolution equations, Ph.D. thesis, Tübingen, 1996.
  • 29. R. Schnaubelt, Sufficient conditions for exponential stability and dichotomy of evolution equations, Forum Math. 11 (1999), 543-566. MR 2001c:34121
  • 30. R. Schnaubelt, Exponential dichotomy of nonautonomous evolution equations, Habilitation thesis, Tübingen, 1999.
  • 31. R. Schnaubelt, A sufficient condition for exponential dichotomy of parabolic evolution equations, in: G. Lumer, L. Weis (eds.), Evolution Equations and their Applications in Physical and Life Sciences (Proceedings Bad Herrenalb, 1998), Marcel Dekker, 2000. MR 2001m:34132
  • 32. R. Schnaubelt, Asymptotically autonomous parabolic evolution equations, J. Evol. Equ. 1 (2001), 19-37. MR 2002e:34095
  • 33. R. Schnaubelt, Asymptotic behaviour of parabolic nonautonomous evolution equations, Report No. 12 (2002), FB Mathematik und Informatik, University of Halle (preprint).
  • 34. K. Schumacher, On the resolvent of linear nonautonomous partial functional differential equations, J. Differential Equations 59 (1985), 355-387. MR 87b:35164
  • 35. H. Tanabe, Equations of Evolution, Pitman, 1979. MR 82g:47032
  • 36. J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer, 1996. MR 98a:35135
  • 37. A. Yagi, Parabolic equations in which the coefficients are generators of infinitely differentiable semigroups II, Funkcial. Ekvac. 33 (1990), 139-150. MR 91h:47039
  • 38. A. Yagi, Abstract quasilinear evolution equations of parabolic type in Banach spaces, Boll. Un. Mat. Ital. B (7) 5 (1991), 351-368. MR 92h:47100

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Additional Information

Roland Schnaubelt
Affiliation: FB Mathematik und Informatik, Martin-Luther-Universität, 06099 Halle, Germany

Keywords: Retarded parabolic evolution equation, asymptotically autonomous, exponential dichotomy, robustness, convergence of solutions, variation of parameters formula, characteristic equation, evolution semigroup
Received by editor(s): January 18, 2002
Received by editor(s) in revised form: March 27, 2003
Published electronically: November 25, 2003
Article copyright: © Copyright 2003 American Mathematical Society

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