Ukrainian Mathematical Journal

, Volume 65, Issue 2, pp 260–276 | Cite as

Robustness of the exponential dichotomies of boundary-value problems for the general first-order hyperbolic systems

  • I. Ya. Kmit
  • L. Recke
  • V. I. Tkachenko

We examine the robustness of exponential dichotomies of boundary-value problems for general linear first-order one-dimensional hyperbolic systems. It is assumed that the boundary conditions guarantee an increase in the smoothness of solutions in a finite time interval, including the reflection boundary conditions. We show that the dichotomy survives in the space of continuous functions under small perturbations of all coefficients in the differential equations.


Hyperbolic System Evolution Operator Continuous Solution Exponential Dichotomy Nonlocal Boundary Condition 
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  1. 1.
    T. A. Akramov, V. S. Belonosov, T. I. Zelenyak, M. M. Lavrentev (Jr.), M. G. Slinko, and V. S. Sheplev, “Mathematical foundations of modeling of catalytic processes: a review,” Theor. Found. Chem. Eng., 34, No. 3, 295–306 (2000).CrossRefGoogle Scholar
  2. 2.
    L. Barreira and C. Valls, “Smooth robustness of exponential dichotomies,” Proc. Amer. Math. Soc., 139, 999–1012 (2011).CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    W. A. Coppel, “Dichotomies in stability theory,” Lect. Notes Math., 629, (1978).Google Scholar
  4. 4.
    Yu. Daleckiy and M. Krein, Stability of Solutions of Differential Equations in Banach Space, Amer. Math. Soc., Providence, RI (1974).Google Scholar
  5. 5.
    D. Henry, “Geometric theory of semilinear parabolic equations,” Lect. Notes Math., 840 (1981).Google Scholar
  6. 6.
    R. Johnson and G. Sell, “Smoothness of spectral subbundles and reducibility of quasiperiodic linear differential systems,” J. Different. Equat., 41, 262–288 (1981).CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    S.-N. Chow and H. Leiva, “Existence and roughness of the exponential dichotomy for skew-product semiflow in Banach spaces,” J. Different. Equat., 120, 429–477 (1995).CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    I. Kmit, “Classical solvability of nonlinear initial boundary problems for first-order hyperbolic systems,” Int. J. Dynam. Syst. Different. Equat., 1, No. 3, 191–195 (2008).CrossRefMathSciNetGoogle Scholar
  9. 9.
    I. Kmit, “Smoothing effect and Fredholm property for first-order hyperbolic PDEs,” Oper. Theory: Adv. Appl., 231, 219–238 (2013).MathSciNetGoogle Scholar
  10. 10.
    I. Kmit, “Smoothing solutions to initial boundary problems for first-order hyperbolic systems,” Appl. Anal., 90, No. 11, 1609–1634 (2011).CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    I. Kmit and G. Hörmann, “Semilinear hyperbolic systems with nonlocal boundary conditions: reflection of singularities and delta waves,” J. Anal. Appl., 20, No. 3, 637–659 (2001).zbMATHGoogle Scholar
  12. 12.
    M. Lichtner, M. Radziunas, and L. Recke, “Well-posedness, smooth dependence, and center manifold reduction for a semilinear hyperbolic system from laser dynamics,” Math. Meth. Appl. Sci., 30, 931–960 (2007).CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    R. Naulin and M. Pinto, “Admissible perturbations of exponential dichotomy roughness,” Nonlin. Anal., 31, 559–571 (1998).CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    K. J. Palmer, “A perturbation theorem for exponential dichotomies,” Proc. Roy. Soc. Edinburgh A, 106, 25–37 (1987).CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    V. A. Pliss and G. R. Sell, “Robustness of exponential dichotomies in infinite-dimensional dynamical systems,” J. Dynam. Different. Equat., 11, 471–513 (1999).CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    R. K. Romanovskii and L. V. Bel’gart, “On the exponential dichotomy of solutions of the Cauchy problem for a hyperbolic system on a plane,” Differents. Uravn., 46, No. 8, 1125–1134 (2010).MathSciNetGoogle Scholar
  17. 17.
    R. Sacker and G. Sell, “Dichotomies for linear evolutionary equations in Banach spaces,” J. Different. Equat., 113, 17–67 (1994).CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    A. M. Samoilenko, Elements of the Mathematical Theory of Multi-Frequency Oscillations, Kluwer, Dordrecht (1991), 327 p.Google Scholar
  19. 19.
    V. I. Tkachenko, “On the exponential dichotomy of impulsive evolution systems,” Ukr. Math. J., 46, No. 4, 441–448 (1994).CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Y. Yi, “A generalized integral manifold theorem,” J. Different. Equat., 102, 153–187 (1993).CrossRefzbMATHGoogle Scholar
  21. 21.
    T. I. Zelenyak, “On stationary solutions of mixed problems relating to the study of certain chemical processes,” Different. Equat., 2, 98–102 (1966).Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • I. Ya. Kmit
    • 1
    • 2
  • L. Recke
    • 2
  • V. I. Tkachenko
    • 3
  1. 1.Institute for Applied Problems in Mechanics and MathematicsUkrainian National Academy of SciencesLvivUkraine
  2. 2.Humboldt UniversityBerlinGermany
  3. 3.Institute of Mathematics of National Academy of Sciences of UkraineKyivUkraine

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