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.
Similar content being viewed by others
References
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).
L. Barreira and C. Valls, “Smooth robustness of exponential dichotomies,” Proc. Amer. Math. Soc., 139, 999–1012 (2011).
W. A. Coppel, “Dichotomies in stability theory,” Lect. Notes Math., 629, (1978).
Yu. Daleckiy and M. Krein, Stability of Solutions of Differential Equations in Banach Space, Amer. Math. Soc., Providence, RI (1974).
D. Henry, “Geometric theory of semilinear parabolic equations,” Lect. Notes Math., 840 (1981).
R. Johnson and G. Sell, “Smoothness of spectral subbundles and reducibility of quasiperiodic linear differential systems,” J. Different. Equat., 41, 262–288 (1981).
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).
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).
I. Kmit, “Smoothing effect and Fredholm property for first-order hyperbolic PDEs,” Oper. Theory: Adv. Appl., 231, 219–238 (2013).
I. Kmit, “Smoothing solutions to initial boundary problems for first-order hyperbolic systems,” Appl. Anal., 90, No. 11, 1609–1634 (2011).
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).
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).
R. Naulin and M. Pinto, “Admissible perturbations of exponential dichotomy roughness,” Nonlin. Anal., 31, 559–571 (1998).
K. J. Palmer, “A perturbation theorem for exponential dichotomies,” Proc. Roy. Soc. Edinburgh A, 106, 25–37 (1987).
V. A. Pliss and G. R. Sell, “Robustness of exponential dichotomies in infinite-dimensional dynamical systems,” J. Dynam. Different. Equat., 11, 471–513 (1999).
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).
R. Sacker and G. Sell, “Dichotomies for linear evolutionary equations in Banach spaces,” J. Different. Equat., 113, 17–67 (1994).
A. M. Samoilenko, Elements of the Mathematical Theory of Multi-Frequency Oscillations, Kluwer, Dordrecht (1991), 327 p.
V. I. Tkachenko, “On the exponential dichotomy of impulsive evolution systems,” Ukr. Math. J., 46, No. 4, 441–448 (1994).
Y. Yi, “A generalized integral manifold theorem,” J. Different. Equat., 102, 153–187 (1993).
T. I. Zelenyak, “On stationary solutions of mixed problems relating to the study of certain chemical processes,” Different. Equat., 2, 98–102 (1966).
Author information
Authors and Affiliations
Additional information
Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 65, No. 2, pp. 236–251, February, 2013.
Rights and permissions
About this article
Cite this article
Kmit, I.Y., Recke, L. & Tkachenko, V.I. Robustness of the exponential dichotomies of boundary-value problems for the general first-order hyperbolic systems. Ukr Math J 65, 260–276 (2013). https://doi.org/10.1007/s11253-013-0776-8
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-013-0776-8