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On Infinite‐Dimensional Dynamic Systems Generated by Non‐linear Hyperbolic Equations
Author(s) -
Shirikyan A.,
Volevich L.
Publication year - 1998
Publication title -
zamm ‐ journal of applied mathematics and mechanics / zeitschrift für angewandte mathematik und mechanik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 51
eISSN - 1521-4001
pISSN - 0044-2267
DOI - 10.1002/zamm.199807815105
Subject(s) - mathematics , mathematical analysis , polynomial , exponential dichotomy , constant (computer programming) , characteristic equation , zero (linguistics) , constant coefficients , exponential function , plane (geometry) , variable (mathematics) , hyperbolic partial differential equation , complex plane , partial differential equation , differential equation , geometry , linguistics , philosophy , computer science , programming language
We consider high‐order non‐linear hyperbolic equations that are small perturbations of an equation with constant coefficients. It is assumed that the unperturbed equation has a characteristic polynomial whose roots with respect to the variable dual to time are outside an open strip containing the imaginary axis. We establish the property of exponential dichotomy for the equation in question. Under the additional condition that the roots of the characteristic polynomial belong to the left half‐plane we prove that the zero solution is asymptotically stable as t → + ∞.