Abstract
We consider an abstract system of coupled nonlinear parabolic-hyperbolic partial differential equations. This system describes, e.g., thermoelastic phenomena in various physical bodies. Several results on the existence of invariant exponentially attracting manifolds for similar problems were obtained earlier. In the present paper, we prove the existence of such an invariant manifold under less restrictive conditions for a broader class of problems.
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References
S. Jiang and R. Racke, “Evolution equations in thermoelasticity,” Monogr. Surv. Pure Appl. Math., 112 (2000).
O. B. Lykova and Yu. A. Mitropol’skii, Integral Manifolds in Nonlinear Mechanics [in Russian], Nauka, Moscow (1973).
A. W. Leung, “Asymptotically stable invariant manifold for coupled nonlinear parabolic-hyperbolic partial differential equations,” J. Different. Equat., 187, 184–200 (2003).
I. D. Chueshov, “A reduction principle for coupled nonlinear parabolic-hyperbolic partial differential equations,” J. Evol. Equat. (to appear).
I. D. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems [in Russian], Acta, Kharkov (1999).
R. Temam, Infinite-Dimensional Dynamic Systems in Mechanics and Physics, Springer, New York (1988).
M. Miclavčič, “A sharp condition for existence of an inertial manifold,” J. Dynam. Different. Equat., 3, 437–456 (1991).
J. Lagnese, Boundary Stabilization of Thin Plates, SIAM, Philadelphia (1989).
J. Mallet-Paret and G. Sell, “Inertial manifolds for reaction-diffusion equations in higher dimension,” J. Amer. Math. Soc., 1, 805–866 (1988).
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 12, pp. 1684–1697, December, 2005.
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Fastovskaya, T.B. Invariant manifolds for coupled nonlinear parabolic-hyperbolic partial differential equations. Ukr Math J 57, 1977–1994 (2005). https://doi.org/10.1007/s11253-006-0043-3
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DOI: https://doi.org/10.1007/s11253-006-0043-3