Geodesic
On the Limit Behavior of the Trajectory Attractor of a Nonlinear Hyperbolic Equation Containing a Small Parameter at the Highest Derivative
Matematičeskie zametki, Tome 85 (2009) no. 5, pp. 745-753.

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We study the trajectory attractor of a nonlinear nonautonomous hyperbolic equation with dissipation depending on a small parameter. The nonlinear function appearing in this equation does not satisfy the Lipschitz condition. It is shown that, as the small parameter tends to zero, the trajectory attractor of the hyperbolic equation converges to the trajectory attractor of the limit parabolic equation in the corresponding topology.
Keywords: nonlinear hyperbolic equation, trajectory attractor, dissipation, Lipschitz condition, Cauchy problem, translation compactness, attracting set. 
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A. S. Lyapin. On the Limit Behavior of the Trajectory Attractor of a Nonlinear Hyperbolic Equation Containing a Small Parameter at the Highest Derivative. Matematičeskie zametki, Tome 85 (2009) no. 5, pp. 745-753. http://geodesic.mathdoc.fr/item/MZM_2009_85_5_a9/

[1] V. V. Chepyzhov, M. I. Vishik, “Perturbation of trajectory attractors for dissipative hyperbolic equations”, The Maz'ya Anniversary Collection, Vol. 2 (Rostock, 1998), Oper. Theory Adv. Appl., 110, Birkhäuser, Basel, 1999, 33–54 | MR | Zbl

[2] A. Haraux, “Two remarks on hyperbolic dissipative problems”, Nonlinear Partial Differential Equations and Their Applications, Collége de France seminar, Vol. VII (Paris, 1983–1984), Res. Notes in Math., 122, Pitman, Boston, MA, 1985, 161–179 | MR | Zbl

[3] X. Mora, J. Solá-Morales, “Existence and nonexistence of finite-dimensional globally attracting invariant manifolds in semilinear damped wave equations”, Dynamics of Infinite-Dimensional Systems (Lisbon, 1986), NATO Adv. Sci. Inst. Ser. F Comput. Systems Sci., 37, Springer-Verlag, Berlin, 1987, 187–210 | MR | Zbl

[4] V. V. Chepyzhov, M. I. Vishik, Attractors for Equations of Mathematical Physics, Amer. Math. Soc. Colloq. Publ., 49, Amer. Math. Soc., Providence, RI, 2002 | MR | Zbl

[5] A. S. Lyapin, “Usrednenie traektornogo attraktora nelineinogo volnovogo uravneniya s bystro ostsilliruyuschei pravoi chastyu”, Matem. zametki, 82:3 (2007), 390–394 | MR | Zbl

[6] A. V. Babin, M. I. Vishik, Attraktory evolyutsionnykh uravnenii, Nauka, M., 1989 | MR | Zbl