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Title: Decay of solutions of some degenerate hyperbolic equations of Kirchhoff type  (English)
Author: Szomolay, Barbara
Language: English
Journal: Commentationes Mathematicae Universitatis Carolinae
ISSN: 0010-2628 (print)
ISSN: 1213-7243 (online)
Volume: 44
Issue: 1
Year: 2003
Pages: 71-84
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Category: math
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Summary: In this paper we study the asymptotic behavior of solutions to the damped, nonlinear vibration equation with self-interaction $$ \ddot{u}= - \gamma \dot{u} + m(\|\nabla u\|^2) \Delta u - \delta |u|^{\alpha }u + f, $$ which is known as degenerate if $m(\cdot )\ge 0$, and non-degenerate if $m(\cdot )\ge m_0 > 0$. We would like to point out that, to the author's knowledge, exponential decay for this type of equations has been studied just for the special cases of $\alpha $. Our aim is to extend the validity of previous results in [5] to $\alpha \ge 0 $ both to the degenerate and non-degenerate cases of $m$. We extend our results to equations with $ \Delta^2$.
Keyword: asymptotic behavior of solutions
Keyword: hyperbolic PDE of degenerate type
MSC: 35B40
MSC: 35L20
MSC: 35L70
MSC: 35L80
MSC: 45K05
MSC: 74H45
idZBL: Zbl 1098.35033
idMR: MR2045846
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Date available: 2009-01-08T19:27:27Z
Last updated: 2012-04-30
Stable URL: http://hdl.handle.net/10338.dmlcz/119368
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Reference: [1] Aassila M.: Some remarks on a second order evolution equation.Electron. J. Diff. Equations, Vol. 1998 (1998), No. 18, pp.1-6. Zbl 0902.35073, MR 1629704
Reference: [2] Aassila M.: Decay estimates for a quasilinear wave equation of Kirchhoff type.Adv. Math. Sci. Appl. 9 1 (1999), 371-381. Zbl 0939.35028, MR 1690380
Reference: [3] Aassila M.: Uniform stabilization of solutions to a quasilinear wave equation with damping and source terms.Comment. Math. Univ. Carolinae 40.2 (1999), 223-226. MR 1732643
Reference: [4] Dix J.G., Torrejón R.M.: A quasilinear integrodifferential equation of hyperbolic type.Differential Integral Equations 6 (1993), 2 431-447. MR 1195392
Reference: [5] Dix J.G.: Decay of solutions of a degenerate hyperbolic equation.Electron. J. Diff. Equations, Vol. 1998 (1998), No. 21, pp.1-10. Zbl 0911.35075, MR 1637075
Reference: [6] Matsuyama T., Ikehata R.: Energy decay for the wave equations II: global existence and decay of solutions.J. Fac. Sci. Univ. Tokio, Sect. IA, Math. 38 (1991), 239-250.
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