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Title: On solutions of quasilinear wave equations with nonlinear damping terms (English)
Author: Park, Jong Yeoul
Author: Bae, Jeong Ja
Language: English
Journal: Czechoslovak Mathematical Journal
ISSN: 0011-4642 (print)
ISSN: 1572-9141 (online)
Volume: 50
Issue: 3
Year: 2000
Pages: 565-585
Summary lang: English
Category: math
Summary: In this paper we consider the existence and asymptotic behavior of solutions of the following problem: \[ u_{tt}(t,x)-(\alpha +\beta \Vert \nabla u(t,x)\Vert _2^2 +\beta \Vert \nabla v(t,x)\Vert _2^2)\Delta u(t,x) +\delta |u_t(t,x)|^{p-1}u_t(t,x) \quad =\mu |u(t,x)|^{q-1}u(t,x), \quad x \in \Omega ,\quad t \ge 0, v_{tt}(t,x)-(\alpha +\beta \Vert \nabla u(t,x)\Vert _2^2+ \beta \Vert \nabla v(t,x)\Vert _2^2) \Delta v(t,x) +\delta |v_t(t,x)|^{p-1}v_t(t,x) \quad =\mu |v(t,x)|^{q-1}v(t,x), \quad x \in \Omega ,\quad t \ge 0, u(0,x)=u_0(x),\quad u_t(0,x)=u_1(x), \quad x \in \Omega , v(0,x)=v_0(x),\quad v_t(0,x)=v_1(x), \quad x \in \Omega , u|_{_{\partial \Omega }}=v|_{_{\partial \Omega }}=0 \] where $q > 1$, $ p \ge 1$, $ \delta >0$, $ \alpha > 0$, $ \beta \ge 0 $, $\mu \in \mathbb R $ and $\Delta $ is the Laplacian in $\mathbb R^N$. (English)
Keyword: quasilinear wave equation
Keyword: existence and uniqueness
Keyword: asymptotic behavior
Keyword: Galerkin method
MSC: 35B35
MSC: 35L15
MSC: 35L70
MSC: 65M60
idZBL: Zbl 1079.35533
idMR: MR1777478
Date available: 2009-09-24T10:35:54Z
Last updated: 2016-04-07
Stable URL:
Reference: [1] E. H. Brito: Nonlinear Initial Boundary Value Problems.Nonlinear Anal. 11 (1987), 125–137. Zbl 0613.34013, MR 0872045, 10.1016/0362-546X(87)90031-9
Reference: [2] C. Corduneanu: Principles of Differential and Integral Equations.Chelsea Publishing Company, The Bronx, New York, 1977. MR 0440097
Reference: [3] R. Ikehata: On the Existence of Global Solutions for some Nonlinear Hyperbolic Equations with Neumann Conditions.T R U Math. 24 (1988), 1–17. Zbl 0707.35094, MR 0999375
Reference: [4] T. Matsuyama, R. Ikehata: On Global Solutions and Energy Decay for the Wave Equations of Kirchhoff type with Nonlinear Damping terms.J. Math. Anal. Appl. 204 (1996), 729–753. MR 1422769, 10.1006/jmaa.1996.0464
Reference: [5] M. Nakao: Asymptotic Stability of the Bounded or Almost Periodic Solutions of the Wave Equations with Nonlinear Damping terms.J. Math. Anal. Appl. 58 (1977), 336–343. MR 0437890, 10.1016/0022-247X(77)90211-6
Reference: [6] K. Narasimha: Nonlinear Vibration of an Elastic String.J. Sound Vibration 8 (1968), 134–146. 10.1016/0022-460X(68)90200-9
Reference: [7] K. Nishihara, Y. Yamada: On Global Solutions of some Degenerate Quasilinear Hyperbolic Equation with Dissipative Damping terms.Funkcial. Ekvac. 33 (1990), 151–159. MR 1065473
Reference: [8] K. Ono: Global Existence, Decay and Blowup of Solutions for some Mildly Degenerate Nonlinear Kirchhoff Strings.J. Differential Equations 137 (1997), 273–301. Zbl 0879.35110, MR 1456598, 10.1006/jdeq.1997.3263
Reference: [9] M. D. Silva Alves: Variational Inequality for a Nonlinear Model of the Oscillations of Beams.Nonlinear Anal. 28 (1997), 1101–1108. Zbl 0871.35064, MR 1422803


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