# Article

 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: http://hdl.handle.net/10338.dmlcz/127594 . 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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