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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
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Category: math
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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
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Date available: 2009-09-24T10:35:54Z
Last updated: 2016-04-07
Stable URL: http://hdl.handle.net/10338.dmlcz/127594
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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
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