| 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:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/127594 |
| . |
| Reference:
|
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| Reference:
|
[2] C. Corduneanu: Principles of Differential and Integral Equations.Chelsea Publishing Company, The Bronx, New York, 1977. MR 0440097 |
| Reference:
|
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| Reference:
|
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| Reference:
|
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| 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:
|
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| . |
