| Title:
|
On the existence of solutions for some nondegenerate nonlinear wave equations of Kirchhoff type (English) |
| Author:
|
Park, Jong Yeoul |
| Author:
|
Bae, Jeong Ja |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
52 |
| Issue:
|
4 |
| Year:
|
2002 |
| Pages:
|
781-795 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $\Omega $ be a bounded domain in ${\mathbb{R}}^n$ with a smooth boundary $\Gamma $. In this work we study the existence of solutions for the following boundary value problem: \[ \frac{\partial ^2 y}{\partial t^2}-M\biggl (\int _\Omega |\nabla y|^2\mathrm{d}x\biggr ) \Delta y -\frac{\partial }{\partial t}\Delta y=f(y) \quad \text{in} Q=\Omega \times (0,\infty ),.1 y=0 \quad \text{in} \Sigma _1=\Gamma _{\!1} \times (0,\infty ), M\biggl (\int _\Omega |\nabla y|^2\mathrm{d}x\biggr ) \frac{\partial y}{\partial \nu } +\frac{\partial }{\partial t}\Bigl (\frac{\partial y}{\partial \nu }\Bigr )=g \quad \text{in} \Sigma _0=\Gamma _{\!0} \times (0,\infty ), y(0)=y_0,\quad \frac{\partial y}{\partial t}\,(0)=y_1 \quad \text{in} \quad \Omega , \qquad \mathrm{(1)}\] where $M$ is a $C^1$-function such that $M(\lambda ) \ge \lambda _0 >0$ for every $\lambda \ge 0$ and $f(y)=|y|^\alpha y$ for $\alpha \ge 0$. (English) |
| Keyword:
|
existence and uniqueness |
| Keyword:
|
Galerkin method |
| Keyword:
|
nondegenerate wave equation |
| MSC:
|
35D05 |
| MSC:
|
35L15 |
| MSC:
|
35L20 |
| MSC:
|
35L70 |
| MSC:
|
35L75 |
| MSC:
|
35L80 |
| MSC:
|
65M60 |
| idZBL:
|
Zbl 1011.35096 |
| idMR:
|
MR1940059 |
| . |
| Date available:
|
2009-09-24T10:56:45Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/127764 |
| . |
| Reference:
|
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| Reference:
|
[2] R. Ikehata: On the existence of global solutions for some nonlinear hyperbolic equations with Neumann conditions.TRU Math. 24 (1988), 1–17. Zbl 0707.35094, MR 0999375 |
| Reference:
|
[3] J. L. Lions: Quelques méthode de résolution des probléme aux limites nonlinéaire.Dunod Gauthier–Villars, Paris (1969). MR 0259693 |
| Reference:
|
[4] T. Matsuyama and 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:
|
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| 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 |
| . |
