| Title:
|
Decay of solutions of some degenerate hyperbolic equations of Kirchhoff type (English) |
| Author:
|
Szomolay, Barbara |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
44 |
| Issue:
|
1 |
| Year:
|
2003 |
| Pages:
|
71-84 |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this paper we study the asymptotic behavior of solutions to the damped, nonlinear vibration equation with self-interaction $$ \ddot{u}= - \gamma \dot{u} + m(\|\nabla u\|^2) \Delta u - \delta |u|^{\alpha }u + f, $$ which is known as degenerate if $m(\cdot )\ge 0$, and non-degenerate if $m(\cdot )\ge m_0 > 0$. We would like to point out that, to the author's knowledge, exponential decay for this type of equations has been studied just for the special cases of $\alpha $. Our aim is to extend the validity of previous results in [5] to $\alpha \ge 0 $ both to the degenerate and non-degenerate cases of $m$. We extend our results to equations with $ \Delta^2$. (English) |
| Keyword:
|
asymptotic behavior of solutions |
| Keyword:
|
hyperbolic PDE of degenerate type |
| MSC:
|
35B40 |
| MSC:
|
35L20 |
| MSC:
|
35L70 |
| MSC:
|
35L80 |
| MSC:
|
45K05 |
| MSC:
|
74H45 |
| idZBL:
|
Zbl 1098.35033 |
| idMR:
|
MR2045846 |
| . |
| Date available:
|
2009-01-08T19:27:27Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119368 |
| . |
| Reference:
|
[1] Aassila M.: Some remarks on a second order evolution equation.Electron. J. Diff. Equations, Vol. 1998 (1998), No. 18, pp.1-6. Zbl 0902.35073, MR 1629704 |
| Reference:
|
[2] Aassila M.: Decay estimates for a quasilinear wave equation of Kirchhoff type.Adv. Math. Sci. Appl. 9 1 (1999), 371-381. Zbl 0939.35028, MR 1690380 |
| Reference:
|
[3] Aassila M.: Uniform stabilization of solutions to a quasilinear wave equation with damping and source terms.Comment. Math. Univ. Carolinae 40.2 (1999), 223-226. MR 1732643 |
| Reference:
|
[4] Dix J.G., Torrejón R.M.: A quasilinear integrodifferential equation of hyperbolic type.Differential Integral Equations 6 (1993), 2 431-447. MR 1195392 |
| Reference:
|
[5] Dix J.G.: Decay of solutions of a degenerate hyperbolic equation.Electron. J. Diff. Equations, Vol. 1998 (1998), No. 21, pp.1-10. Zbl 0911.35075, MR 1637075 |
| Reference:
|
[6] Matsuyama T., Ikehata R.: Energy decay for the wave equations II: global existence and decay of solutions.J. Fac. Sci. Univ. Tokio, Sect. IA, Math. 38 (1991), 239-250. |
| . |
