# Decay of solutions of some degenerate hyperbolic equations of Kirchhoff type

Commentationes Mathematicae Universitatis Carolinae (2003)

- Volume: 44, Issue: 1, page 71-84
- ISSN: 0010-2628

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topSzomolay, Barbara. "Decay of solutions of some degenerate hyperbolic equations of Kirchhoff type." Commentationes Mathematicae Universitatis Carolinae 44.1 (2003): 71-84. <http://eudml.org/doc/249197>.

@article{Szomolay2003,

abstract = {In this paper we study the asymptotic behavior of solutions to the damped, nonlinear vibration equation with self-interaction \[ \ddot\{u\}= - \gamma \dot\{u\} + m(\Vert \nabla u\Vert ^2) \Delta u - \delta |u|^\{\alpha \}u + f, \]
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$.},

author = {Szomolay, Barbara},

journal = {Commentationes Mathematicae Universitatis Carolinae},

keywords = {asymptotic behavior of solutions; hyperbolic PDE of degenerate type; asymptotic behavior of solution; hyperbolic partial differential equation of degenerate type; exponential decay},

language = {eng},

number = {1},

pages = {71-84},

publisher = {Charles University in Prague, Faculty of Mathematics and Physics},

title = {Decay of solutions of some degenerate hyperbolic equations of Kirchhoff type},

url = {http://eudml.org/doc/249197},

volume = {44},

year = {2003},

}

TY - JOUR

AU - Szomolay, Barbara

TI - Decay of solutions of some degenerate hyperbolic equations of Kirchhoff type

JO - Commentationes Mathematicae Universitatis Carolinae

PY - 2003

PB - Charles University in Prague, Faculty of Mathematics and Physics

VL - 44

IS - 1

SP - 71

EP - 84

AB - In this paper we study the asymptotic behavior of solutions to the damped, nonlinear vibration equation with self-interaction \[ \ddot{u}= - \gamma \dot{u} + m(\Vert \nabla u\Vert ^2) \Delta u - \delta |u|^{\alpha }u + f, \]
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$.

LA - eng

KW - asymptotic behavior of solutions; hyperbolic PDE of degenerate type; asymptotic behavior of solution; hyperbolic partial differential equation of degenerate type; exponential decay

UR - http://eudml.org/doc/249197

ER -

## References

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- Dix J.G., Decay of solutions of a degenerate hyperbolic equation, Electron. J. Diff. Equations, Vol. 1998 (1998), No. 21, pp.1-10. Zbl0911.35075MR1637075
- 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. (1991)

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