Decay of solutions of some degenerate hyperbolic equations of Kirchhoff type

Barbara Szomolay

Commentationes Mathematicae Universitatis Carolinae (2003)

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

Abstract

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In this paper we study the asymptotic behavior of solutions to the damped, nonlinear vibration equation with self-interaction u ¨ = - γ u ˙ + m ( ∥ ∇ u ∥ 2 ) Δ u - δ | u | α u + f , which is known as degenerate if m ( · ) ≥ 0 , and non-degenerate if m ( · ) ≥ 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 α . Our aim is to extend the validity of previous results in [5] to α ≥ 0 both to the degenerate and non-degenerate cases of m . We extend our results to equations with Δ 2 .

How to cite

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Szomolay, 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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  1. Aassila M., Some remarks on a second order evolution equation, Electron. J. Diff. Equations, Vol. 1998 (1998), No. 18, pp.1-6. Zbl0902.35073MR1629704
  2. Aassila M., Decay estimates for a quasilinear wave equation of Kirchhoff type, Adv. Math. Sci. Appl. 9 1 (1999), 371-381. (1999) Zbl0939.35028MR1690380
  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. (1999) MR1732643
  4. Dix J.G., Torrejón R.M., A quasilinear integrodifferential equation of hyperbolic type, Differential Integral Equations 6 (1993), 2 431-447. (1993) MR1195392
  5. 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
  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. (1991) 

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