Geodesic
Decay of solutions of some degenerate hyperbolic equations of Kirchhoff type
Commentationes Mathematicae Universitatis Carolinae, Tome 44 (2003) no. 1, pp. 71-84.

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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$.
Classification : 35B40, 35L20, 35L70, 35L80, 45K05, 74H45
Keywords: asymptotic behavior of solutions; hyperbolic PDE of degenerate type 
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     title = {Decay of solutions of some degenerate hyperbolic equations of {Kirchhoff} type},
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Szomolay, Barbara. Decay of solutions of some degenerate hyperbolic equations of Kirchhoff type. Commentationes Mathematicae Universitatis Carolinae, Tome 44 (2003) no. 1, pp. 71-84. http://geodesic.mathdoc.fr/item/CMUC_2003__44_1_a5/