Barbara Szomolay
Decay of solutions of some degenerate hyperbolic equations of Kirchhoff type

Comment.Math.Univ.Carolinae 44,1 (2003) 71-84.

Abstract:In this paper we study the asymptotic behavior of solutions to the damped, nonlinear vibration equation with self-interaction $$ \mathaccent "707F {u}= - \gamma \dot {u} + m(\|\nabla u\|^2) \Delta u - \delta |u|^{\alpha }u + f, $$ 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$.

Keywords: asymptotic behavior of solutions, hyperbolic PDE of degenerate type
AMS Subject Classification: 35B40, 35L80