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In this paper we study the asymptotic behavior of solutions to the damped, nonlinear vibration equation with self-interaction $$\ddot{u}=-\gamma \dot{u}+{m(\parallel \nabla u\parallel }^2{)\Delta u-\delta |u|}^{\alpha }u+f,$$ which is known as degenerate if $m(·)\ge 0$, and non-degenerate if $m(·)\ge m_0>0$. 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...