Summary: In this paper, first we prove the existence of global solutions in Sobolev spaces for the initial boundary value problem of the wave equation of
\[ (|u'|^{l-2}u')'-\Delta_{\varphi}u+\sigma(t) g(u')=0 \quad\text{in } \Omega\times \mathbb R_+, \]
where \(\Delta_{\varphi}=\sum_{i=1}^n \partial_{x_i}l(\varphi (|\partial_{x_i}|^2)\partial_{x_i})\). Then we prove general stability estimates using multiplier method and general weighted integral inequalities. Without imposing any growth condition at the origin on \(g\) and \(\varphi\), we show that the energy of the system is bounded above by a quantity, depending on \(\varphi, \sigma\) and \(g\), which tends to zero (as time approaches infinity). These estimates allows us to consider large class of functions \(g\) and \(\varphi\) with general growth at the origin. We give some examples to illustrate how to derive from our general estimates the polynomial, exponential or logarithmic decay. The results of this paper improve and generalize many existing results in the literature, and generate some interesting open problems.