This work examines a nonlinear wave equation characterized by a damping term whose exponent varies, along with a delay component that changes over time. The mathematical representation of the problem is provided below. \(\nu _{tt} - \Delta _x \nu + \gamma _1(t) \, \nu _t|\nu _t|^{\varpi (x)-2} + \gamma _2(t) \, \nu _t(x, t - \varrho (t))| \nu _t(x, t-\varrho (t))|^{\varpi (x)-2} = 0.\) To demonstrate the existence of global solutions, we impose suitable conditions on the functions \(\gamma _1\) , \(\gamma _2\) , \(\varpi\) and \(\varrho\) . This is achieved through the compactness method, specifically utilizing the Faedo–Galerkin approach. Furthermore, by applying the multiplier technique in conjunction with an integral inequality of Komornik type, we establish a quantitative assessment for how swiftly the system’s energy diminishes over time.
