A note on the dispersive-dissipative wave equation having strong damping and nonlinear sources

In this paper, we delve into an initial-boundary value problem of the dispersive-dissipative wave equation having strong damping, $ u_{tt}-\Delta u-\Delta u_{t}+\Delta^{2} u-\Delta u_{tt} = a|u|^{p-1} u-b|u|^{q-1} u \quad \text{in}\ \Omega\times(0, T_{\max}), $where $ \Omega $ is an open bounded domain in $ \mathbb{R}^{n} $ with smooth boundary; $ T_{\max} $ is the maximal existence time; $ a, b>0 $, $ p, q>1 $ are constants. Through the combination of the contraction mapping principle and the Faedo-Galerkin method, we achieve the local existence and uniqueness of the weak solution. For $ p>q $ or $ p = q $ and $ a>b $, we obtain the existence of the global weak solution, and the energy decay estimate. Moreover, based on the auxiliary function approach, we demonstrate that the weak solutions blow up in finite time under distinct conditions related to the initial energy. Furthermore, we succeed in obtaining lower and upper bounds for blow-up time (life span). For $ p<q $ or $ p = q $ and $ a\leq b $, we ascertain that all the weak solutions are globally bounded.