Critical blow-up exponent for a doubly dispersive quasilinear wave equation

In this paper, we study an initial-boundary value problem of the doubly dispersive quasilinear wave equation \(\begin{aligned} u_{tt}-\textrm{div}(|\nabla u|^{p-2}\nabla u)+\Delta ^{2} u-\Delta u_{tt}=|u|^{q-2} u\log |u| \quad \text {in}\ \Omega \times (0,T_{\max }), \end{aligned}\) $$\begin{array}{c}\hfill u_{\mathit{tt}}-{\text{div}(|\mathrm{\nabla }u|}^{p-2}\mathrm{\nabla }u)+{\mathrm{\Delta }}^{2}u-\mathrm{\Delta }{u}_{\mathit{tt}}={|u|}^{q-2}ulog|u|\text{in}\mathrm{\Omega }\times (0,T_{max}),\end{array}$$ where \(\Omega \) $\mathrm{\Omega }$ is an open bounded domain in \({\mathbb {R}}^{n}\) ${\mathbb{R}}^n$ with smooth boundary; \(T_{\max }(\le +\infty )\) $T_{max}(\le +\infty )$ denotes the maximal existence time; \(p,q>2\) $p,q>2$ are constants. We denote \(q=p\) $q=p$ the critical exponent for blow-up solutions. For \(q<p\) $q<p$ , we prove that all the weak solutions are globally bounded even if the initial energy is negative. For \(q\ge p\) $q\ge p$ , we obtain the optimal classification of initial data on the existence of global and blow-up solutions, which is divided into the subcritical, critical, and super critical initial energy in the framework of potential well. By constructing new auxiliary functions, we obtain the upper bounds of blow-up time for different norms.