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

$$\begin{aligned} u_{tt}-\textrm{div}(|\nabla u|^{p-2}\nabla u)+\Delta ^{2} u-\Delta u_{tt}=|u|^{q-2} u\log |u| \quad \text {in}\ \Omega \times (0,T_{\max }), \end{aligned}$$

where \(\Omega \) is an open bounded domain in \({\mathbb {R}}^{n}\) with smooth boundary; \(T_{\max }(\le +\infty )\) denotes the maximal existence time; \(p,q>2\) are constants. We denote \(q=p\) the critical exponent for blow-up solutions. For \(q<p\), we prove that all the weak solutions are globally bounded even if the initial energy is negative. For \(q\ge p\), 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.