Classification of decay estimate in a 4th-order quasilinear hyperbolic equation with strong damping

Abstract

This paper deals with a 4th-order quasilinear hyperbolic equation involving strong damping and superlinear source,

subject to homogeneous Navier boundary condition, where $\Omega$ is an open bounded domain in ${\mathbb{R}}^n$ $(n>2)$; $p>m\ge r\ge 2$; ${\Delta }_{m}u:=\mathrm{div}(|\nabla u{|}^{m-2}\nabla u)$; and ${\Delta }_r{u}_t:=\mathrm{div}(|\nabla u_t{|}^{r-2}\nabla u_t)$. For the positive initial energy case, we obtain the existence of global solutions, where the decay estimates are divided into five kinds for all the exponent regions. When the initial energy is negative, we arrive at the upper and lower bounds of blow-up time. The $L^2$ inner product $(u_1,u_0)>0$ of the initial data is not a necessary condition on the existence of blow-up solutions in the region $\{p>m>2=r\}$.