Qualitative Analysis for a Fourth-Order Wave Equation With Exponential-Type Nonlinearity

This paper is concerned with the properties of solutions to the following fourth-order wave equation with exponential-type nonlinearity:

$u_{tt}+{\Delta }^{2}u+u+\omega {\Delta }^2{u}_t+\mu u_t=f\left(u\right),$

where the exponential nonlinearity $f$ is classified as either subcritical growth or critical growth at infinity, based on the Adams-type inequality. By utilizing the potential well theory and Adams-type inequality, we first prove the existence, stability, and blow-up of solutions at critical energy $E\left(t_0\right)=d$. Subsequently, when $\mu >-\omega {\lambda }_1$ and $f$ is subcritical, we provide a sufficient condition for finite time blow-up with arbitrarily positive initial energy, which permits the $L^2$-inner product of the initial displacement and the initial velocity to be negative. The upper bounds of the blow-up time are then estimated via the concavity method. Furthermore, when $f$ is critical, we also establish a criterion for finite time blow-up with negative initial energy. Finally, in the specific case where $\omega =\mu =0$ and the initial energy $E\left(0\right)>0$ can be arbitrarily large, we employ an improved concavity method, in conjunction with the contradiction argument, to derive a weaker sufficient condition for finite time blow-up.