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

ABSTRACT This paper is concerned with the properties of solutions to the following fourth‐order wave equation with exponential‐type nonlinearity: where the exponential nonlinearity 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 . Subsequently, when and is subcritical, we provide a sufficient condition for finite time blow‐up with arbitrarily positive initial energy, which permits the ‐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 is critical, we also establish a criterion for finite time blow‐up with negative initial energy. Finally, in the specific case where and the initial energy 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.