Global Existence and Blow-Up for a Kirchhoff-Type Hyperbolic Problem with Logarithmic Nonlinearity

This paper deals with a Kirchhoff-type hyperbolic problem with logarithmic nonlinearity. Firstly, by employing the classical Galerkin method and potential well method, we show the existence of the global weak solution with subcritical initial energy and critical initial energy. Secondly, we obtain the finite time blow up results with subcritical initial energy and critical initial energy by using the concavity method. In addition, under some suitable conditions, we also estimate the upper and lower bounds of the blow-up time for the blow-up solution by using the differential inequality technique. Finally, we establish a new finite time blow up condition which is independent of the depth of the potential well and we obtain the upper bound of the blow-up time for the blow-up solution. Furthermore, we also prove the weak solution can blow up in finite time at arbitrary initial energy level by using this condition.