Asymptotic behavior for a singular parabolic $ p $-biharmonic equation with gradient-type logarithmic nonlinearity

This paper deals with the initial boundary value problem for a singular parabolic $ p $-biharmonic equation with gradient-type logarithmic nonlinearity. By the cut-off technique, combined with the methods of Faedo-Galerkin approximation and multiplier, we establish the local solvability. Further, by virtue of the family of potential wells and the modified differential inequality technique, we derive the threshold results of global existence and nonexistence with low initial energy conditions ($ J(u_{0})\leq d $), the decay property, blow-up with negative initial energy, the estimates of blow-up rate, and the bounds of life span of blow-up solutions with subcritical initial energy. Furthermore, we adopt the Hardy-Sobolev inequality to show the finite time blow-up result with arbitrary initial energy.