We in this manuscript restudy the long time decay, extinction and blow-up for a singular p-biharmonic parabolic equation with logarithmic nonlinearity, which appears in many branches of physics. In the framework of potential well theory and the existence of global solution, by a way of establishing a nonlinearly integral inequality without non-increasing condition, we prove that W 2 , p -norm for the weak solutions is non-increasing, and establish two decay and extinction theorems that incorporate two kinds of polynomial decay, two kinds of exponential decay and two kinds of finite time extinction. By a way of establishing a improved Hardy-Sobolev inequality and applying a non-concavity method, we establish the four blow-up theorems independent of the potential well depth with J ( u 0 ) < 𝔐 ( 0 ) as the blow-up criterion, where two of them are finite time blow-up, one is at least exponential growth and blows up at least at infinity, the last one blows up at infinity, where 𝔐( t) is a nonlinear function of ∫ R n | u ( x , t ) | 2 | x | s d x . These generalize previous research results from three aspects: long time decay, extinction and blow-up.

In this note, we consider the initial boundary value problem for a parabolic equation with logarithmic nonlinearity, which has been studied by Chen et al. (J. Math. Anal. Appl. 2015, 422, 84-98) and Han (J. Math. Anal. Appl. 2019, 474, 513-517). On the one hand, we not only prove the existence of doubly exponential decay solutions, but also find its threshold, and obtain the solutions with ∥ u 0 ∥ 2 2 → 0 + is always zero. On the other hand, we also prove the existence of doubly exponential growth solutions. The reseach results in this note extend previous results from both decay and growth.