The Decay and Extinction of W2,p$$ {W}^{2,p} $$‐Norm and New Blow‐Up Phenomena for a Singular p‐Biharmonic Parabolic Equation With Logarithmic Nonlinearity

ABSTRACT 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 ‐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 an improved Hardy–Sobolev inequality and applying a non‐concavity method, we establish the four blow‐up theorems independent of the potential well depth with 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 is a nonlinear function of . These generalize previous research results from three aspects: long time decay, extinction and blow‐up.