A blow-up criterion for a $p$-Laplacian-type pseudo-parabolic equation involving singular potential and logarithmic source

The main goal of this paper was to develop new skills to investigate the blow-up properties of solutions to an initial-boundary value problem for a $p$-Laplacian-type pseudo-parabolic equation with singular potential and logarithmic source. By establishing a crucial reverse Sobolev inequality, we derived that the solutions blow up in finite time under a new blow-up criterion and we estimated the lifespan of the weak solutions from both above and below. It is worthy to point out that our blow-up criterion implies that this problem admits finite time blow-up solutions at an arbitrarily high initial energy level. From methods to results, we partially extended some results obtained in recent literatures.