Classification of Initial Energy in a Pseudo-parabolic Equation with Variable Exponents and Singular Potential

This paper deals with a pseudo-parabolic equation with singular potential and variable exponents. First, we determine the existence and uniqueness of weak solutions in Sobolev spaces with variable exponents. Second, in the frame of variational methods, we classify the blow-up and the global existence of solutions completely using the initial energy. Third, we obtain lower and upper bounds of blow-up time for all possible initial energy. The results in this paper are compatible with the corresponding problems with constant exponents. Part results of the paper extend the recent ones in Lian et al. (J Differ Equ 269:4914-4959, 2020), Xu and Su (J Funct Anal 264:2732-2763, 2013), and Liu and Yu (J Funct Anal 274:1276-1283, 2018).