Global and Non-Global Solutions for Pseudo-Parabolic Equation with Singular Potential

Abstract

This paper considers initial boundary value to a pseudo-parabolic equation with singular potential $\frac{u_t}{|x|^s}-\Delta u_t-\Delta u =|u|^{p-2}u$ with $2<p<\frac{2N}{N−2},$ which was studied in [1] by Lian et al. They dealt with the global existence, asymptotic behavior with low initial level $J(u_0)≤d$ and got the blow-up conditions of solutions with low and high initial level. In this paper, we give a new blow-up result which independent of the initial Nehari functional $I(u_0)$, and estimate the lower bound for blow-up time under some conditions. Finally, the precise exponential decay estimate is obtained for global solution with some conditions.