Bounds for blow-up time in a semilinear pseudo-parabolic equation with nonlocal source

This paper considers the following semilinear pseudo-parabolic equation with a nonlocal source: ut−△ut−△u=up(x,t)∫Ωk(x,y)up+1(y,t)dy,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ u_{t}-\triangle u_{t}-\triangle u=u^{p}(x,t) \int_{\Omega}k(x,y)u^{p+1}(y,t)\,dy, $$\end{document} and it explores the characters of blow-up time for solutions, obtaining a lower bound as well as an upper bound for the blow-up time under different conditions, respectively. Also, we investigate a nonblow-up criterion and compute an exact exponential decay.