Bounds for blow-up solutions of a semilinear pseudo-parabolic equation with a memory term and logarithmic nonlinearity in variable space

Abstract

In this article, we investigate the initial boundary value problem for a pseudo-parabolic equation under the influence of a linear memory term and a logarithmic nonlinear source term \[ u_{t}-\Delta u_{t}+\int _{0}^{t}g( t-s) \Delta u( x,s) \mathrm {d}s-\Delta u\]\[=|u|^{p(\cdot ) -2}u\ln (|u|), \]with a Dirichlet boundary condition. Under appropriate assumptions about the relaxation function $g$, the initial data $u_{0}$ and the function exponent $p$, we not only set the lower bounds for the blow-up time of the solution when blow-up occurs, but also by assuming that the initial energy is negative, we give a new blow-up criterion and an upper bound for the blow-up time of the solution.