Lifespan of solutions to a damped fourth-order wave equation with logarithmic nonlinearity

TL;DR

This paper investigates the lifespan of solutions to a damped fourth-order wave equation with a logarithmic nonlinearity, establishing blow-up criteria, and providing bounds for the blow-up time based on initial energy levels.

Contribution

It introduces new criteria for finite time blow-up and derives bounds for the lifespan of solutions using a novel auxiliary functional and damping effects.

Findings

Finite time blow-up criteria established

Upper bounds for blow-up time derived

Lower bounds for blow-up time obtained

Abstract

This paper is devoted to the lifespan of solutions to a damped fourth-order wave equation with logarithmic nonlinearity $$u_{tt}+\Delta^2u-\Delta u-\omega\Delta u_t+\alpha(t)u_t=|u|^{p-2}u\ln|u|.$$ Finite time blow-up criteria for solutions at both lower and high initial energy levels are established, and an upper bound for the blow-up time is given for each case. Moreover, by constructing a new auxiliary functional and making full use of the strong damping term, a lower bound for the blow-up time is also derived.