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

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.