Bounds for lifespan of solutions to strongly damped semilinear wave equations with logarithmic sources and arbitrary initial energy

Abstract

<p style='text-indent:20px;'>The main aim of this paper is to deal with the upper and lower bounds for blow-up time of solutions to the following equation: <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ u_{tt}-\Delta u-\Delta u_{t} = |u|^{p-2}u\log|u|, $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>which has been studied in [<xref ref-type="bibr" rid="b5">5</xref>]. For high initial energy, it is well known that the classical potential well method is not effective. In order to overcome this difficulty, the authors apply the new energy estimate method to establish the lower bound of the <inline-formula><tex-math id="M1">\begin{document}$ L^{2}(\Omega) $\end{document}</tex-math></inline-formula> norm of the solution. Furthermore, the authors construct a new control functional and combine energy inequalities with the concavity argument to prove that the solution blows up in finite time for high initial energy. Meanwhile, an estimate of the upper bound of blow-up time is also obtained. Finally, a lower bound for blow-up time is obtained by introducing a new control functional. These results fill the gap of [<xref ref-type="bibr" rid="b5">5</xref>].