Studying the damped wave equation with localized initial data in the textitscale-invariant case with mass term and two combinedities

A blow-up result for the wave equation with localized initial data: the scale-invariant damping and mass term with combined nonlinearities

We study the damped wave equation with localized initial data, in the \textit{scale-invariant case} with mass term and two combined nonlinearities.Under some assumptions on the mass and damping coefficients, and respectively, we show that blow-up region and the lifespan bound of the solution of remain the same as the ones obtained in \cite{our2} in the case of a mass-free wave equation, it i.e.