Improvement on the blow-up of the wave equation with the scale-invariant damping and combined nonlinearities

We consider in this article the damped wave equation, in the $\mathit{\text{scale-invariant}}\mathit{\text{case}}$ with combined two nonlinearities, which reads as follows: \begin{displaymath} \d (E) \hspace{1cm} u_{tt}-\Delta u+\frac{\mu}{1+t}u_t=|u_t|^p+|u|^q, \quad \mbox{in}\ \R^N\times[0,\infty), \end{displaymath} with small initial data.\ Compared to our previous work \cite{Our}, we show in this article that the first hypothesis on the damping coefficient $\mu$, namely $\mu <\frac{N(q-1)}{2}$, can be removed, and the second one can be extended from $(0,{\mu }_{\ast }\mathrm{/}2)$ to $(0,{\mu }_{\ast })$ where ${\mu }_{\ast }>0$ is solution of $(q-1)((N+{\mu }_{\ast }-1)p-2)=4$. Indeed, owing to a better understanding of the influence of the damping term in the global dynamics of the solution, we think that this new interval for $\mu$ describe better the threshold between the blow-up and the global existence regions. Moreover, taking advantage of the techniques employed in the problem $(E)$, we also improve the result in \cite{LT2,Palmieri} in relationship with the Glassey conjecture for the solution of $(E)$ but without the nonlinear term $\mathrm{\mid }u{\mathrm{\mid }}^q$. More precisely, we extend the blow-up region from $p\in (1,p_G(N+\sigma )]$, where $\sigma$ is given by \eqref{sigma} below, to $p\in (1,p_G(N+\mu )]$ giving thus a better estimate of the lifespan in this case.