Blow-up for wave equation with the scale-invariant damping and combined nonlinearities

In this article, we study the blow-up of the damped wave equation in the $\mathit{\text{scale-invariant}}\mathit{\text{case}}$ and in the presence of two nonlinearities. More precisely, we consider the following equation: $$u_{tt}-\mathrm{\Delta }u+\frac{\mu }{1+t}{u}_t=\mathrm{\mid }{u}_t{\mathrm{\mid }}^p+\mathrm{\mid }u{\mathrm{\mid }}^q,in{\mathbb{R}}^N\times [0,\mathrm{\infty }),$$ with small initial data.\ For $\mu <\frac{N(q-1)}{2}$ and $\mu \in (0, \mu_)$,where$\mu_>0$ is depending on the nonlinearties’ powers and the space dimension (${\mu }_{\ast }$ satisfies $(q-1)((N+2{\mu }_{\ast }-1)p-2)=4$), we prove that the wave equation, in this case, behaves like the one without dissipation ($\mu =0$). Our result completes the previous studies in the case where the dissipation is given by $\frac{\mu }{(1+t{)}^{\beta }}{u}_t;\beta >1$ (\cite{LT3}), where, contrary to what we obtain in the present work, the effect of the damping is not significant in the dynamics. Interestingly, in our case, the influence of the damping term $\frac{\mu }{1+t}{u}_t$ is important.