*Mathematical Methods in the Applied Sciences*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
\textit{scale-invariant case} and in the presence of
two nonlinearities. More precisely, we consider the following equation:
\begin{displaymath} \d
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.\\ For $\mu
< \frac{N(q-1)}{2}$ and
$\mu \in (0, \mu_*)$,
where $\mu_*>0$ is depending on the
nonlineartiesâ€™ powers and the space dimension ($\mu_*$
satisfies
$(q-1)\left((N+2\mu_*-1)p-2\right)
= 4$), 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.

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