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

Makram Hamouda, Mohamed Ali Hamza

We are interested in this article in studying the damped wave equation with
localized initial data, in the \textit{scale-invariant case} with mass term and
two combined nonlinearities. More precisely, we consider the following
equation: $$ (E) {1cm} u_{tt}-\Delta
u+\frac{\mu}{1+t}u_t+\frac{\nu^2}{(1+t)^2}u=|u_t|^p+|u|^q, \quad \mbox{in}\
\mathbb{R}^N\times[0,\infty), $$ with small initial data. Under some
assumptions on the mass and damping coefficients, $\nu$ and $\mu>0$,
respectively, we show that blow-up region and the lifespan bound of the
solution of $(E)$ remain the same as the ones obtained in \cite{Our2} in the
case of a mass-free wave equation, it i.e. $(E)$ with $\nu=0$.
Furthermore, using in part the computations done for $(E)$, we enhance the
result in \cite{Palmieri} on the Glassey conjecture for the solution of $(E)$
with omitting the nonlinear term $|u|^q$. Indeed, the blow-up region is
extended from $p \in (1, p_G(N+\sigma)]$, where $\sigma$ is given by (1.12)
below, to $p \in (1, p_G(N+\mu)]$ yielding, hence, a better estimate of the
lifespan when $(\mu-1)^2-4\nu^2<1$. Otherwise, the two results coincide.
Finally, we may conclude that the mass term {\it has no influence} on the
dynamics of $(E)$ (resp. $(E)$ without the nonlinear term $|u|^q$), and the
conjecture we made in \cite{Our2} on the threshold between the blow-up and the
global existence regions obtained holds true here.