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

Abstract

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: $$ (E) {1cm} u_{tt}-Δu+\fracμ{1+t}u_t+\frac{ν^2}{(1+t)^2}u=|u_t|^p+|u|^q, \quad \mbox{in}\ \mathbb{R}^N\times[0,\infty), $$ 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.

arXiv admin note: text overlap with arXiv:2008.02109