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

TL;DR

This paper investigates the blow-up behavior of solutions to a scale-invariant damped wave equation with mass and combined nonlinearities, showing that the mass term does not affect blow-up regions and extending previous lifespan estimates.

Contribution

It demonstrates that the mass term does not influence blow-up regions and refines lifespan bounds for the damped wave equation with combined nonlinearities.

Findings

Blow-up region remains unchanged with mass term.

Extended the blow-up region for certain parameters.

Mass term has no influence on solution dynamics.

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