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

TL;DR

This paper investigates the blow-up behavior of solutions to a damped wave equation with scale-invariant damping and combined nonlinearities, revealing the damping's significant influence under certain conditions.

Contribution

It establishes the blow-up results for the wave equation with scale-invariant damping, showing the damping's impact is significant in this setting, unlike in previous studies with different damping.

Findings

Wave equation behaves like the undamped case for certain parameters.

Damping influence is significant in the scale-invariant case.

Blow-up occurs under specified conditions on parameters.

Abstract

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