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

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: $$u_{tt}-Δu+\fracμ{1+t}u_t=|u_t|^p+|u|^q, \quad \mbox{in}\ \R^N\times[0,\infty), $$ with small initial data.\\ For $μ< \frac{N(q-1)}{2}$ and $μ\in (0, μ_*)$, where $μ_*>0$ is depending on the nonlinearties' powers and the space dimension ($μ_*$ satisfies $(q-1)\left((N+2μ_*-1)p-2\right) = 4$), we prove that the wave equation, in this case, behaves like the one without dissipation ($μ=0$). Our result completes the previous studies in the case where the dissipation is given by $\fracμ{(1+t)^β}u_t; \ β>1$ (\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 $\fracμ{1+t}u_t$ is important.