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 $\mu <\frac{N(q-1)}{2}$ and $\mu \in (0,{\mu }_{\ast })$, where ${\mu }_{\ast }>0$ is depending on the nonlinearties' powers and the space dimension (${\mu }_{\ast }$ satisfies $(q-1)\left((N+2{\mu }_{\ast }-1)p-2\right)=4$), 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.

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Cited by in corpus (1)

• Blow-up and lifespan estimates for a damped wave equation in the Einstein-de Sitter spacetime with nonlinearity of derivative type