Blow up for the wave equation with a nonlinear dissipation of cubic convolution type in \(\mathbb R^N\) (Q1421274)

The author deals with the following equation \[ \begin{gathered} u_{tt}+\lambda u+ u_t(V_\gamma* u^2_t)=\Delta u+ a| u|^{p-1} u\quad\text{in }\mathbb{R}^N\times (0,\infty),\\ u(x,0)= u_0(x),\quad u_t(x,0)= u_1(x),\end{gathered}\tag{1} \] where \((V_\gamma* u^2_t)(x,t)= \int_{\mathbb{R}^N} V_\gamma(x- y)u^2_t(y,t)\,dy\) and \(V_\gamma(x)=| x|^{-\gamma}\), \(0< \gamma<\mathbb{N}\), \(\lambda\geq 0\), \(a>0\), \(p> 1\). For the case \(a\neq 0\), \(a> 0\), that is the presence of a source of power type, and under suitable assumptions on the data of (1), the author proves a blow-up result in finite time.