Blow-up results for damped wave equation with fractional Laplacian and non linear memory

"The goal of this paper is to study the nonexistence of nontrivial solutions of the following Cauchy problem $$\left\{ \begin{array}{ll} u_{tt}+(-\Delta)^{\beta/2} u+u_{t}=\displaystyle\int_{0}^{t}\left(t-\tau \right) ^{-\gamma}\left\vert u(\tau ,\cdot) \right\vert^{p}d\tau,\\ \cr u(0,x)=u_{0}(x),\quad u_t(0,x)=u_1(x),\quad x\in\mathbb{R}^n, \end{array}\right.$$ where $p>1,\ 0<\gamma <1,\,\, \beta\in(0,2) $ and $(-\Delta)^{\beta/2} $ is the fractional Laplacian operator of order $\frac{\beta}{2}$. Our approach is based on the test function method."