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

Abstract

"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."