Blow up for Initial-Boundary Value Problem of Wave Equation with a Nonlinear Memory in 1-D

The present paper is devoted to studying the initial-boundary value problem of a 1-D wave equation with a nonlinear memory: $$u_{tt}-u_{xx}=\frac{1}{\Gamma(1-\gamma)}\int_0^t(t-s)^{-\gamma}|u(s)|^p\rmd s.$$ The blow up result will be established when $p>1$ and $0<\gamma<1$, no matter how small the initial data are, by introducing two test functions and a new functional.