We consider a non-linear wave equation with an internal fractional damping, a polynomial source and an infinite memory. Using the semi-group theory, we get the existence of a local weak solution. Moreover, we show under some conditions, local solutions may blow up a in finite time; this is achieved by constructing a suitable Lyapunov functional.

In the present work, we consider a nonlinear wave equation with a time delay condition of fractional type and a logarithmic source. Using the semigroup theory, we get the existence of a local weak solution. Moreover, we show that this solution blows up in finite time with negative initial-energy.