Exponential stability and blow up for a problem with Balakrishnan–Taylor damping

This work is devoted to the study of a nonlinear viscoelastic Kirchhoff equation with Balakrishnan–Taylor damping. We show that the weak dissipation produced by the memory term is strong enough to stabilize solutions exponentially. Also, we show that a nonlinear source of polynomial type is able to force solutions to blow up in finite time even in presence of a stronger damping.