In this paper, we prove the weighted Lorentz and weighted Orlicz estimates for the weak solutions to the higher order parabolic systems with the leading coefficients satisfying a small BMO norm condition. As a byproduct, we obtain the weighted estimates for the higher order elliptic systems.

This work deals with a class of nonlinear viscoelastic wave equations with strong damping and variable exponent sources. The main interest of this paper compared to many previous works in the literature is able to use the idea of potential well method to study both the finite time blow-up for solutions starting from the unstable sets and decay estimate for global solutions starting in the potential wells.