Global and blow-up solutions for a quasilinear hyperbolic equation with strong damping

In this paper, we study a quasilinear hyperbolic equation with strong damping. Firstly, by use of the successive approximation method and a series of classical estimates, we prove the local existence and uniqueness of a weak solution. Secondly, via some inequalities, the potential method and the concave method, we derive the asymptotic and blow-up behavior of the weak solution with different conditions.