Critical exponent for semi-linear structurally damped wave equation of derivative type

The main purpose of this paper is to study the following semi-linear structurally damped wave equation with nonlinearity of derivative type: u(tt) - Delta u + mu(-Delta)(sigma/2)u(t) = vertical bar u(t vertical bar)(p), u(0,x) = u(0)(x), u(t)(0,x) = u(1)(x), with mu > 0, n >= 1, sigma is an element of(0,2], and p > 1. In particular, we would like to prove the nonexistence of global weak solutions by using a new test function and suitable sign assumptions on the initial data in both the subcritical case and the critical case.