  We would like to prove a blow-up result for semi-linear structurally damped
$\sigma$-evolution equations, where $\sigma \ge 1$ and $\delta\in [0,\sigma)$
are assumed to be any fractional numbers. To deal with the fractional Laplacian
operators $(-\Delta)^\sigma$ and $(-\Delta)^\delta$ as well-known non-local
operators, in general, it seems difficult to apply the standard test function
method directly. For this reason, in this paper we shall construct new test
functions to overcome this difficulty.
