A blow-up result for semi-linear structurally damped $\sigma$-evolution equations

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.