Nonexistence of global weak solutions to semilinear wave equations involving time-dependent structural damping terms

We consider a semilinear wave equation involving a time-dependent structural damping term of the form ${\displaystyle \frac{1}{{(1+t)}^{\beta }}(-\mathrm{\Delta }{)}^{\sigma \mathrm{/}2}{u}_t}$. Our results show the influence of the parameters $\beta ,\sigma$ on the nonexistence of global weak solutions under assumptions on the given system data.