Blow‐up results for a semi‐linear structural damped wave model with nonlinear memory

In this article, we study the nonexistence of global solutions to a semi-linear structurally damped wave equation with nonlinear memory in R n $\mathbb {R}^n$ for any space dimension n ≥ 1 $n\ge 1$ and for the initial arbitrarily data being subject to a positivity assumption. We intend to apply the method of test function to establish blow-up results and to overcome some difficulties as well caused by the well-known fractional Laplacian ( − Δ ) σ / 2 $(-\Delta )^{\sigma /2}$ in structural damping terms.