A competition on blow-up for semilinear wave equations with scale-invariant damping and nonlinear memory term

<p style='text-indent:20px;'>In this paper, we investigate blow-up of solutions to semilinear wave equations with scale-invariant damping and nonlinear memory term in <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^n $\end{document}</tex-math></inline-formula>, which can be represented by the Riemann-Liouville fractional integral of order <inline-formula><tex-math id="M2">\begin{document}$ 1-\gamma $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M3">\begin{document}$ \gamma\in(0, 1) $\end{document}</tex-math></inline-formula>. Our main interest is to study mixed influence from damping term and the memory kernel on blow-up conditions for the power of nonlinearity, by using test function method or generalized Kato's type lemma. We find a new competition, particularly for the small value of <inline-formula><tex-math id="M4">\begin{document}$ \gamma $\end{document}</tex-math></inline-formula>, on the blow-up range between the effective case and the non-effective case.</p>