On a system of semilinear damped $\sigma$-evolution equations with different damping types in the critical case

In this paper, we study the non-symmetric system of semilinear damped $\sigma$-evolution equations with different damping types, where two power exponents of nonlinearities belong to the critical curve, by using moduli of continuity in nonlinear terms. Our goal is to determine the sharp conditions on these moduli of continuity that guarantee the global (in time) existence of Sobolev solutions or, conversely, lead to finite-time blow-up. Furthermore, by employing the analysis introduced in the proof of our blow-up result, we provide a positive answer to an open problem for the symmetric models posed in the literature.