Blow-up of solutions for coupled wave equations with damping terms and derivative nonlinearities

This work was concerned with the weakly coupled system of semi-linear wave equations with time dependent speeds of propagation, damping terms, and derivative nonlinear terms in generalized Einstein-de Sitter space-time on ${\mathbb{R}}^n$. Under certain assumptions about the indexes $k_1,k_2$, coefficients ${\mu }_1,{\mu }_2$, and nonlinearity exponents $p,q$, applying the iteration technique, finite time blow-up of local solutions to the small initial value problem of the coupled system was investigated. Blow-up region and upper bound lifespan estimate of solutions to the problem were established. Compared with blow-up results in the previous literature, the new ingredient relied on that the blow-up region of solutions obtained in this work varies due to the influence of coefficients $k_1,k_2$.