Modified different nonlinearities for weakly coupled systems of semilinear effectively damped waves with different time-dependent coefficients in the dissipation terms

We prove the global existence of small data solution in all space dimension for weakly coupled systems of semi-linear effectively damped wave, with different time-dependent coefficients in the dissipation terms. Moreover, nonlinearity terms $f(t,u)$ and $g(t,v)$ satisfying some properties of the parabolic equation. We study the problem in several classes of regularity.