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 spaces of all dimensions $n\geq 1$ n ≥ 1 for weakly coupled systems of semilinear effectively damped wave, with different time-dependent coefficients in the dissipation terms. Moreover, we assume that the nonlinearity terms $f(t,u) $ f ( t , u ) and $g(t,v) $ g ( t , v ) satisfy some properties of parabolic equations. We study the problem in several classes of regularity.