Abstract
We prove the global existence of small data solution in all spaces of all dimensions
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
)
and
g
(
t
,
v
)
satisfy some properties of parabolic equations. We study the problem in several classes of regularity.