The asymptotic behavior solutions to damped Navier-Stokes equations in unbounded domain

This paper is concerned with time decay rates of the weak solutions to the 3D Navier-Stokes equations with nonlinear damping |u|
^p-1
u in R
³
₊
. With the aid of the spectral decomposition of the Stokes operator and L
^p
- L
^q
estimates, it shows that the weak solution decay in L
²
norm like ct
^{-3/2r + 3/4}
with the initial velocity u
₀
ϵ L
²
_σ
∩ L
^r
for 1 <; r ≤ 2. And if also satisfy ∫
_{R 3 +}
|x
₃
u
₀
(x)|
^r
, and then we have the decay in L
²
norm like ct
^{-3/2r + 3/4}
.