We study large time asymptotics of small solutions to the Cauchy problem for nonlinear damped wave equations with a critical nonlinearity
{
∂
t
2
u
+
∂
t
u
−
Δ
u
+
λ
u
1
+
2
n
=
0
,
x
∈
R
n
,
t
>
0
,
u
(
0
,
x
)
=
ε
u
0
(
x
)
,
∂
t
u
(
0
,
x
)
=
ε
u
1
(
x
)
,
x
∈
R
n
,
\begin{equation*} \left \{ \begin {array}{c} \partial _{t}^{2}u+\partial _{t}u-\Delta u+\lambda u^{1+\frac {2}{n}}=0,\text { }x\in \mathbf {R}^{n},\text { }t>0, u(0,x)=\varepsilon u_{0}\left ( x\right ) ,\partial _{t}u(0,x)=\varepsilon u_{1}\left ( x\right ) ,x\in \mathbf {R}^{n}, \end{array} \right . \end{equation*}
where
ε
>
0
,
\varepsilon >0,
and space dimensions
n
=
1
,
2
,
3
n=1,2,3
. Assume that the initial data
u
0
∈
H
δ
,
0
∩
H
0
,
δ
,
u
1
∈
H
δ
−
1
,
0
∩
H
−
1
,
δ
,
\begin{equation*} u_{0}\in \mathbf {H}^{\delta ,0}\cap \mathbf {H}^{0,\delta },\text { }u_{1}\in \mathbf {H}^{\delta -1,0}\cap \mathbf {H}^{-1,\delta }, \end{equation*}
where
δ
>
n
2
,
\delta >\frac {n}{2},
weighted Sobolev spaces are
H
l
,
m
=
{
ϕ
∈
L
2
;
‖
⟨
x
⟩
m
⟨
i
∂
x
⟩
l
ϕ
(
x
)
‖
L
2
>
∞
}
,
\begin{equation*} \mathbf {H}^{l,m}=\left \{ \phi \in \mathbf {L}^{2};\left \Vert \left \langle x\right \rangle ^{m}\left \langle i\partial _{x}\right \rangle ^{l}\phi \left ( x\right ) \right \Vert _{\mathbf {L}^{2}}>\infty \right \} , \end{equation*}
⟨
x
⟩
=
1
+
x
2
.
\left \langle x\right \rangle =\sqrt {1+x^{2}}.
Also we suppose that
λ
θ
2
n
>
0
,
∫
u
0
(
x
)
d
x
>
0
,
\begin{equation*} \lambda \theta ^{\frac {2}{n}}>0,\int u_{0}\left ( x\right ) dx>0, \end{equation*}
where
θ
=
∫
(
u
0
(
x
)
+
u
1
(
x
)
)
d
x
.
\begin{equation*} \text { }\theta =\int \left ( u_{0}\left ( x\right ) +u_{1}\left ( x\right ) \right ) dx\text {.} \end{equation*}
Then we prove that there exists a positive
ε
0
\varepsilon _{0}
such that the Cauchy problem above has a unique global solution
u
∈
C
(
[
0
,
∞
)
;
H
δ
,
0
)
u\in \mathbf {C}\left ( \left [ 0,\infty \right ) ;\mathbf {H}^{\delta ,0}\right )
satisfying the time decay property
‖
u
(
t
)
−
ε
θ
G
(
t
,
x
)
e
−
φ
(
t
)
‖
L
p
≤
C
ε
1
+
2
n
g
−
1
−
n
2
(
t
)
⟨
t
⟩
−
n
2
(
1
−
1
p
)
\begin{equation*} \left \Vert u\left ( t\right ) -\varepsilon \theta G\left ( t,x\right ) e^{-\varphi \left ( t\right ) }\right \Vert _{\mathbf {L}^{p}}\leq C\varepsilon ^{1+\frac {2}{n}}g^{-1-\frac {n}{2}}\left ( t\right ) \left \langle t\right \rangle ^{-\frac {n}{2}\left ( 1-\frac {1}{p}\right ) } \end{equation*}
for all
t
>
0
,
t>0,
1
≤
p
≤
∞
,
1\leq p\leq \infty ,
where
ε
∈
(
0
,
ε
0
]
.
\varepsilon \in \left ( 0,\varepsilon _{0}\right ] .