Damped wave equation with a critical nonlinearity
Authors: Hayashi Nakao , Kaikina Elena , Naumkin Pavel 
Published: November 2005

We study large time asymptotics of small solutions to the Cauchy problem for nonlinear damped wave equations with a critical nonlinearity
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout Enlarged left-brace 1st Row partial-differential Subscript t Superscript 2 Baseline u plus partial-differential Subscript t Baseline u minus normal upper Delta u plus lamda u Superscript 1 plus StartFraction 2 Over n EndFraction Baseline equals 0 comma x element-of bold upper R Superscript n Baseline comma t greater-than 0 comma u left-parenthesis 0 comma x right-parenthesis equals epsilon u 0 left-parenthesis x right-parenthesis comma partial-differential Subscript t Baseline u left-parenthesis 0 comma x right-parenthesis equals epsilon u 1 left-parenthesis x right-parenthesis comma x element-of bold upper R Superscript n Baseline comma EndLayout">

{

∂

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
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u 0 element-of bold upper H Superscript delta comma 0 Baseline intersection bold upper H Superscript 0 comma delta Baseline comma u 1 element-of bold upper H Superscript delta minus 1 comma 0 Baseline intersection bold upper H Superscript negative 1 comma delta Baseline comma">

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
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper H Superscript l comma m Baseline equals left-brace phi element-of bold upper L squared semicolon double-vertical-bar mathematical left-angle x mathematical right-angle Superscript m Baseline mathematical left-angle i partial-differential Subscript x Baseline mathematical right-angle Superscript l Baseline phi left-parenthesis x right-parenthesis double-vertical-bar Subscript bold upper L squared Baseline greater-than normal infinity right-brace comma">

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
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar u left-parenthesis t right-parenthesis minus epsilon theta upper G left-parenthesis t comma x right-parenthesis e Superscript minus phi left-parenthesis t right-parenthesis Baseline double-vertical-bar Subscript bold upper L Sub Superscript p Baseline less-than-or-equal-to upper C epsilon Superscript 1 plus StartFraction 2 Over n EndFraction Baseline g Superscript negative 1 minus StartFraction n Over 2 EndFraction Baseline left-parenthesis t right-parenthesis mathematical left-angle t mathematical right-angle Superscript minus StartFraction n Over 2 EndFraction left-parenthesis 1 minus StartFraction 1 Over p EndFraction right-parenthesis">

‖
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 ] .