-raptor
This paper is concerned with the Cauchy
problem of nonlinear wave equations with
potential, strong, and nonlinear
damping terms. Firstly, by using variational
calculus
and compactness lemma, the
existence of standing waves of the ground states
is
obtained. Then the instability of
the standing wave is shown by applying
potential-well arguments and concavity
methods. Finally, we show how small the
initial data are for the global
solutions to exist.
Keywords:
wave equations; nonlinear damping
terms; strong damping terms;
global
existence; blow-up
Introduction
Consider the
Cauchy problem for nonlinear wave equations with
potential, strong, and
nonlinear
damping terms,
{
utt
?Δ
u
?
ω
Δ
ut
+
V
(
x
)
u
+|
ut
|
m
?<
/p>
2
ut
=|
u<
/p>
|
p
?
2
u
,
u
(0,
x
)=
u
0,
ut
(0,
x
)=<
/p>
u
1
,in [0,
T
)×
Rn
,in
Rn
,
(1)
where
p
>2
,
m
≥2
,
T
>0
,
ω
>0
,
<
/p>
u
0
∈
H
1(
Rn
),
u
1
∈
L
2(
Rn
),
(2)
and
2<
p
≤
2
nn
?
2,for
n
≥3;
p
>2,for
n
≤2.<
/p>
(3)
With the
absence of the strong damping term
Δ
ut
, and the
damping
term
ut
(see [
1
]), (1.1) can be
viewed as an interaction between one or
more discrete oscillators and a field
or continuous medium
[
2
].
For the case of linear damping (
ω
=0
,
m
=2
) and
nonlinear sources, Levine
[
3
] showed
that
the solutions to (1.1) with negative initial
energy blow-up for the abstract version.
For the nonlinear damping and source
terms (
ω
=0
,
m
>2
,
p
>2
,
V
(
x
)=0
)
, the abstract
version has been
considered by many researchers [
4
]
–
[
12
]. For instance, Georgiev
and
Todorova [
4
] prove that if
m
≥
p
, a
global weak solution exists for any initial data,
while if
2<
m
<
p
the solution blows
up in finite time when the initial energy is
sufficiently negative. In
[
5
], Todorova considers the
additional restriction on
m
.
Ikehata[
6
]
considers the solutions of (1.1) with small
positive initial energy, using the
so-
called
‘potential
-
well’ theory. The
case of strong damping (
ω
>0
,
m
=2
,
V
(
x
)=0
)
and nonlinear source terms (
p
>2
) has been
studied by Gazzola and Squassina in
[
1
].
They prove
the global existence of solutions with initial
data in the potential well and
show
that every global solution is uniformly bounded in
the natural phase space.
Moreover, they
prove finite time blow-up for solutions with high
energy initial data.
However, they do
not consider the case of a nonlinear damping term
(
ω
>0
,
m
>2
,
p
>2
).
To the best of our knowledge, little
work has been carried out on the existence and
instability of the standing wave for
(1.1). In this paper, we study the existence of a
standing wave with ground state
(
ω
=1
), which is
the minimal action solution of the
following elliptic equation:
?Δ
?
+
V
(
x
)
?
=|
?
|
p
?
2
?
.
(4)
Based on the characterization
of the ground state and the local
well-
posedness
theory[
7
],
we
investigate
the
instability
of
the
standing
wave for the Cauchy problem (1.1).
Finally, we derive a sufficient
condition of global existence of
solutions to the Cauchy problem (1.1)
by
using
the
relation
between
initial
data
and
the
ground
state
solution
of
(1.4).
It
should
be
pointed
out
that
these
results
in
the
present
paper
are
unknown to (1.1) before.
For
simplicity, throughout this paper we denote
∫
Rn
?
dx
by
∫
?
dx
and arbitrary
positive constants by
C
.
Preliminaries
and statement of main results
We define
the energy space
H
in the
course of nature as
H
:={
φ
∈
H
1(
Rn
),
∫
V
(
x
)|
φ
|
dx
<∞
}.
(5)
By its definition,
H
is a Hilbert space,
continuously embedded
in
H
1(
Rn
)
,
when endowed with the inner product as
follows:
?
φ
,
?
?
H
< br>:=
∫(
?
φ
< br>?
?
?+
V
(
x
)
φ
?
?)
dx
,
(6)
whose associated norm is
denoted by
∥
?
∥
H
. If
φ
∈
H
,
then
∥
φ
∥
H
=(
∫
|
?
φ
|2
d
x
+
∫
V
(<
/p>
x
)|
φ
|2<
/p>
dx
)12.
(7)
Throughout this paper, we make the
following assumptions on
V
(<
/p>
x
)
:
???????
inf
x
∈
RnV
(
x
)=
V
?(
x
)>0,
V
(
x
< br>) is a
C
1 bounded
measurable
function on
Rn
,lim
x
→∞
V<
/p>
(
x
)=∞.
(
8)
According
to
[
1
]
and
[
7
],
we
have
the
following
local
well-posedness
for
the Cauchy problem (1.1).
Proposition 2.1
If
(1.2)
and
(1.3)
hold
,
then there
exists a unique solution
u
(
t
,
x
)
of the Cauchy
problem
(1.1)
on a maximal time interval
[0,
T
)
,
for some
T
∈
(0,∞)
(
maximal
existence time
)
such that
u
(
t
,
x
< br>)
∈
C
([0,
T
);
H
1(
Rn
))
∩
C
1([0,
T
);
L
2(
Rn
))
∩
C
2([0,
T
);
H
?
1(
Rn
))
,
ut
(
t
,
x
)
∈
C
([0,
T<
/p>
);
H
1(
Rn
))
∩
Lm
(
[0,
T
)×
Rn
),
(9)
and
either
T
=∞
or
T
<∞
and
lim
t
→
T
?
∥
u
∥
H
1
=∞
.
Remark 2.2
From
Proposition 2.1, it follows that
m
=
p
is
the critical case, namely for
p
≤
m
, a
weak
solution exists globally in time
for any compactly supported initial data; while
for
m
<
p
,
blow-up of the solution to the Cauchy problem
(1.1) occurs.
We define the functionals
S
(
?
):=
12
∫
|
?
?
|2
dx
+12
∫
V
(
x
)
|
?
|2
dx
?
1
p
∫
|<
/p>
?
|
pdx
,<
/p>
(10)
R
(
?
):=
∫
|
?
?
|2
dx
+
∫
V
(
x
)|
?
|2
dx
?∫
|
?
|
pdx
,
(11
)
for
?
∈
H
1(
Rn
)
, and we define the set
M
:={
?
∈
H
1
?
{0};<
/p>
R
(
?
)=0}
.
(12)
We consider
the constrained variational problem
dM<
/p>
:=inf{sup
λ
≥0
S
(
λ
?
< br>):
R
(
?
)<0,
?
∈
H
< br>1
?
{0}}.
(13)
For the Cauchy problem (1.1),
we define unstable and stable sets,
K
1
and
K
2
, as
follows:
K
1
≡
{
?
∈
< br>H
1(
Rn
)
< br>∣
R
(
?
)<0,
S
(
?
)<
dM
},
K
< br>2
≡
{
?
∈
H
1(
Rn
)
∣
R
(
?
)>0,
S
(
?
)<
dM
}
∪
{0}.
(14)
The main results of this paper are the
following.
Theorem 2.3
There
exists
Q
∈
< br>M
such that
(a1)
S
(
Q
)=inf
MS
(
?
)=
dM
;
(a2)
Q
is a ground
state solution of
(1.4).
From Theorem 2.3, we have the
following.
Lemma 2.4
Let
Q
(
x
)
be the ground state of
(1.4).
If
(1.3)
holds
,
then
S
(
Q
)=min
MS
< br>(
?
).
(15)
Theorem 2.5
Assume
that
(1.2)-(1.3)
hold and
the initial energy
E
(0)
satisfies
E
(
0)=<12
∫
|
u
< br>1|2
dx
+12(
∫
|
?
u
0|2
dx
+
∫
V
(
x
)|
u
0|2
dx
)
?
1
p
∫
|
u
0|
pdxp
?
2
2
p
(
∫
|
?
Q
|2
dx
+
∫
V
(
x
)|
Q
|2
dx
).
(16)
(b1)
If
2<
m
<
p
,
and there exists
t
0
∈
[0,
T
)
such that
u
(
t
0)
∈
K
1
,
then the
solution
u
(
x
,
t<
/p>
)
of the Cauchy
problem
(1.1)
blows up in a
finite time
.
(b2)
If
2<
p
≤
m
,
there exists
t
0
∈
[0,
T
)
such that
u
(
t
0)
∈
K
2
,
then the
solution
u
(
x
,
t
)
of the Cauchy problem
(1.1)
globally exists
on
[0,∞)
.
Moreover
,
for
t
∈
[0,∞)
, <
/p>
u
(
x
,
t
)
satisfies
∥
ut
∥
22+
p
?
2
p
(
∫
|
?
u
0|2
dx
+
∫
V
(
x<
/p>
)|
u
0|2
d
x
)<
p
?
2
p
(
∫
|
?
Q
|2
dx
+
∫
V
(
x
)|
Q
|2
dx
).
(17)
Variational characterization of the
ground state
In this section, we prove
Theorem 2.3.
Lemma 3.1
The
constrained variational problem
dM
:=inf{sup
λ
≥
0
S
(
λ
?<
/p>
):
R
(
?
)<0,
?
∈
H<
/p>
1
?
{0}},
(18)
is equivalent
to
d
1:=inf{sup
λ
≥0
S
(
λ
?
):
R
(
?
)=0,
?
∈
H
1
?
{0}}=inf
?
∈
MS
(
?
),
(1
9)
and
dM
provided
2<
p
≤
2
nn
?
2
as well
as
2≤
m
<
p
.
Proof
Let
?
∈
H
1
. Since <
/p>
S
(
λ
?
)=
λ
22(
∫
|
?
?
|2
dx
+
∫
V
(
x
)|
?
|2
dx
)
?
λpp
∫
|
?
|
pdx
,
(20)
it follows that
ddλ
S
(
λ
?
)=
λ
(
∫
|
?
?
|2
dx
+
∫
V
(
x
)|
?
|2
dx
)
?
λ
p
?
< br>1
∫
|
?
|
pdx
.
(21)
Thus by
2≤
m<
/p>
<
p
, one sees that
there exists some
λ
1
≥0
such that
sup
λ
≥0
S
(
λ
?
)=
S
(
λ
1
?
)=
λ
21(12
∫
|
?
?
|2
dx
+12
∫
V
(
x
)|
?
|2
dx
?
λp
?
21
p
∫
|
?
|
pdx
),
(22)
where
λ
1
uniquely
depends on
?
and
satisfies
∫
|
?
?
|2
dx
+
∫
V
(
x
)|
?
|2
dx
?
λ
p
< br>?
21
∫
|
?
|
pdx
=0.
(23)
Since
d
2
dλ
2
S
(
λ
?
)=
∫
|
?
?
|2
dx
+
∫
V
(
x
< br>)|
?
|2
dx
?
pλ
p
?
< br>2
∫
|
?
|
pdx
,
(24)
which
together
with
p
>2
and
(3.6)
implies
that
d
2
dλ
2
S
(
λ
?
)|
λ
=
λ
1<0
,
we
have
sup
λ
≥0
S
< br>(
λ
?
)=
p
?
22
p
(
∫
|
?
?
|2
dx
+
∫
V
(
x
)|<
/p>
?
|2
dx
)<
/p>
pp
?
2(
∫<
/p>
|
?
|
pdx<
/p>
)2
p
?
2.<
/p>
(25)
Therefore,
the above estimates lead to
dM
==inf{sup
λ
≥0
S
(
λ
?
):
R
(
?
)<0,
?
∈
H
1
?
{0}}inf{
p
?
22
p
(<
/p>
∫
|
?
?
|2
dx
+
∫
V
(
x
)|
?
|2
dx
)
pp
?
2(
∫
|
?
|
pdx
)2
p
?
2:
R
(
?
)<0}.
(26)
It is easy to
see that
d
1==inf{
sup
λ
≥0
S
(
λ
?
):
R
(
?
)=0,
?
∈
H
1
?
{0}}inf
?
∈
< br>M
{
p
?
22
p
(
∫
|
?
?
|2
d
x
+
∫
V
(<
/p>
x
)|
?
|2<
/p>
dx
)}.
(27)
From (2.3)-(2.5), on
M
one has
S
(
?
)=
< br>p
?
22
p
(
∫
|
?
?
|2
dx
+
∫
V
(
x
)|
?
|2
dx
)
.
(28)
It follows
that
d
1=inf
?
∈
MS
(
?
)
and
S
(
?
)>0
on
M
.
Next we establish the equivalence of
the two minimization problems (3.1) and (3.2).
For any
?
0
∈
H
1
and
R
(
?
0)<0
, let
?
β
(
x
)=
β
?
0
. There exists a
β<
/p>
0
∈
(0,1)
such
that
R
(
?
β
0)=0
, and
from (3.8) we get
sup
λ
≥0
S
(
λ
?
β
0)===
p
?
22
p
(
∫
|
?
?
< br>β
0|2
dx
+
∫
V
(
x
)|
?
β
0|2
< br>dx
)
pp
?
< br>2(
∫
|
?
β
0
|
pdx
)2
p
?
2
p
?
22
pβ
2
pp
?
20(
∫
|
?
?
0|2
dx
+
∫
V
(
x
)|
?
0|2
dx
)
pp
?
2
β
2
pp
?
20(
∫
|
?
0|
pdx
)2
p
?
2
p
?
22
p
(
∫
|
?<
/p>
?
0|2
dx
+
∫
V
(
x
)|
?
0|2
dx
)
pp
?
2(
∫
|
?
0|<
/p>
pdx
)2
p
?
2.
(29)
Consequently
the
two
minimization
problems
(3.8)
and
(3.9)
are
equivalent,
that is, (3.1)
and (3.2) are equivalent.
Finally, we prove
dM
>0
by showing
d
1>0
in terms of
the above equivalence.
Since
2<
p
≤
2
nn
?
2
, the Sobolev
embedding inequality yields
∫
|
?
|
pdx
≤
C
(
∫
|
?
?
|2
dx
+
∫
V
(
x
)|
?
|2
dx
)
p
2.
(30)
From
R
(
?
)=0
, it follows that
∫
|
?
?
|2
< br>dx
+
∫
V
(
x
)|
?
|2
dx
=
∫
|
?
|
pdx
≤
C
(
∫
|
?
?
|2
dx
+
∫
V
(
x
)|
?
|2
dx
)
p
2,
(31
)
which together with
p
>2
implies
∫
|
?
?
|2
dx
+
∫
V
(
x
)|
?
|2
dx
≥
C
>0.
< br>(32)
Therefore, from
(3.10), we get
S
(
?
)≥
C
>0
,
?
∈
M
.<
/p>
(33)
Thus
from
the
equivalence
of
the
two
minimization
problems
(3.1)
and
(3.2)
one concludes that
dM
>0
for
< br>2<
p
≤
2
nn
?
2
.
This completes the proof of Lemma
3.1.
□
Proposition 3.2
S
is bounded below on M
and
dM
>0
.
Proof
From (2.3)-(2.6), on
M
one has
S
(
?
)=
p
< br>?
22
p
(
∫
|
?
?
|2
dx
+
∫
V
(
x
)|
?
|2
dx
).
(34)
It
follows
that
S
(
?
)>0
on
M
.
So
S
is
bounded
below
on
M
.
From
(2.6)
we
have
dM
>0
.
□
Proposition 3.3
Let
?
λ
(
x
)=
λ
?<
/p>
(
x
)
,
for
?
∈
H
1
?
{0}
and
λ
>0
.
Then there exists a
unique
μ
>0
(
depending on
?
)
such that
R
(
?
μ
)=0
.
Moreover
,
R
(
?
λ
)>0,
for
λ
∈
(0,
μ
);
R
(
?
λ
)<0,
for
λ
∈
(
μ
,∞);
(35)
and
S
(
?
μ
)≥
S<
/p>
(
?
λ
),
?
λ
>0.
(36
)
Proof
By (2.3)
and (2.4), we have
S
(
?
λ
)=
λ
< br>22(
∫
|
?
< br>?
|2
dx
+
< br>∫
V
(
x
)|
?
|2
dx
)
?
λpp
∫
|
?
|
pdx
,
R
(
?
λ
)=
λ
2(
∫
|
?
?
|<
/p>
2
dx
+
∫
V
(
x
)|
?
|2
dx
)
?
λ
p
∫
|
?
|
pdx
.
(37)
From
the
definition
of
M
,
there
exists
a
unique
μ
>0
such
that
R
(
?
μ
)=0
.
Moreover,
R
(
?
λ
)>0,for <
/p>
λ
∈
(0,
μ<
/p>
);
R
(
?
λ
)<0,for
λ
∈
(
μ
,∞).
< br>(38)
Since
ddλ
S
(
?
λ
)=
λ
?
1
R
(
?
λ
),
(39)
and
R
(
?
μ
)=0
, we have
S
(
?
μ
)≥
S
(
?
λ
)
,
?
λ
>0
.
□
Next, we solve
the variational problem (2.6).
We first
give a compactness lemma in
[
8
].
Lemma 3.4
Let
1≤
p
<
N
+2
N
?
2
when
N
≥3
and
1≤
p
<∞
w
hen
N
=1,2
.
Then the
embedding
H
?
Lp
+1
is compact
.
In the
following, we prove Theorem 2.3.
Proof
of Theorem 2.3
According to Proposition
3.2, we let
{
?
n
,
n
∈
N
}
?
M
be a
minimizing sequence for
(2.6), that is,
R
(
?
n
)=0,
S
(
?<
/p>
n
)→
dM
.<
/p>
(40)
From (3.15)
and (3.16), we know
∥
?
?
n
∥
22
is bounded for all
n
∈
N
.
Then
there exists a subsequence
{
?
nk
,
< br>k
∈
N
}
?
{
?
n
,
n
∈
N
}
, such that
{
?
nk
}
?
?
∞
weakly in
H
1.
(41)
For simplicity, we still denote
{
?
nk
,
< br>k
∈
N
}
by
{
?
n
,
n
∈
N
}
. So we have
?
n
?
?
∞
weakly in
H
1.
< br>(42)
By Lemma 3.4, we
have
?
n
→
?
∞
strongly in <
/p>
L
2(
Rn
),
(43)
?
n
→
?
∞
st
rongly in
Lp
(
Rn
).
(44)
Next, we prove that
?
∞≠0
by
contradiction. If
?
∞≡0
, from (3.18)
and
(3.19), we have
?
n
→0strongly
in
L
2(
Rn
),
(45)
?
n
→0strongly
in
Lp
(
Rn
).
(46)
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