Global well-posedness for the semilinear wave equation with time dependent damping in the overdamping case
Authors: Ikeda Masahiro , Wakasugi Yuta 
Published: September 2019

We study the global existence of solutions to the Cauchy problem for the wave equation with time-dependent damping and a power nonlinearity in the overdamping case:
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row StartLayout Enlarged left-brace 1st Row 1st Column u Subscript t t Baseline minus normal upper Delta u plus b left-parenthesis t right-parenthesis u Subscript t Baseline equals upper N left-parenthesis u right-parenthesis comma 2nd Column a m p semicolon t element-of left-bracket 0 comma upper T right-parenthesis comma x element-of double-struck upper R Superscript d Baseline comma 2nd Row 1st Column u left-parenthesis 0 right-parenthesis equals u 0 comma u Subscript t Baseline left-parenthesis 0 right-parenthesis equals u 1 comma 2nd Column a m p semicolon x element-of double-struck upper R Superscript d Baseline period EndLayout EndLayout">

{

u

t
t

−
Δ
u
+
b
(
t
)

u
t

=
N
(
u
)
,

a
m
p
;
t
∈
[
0
,
T
)
,
 
x
∈

R

d

,

u
(
0
)
=

u
0

,
 

u
t

(
0
)
=

u
1

,

a
m
p
;
x
∈

R

d

.

\begin{align*}\left \{\begin {array}{ll} u_{tt} - \Delta u + b(t) u_t = N(u),&t\in [0,T),\ x\in \mathbb {R}^d,\\ u(0) = u_0,\ u_t(0) = u_1,&x\in \mathbb {R}^d. \end{array}\right . \end{align*}

Here,

b
(
t
)

b(t)

is a positive

C
1

C^1

-function on

[
0
,
∞
)

[0,\infty )

satisfying
\[

b
(
t

)

−
1

∈

L
1

(
0
,
∞
)
,

b(t)^{-1} \in L^1(0,\infty ),

\]
whose case is called overdamping.

N
(
u
)

N(u)

denotes the

p
p

th order power nonlinearities. It is well known that the problem is locally well-posed in the energy space

H
1

(

R

d

)
×

L
2

(

R

d

)

H^1(\mathbb {R}^d)\times L^2(\mathbb {R}^d)

in the energy-subcritical or energy-critical case

1
≤
p
≤

p
1

1\le p\le p_1

, where

p
1

:=
1
+

4

d
−
2

p_1:=1+\frac {4}{d-2}

if

d
≥
3

d\ge 3

or

p
1

=
∞

p_1=\infty

if

d
=
1
,
2

d=1,2

. It is known that when

N
(
u
)
:=
±

|

u

|

p

N(u):=\pm |u|^p

, small data blow-up in

L
1

L^1

-framework occurs in the case

b
(
t

)

−
1

∉

L
1

(
0
,
∞
)

b(t)^{-1} \notin L^1(0,\infty )

and

1
>
p
>

p
c

(
>

p
1

)

1>p>p_c(> p_1)

, where

p
c

p_c

is a critical exponent, i.e., threshold exponent dividing the small data global existence and the small data blow-up. The main purpose in the present paper is to prove the global well-posedness to the problem for small data
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis u 0 comma u 1 right-parenthesis element-of upper H Superscript 1 Baseline left-parenthesis double-struck upper R Superscript d Baseline right-parenthesis times upper L squared left-parenthesis double-struck upper R Superscript d Baseline right-parenthesis">

(

u
0

,

u
1

)
∈

H
1

(

R

d

)
×

L
2

(

R

d

)

(u_0,u_1)\in H^1(\mathbb {R}^d)\times L^2(\mathbb {R}^d)

in the whole energy-subcritical case, i.e.,

1
≤
p
>

p
1

1\le p>p_1

. This result implies that the small data blow-up does not occur in the overdamping case, different from the other case

b
(
t

)

−
1

∉

L
1

(
0
,
∞
)

b(t)^{-1}\notin L^1(0,\infty )

, i.e., the effective or noneffective damping.