International Journal of Modern Nonlinear Theory and Application, 2013, 2, 186-193
http://dx.doi.org/10.4236/ijmnta.2013.23026 Published Online September 2013 (http://www.scirp.org/journal/ijmnta)
Pointwise Estimates for Solutions to a
System of Radiating Gas
Shikuan Mao, Yongqin Liu
School of Mathematics and Physics, North China Electric Power University, Beijing, China
Email: shikuanmao@ncepu.edu.cn, yqliu2@ncepu.edu.cn
Received August 22, 2013; revised September 8, 2013; accepted September 13, 2013
Copyright © 2013 Shikuan Mao, Yongqin Liu. This is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
In this paper we focus on the initial value problem of a hyperbolic-elliptic coupled system in multi-dimensional space of
a radiating gas. By using the method of Green function combined with Fourier analysis, we obtain the pointwise decay
estimates of solutions to the problem.
Keywords: Radiating Gas; Initial Value Problem; Pointwise Estimates
1. Introduction
In this paper we consider the initial value problem
 
2
0
div 0,,0,
div0,, 0,
,0 ,,
n
t
n
n
uau q x t
qq uxt
uxu xx
 
 

(1.1)
here is a constant vector, and
are unknown functions of
n
a
1,,
n
qq

,,xx
,uuxt
0.t

,qxt
n
x
11n
n and Typically,
represent the velocity and radiating heat flux of the gas
respectively.
,uq
The system (1.1) is a simplified version of the model
for the motion of radiating gas in n-dimensional space.
More precisely, in a certain physical situation, the system
(1.1) gives a good approximation to the following system
describing the motion of radiating gas, which is a quite
general model for compressible gas dynamics where heat
radiative transfer phenomena are taken into account,

 
22
4
12
div 0,
div 0,
div 0,
22
div 0,
t
t
t
u
uuupI
uu
euepu
qaqa





 
 
 
 
 
 
 
 
 
 
 
q
(1.2)
where ρ, u, p, e and θ are respectively the mass density,
velocity, pressure, internal energy and absolute tempera-
ture of the gas, while q is the radiative heat flux, and a1
and a2 are given positive constants depending on the gas
itself. The first three equations are motivated by the usual
Euler system, which describe the in-viscid flow of a
compressible fluid and express conservation of mass,
momentum and energy respectively. We refer to the book
of Courant and Friedrichs [1] for a detailed derivation of
several models in compressible gas-dynamics. The
physical motivation of the fourth equation, which takes
into account of heat radiation phenomena, is given in [2].
Moreover, the simplified model (1.1) was first recovered
by Hamer (see [3]), and for the reduction of system (1.2)
to system (1.1), see [2-4].
Concerning the investigation on the hyperbolic-elliptic
coupled system in one-dimensional radiating gas, we
refer to [5,6]. In the case of the muti-dimensional case,
Francesco in [7] obtained the global well-posedness of
the system (1.1) and analyzed the relaxation limits. Re-
cently, in [8], Liu and Kawashima investigated the decay
rate to diffusion wave for the initial value problem (1.1)
in n(n 1)-dimensional space by using a time-weighted
energy method.
The rest of the paper is arranged as follows. Section 2
gives the full statement of our main theorem. In Section 3,
we give estimates on the Green function by Fourier
analysis which will be used in Section 5. Section 4 gives
the global existence of solutions to the problem (2.3). In
Section 5, we obtain the pointwise decay estimates of
solutions.
Before closing this section, we give some notations to
be used below. Let
f
denote the Fourier transform
of
f
defined by
C
opyright © 2013 SciRes. IJMNTA
S. K. MAO, Y. Q. LIU 187

  
2
·
1
ˆ:e
2
n
n
ix d,
f
f


fxx

and we de-
note its inverse transform as
1.

1
ppn
LL p is the usual Lebesgue space
with the norm ·
p
L
. Let be a nonnegative integer,
s
then

s
sn
HH denotes the Sobolev space of
functions, equipped with the norm
2
L
1
2
2
2
0
:.
s
sk
x
HL
k
ff




For
0,k

1
11
:;01,
n
n
n
k
k
k
xi
xx
i
kink

  


ksn
.
i
k
For non-
negative integer , denotes the space
of -times continuously differentiable functions on the
interval
k

;CIH
k
with values in the Sobolev space

.
nss
HH
Finally, in this paper, we denote every positive con-
stant by the same symbol or c without confusion.
[·] is the Gauss’s symbol.
C
2. Main Theorems and Proof
For simplicity, without loss of generality, we
choose 11
,,
22
n
a



in (1.1). That is, we will con-
sider the following initial value problem:
 
1
0
div 0,, 0,
div0,, 0,
,0 ,.
i
nn
tx
i
n
n
uuuq x t
qq uxt
uxu xx

 
 

(2.3)
Our Main results are the following:
Theorem 2.1. Let

3,1,
22,2,
n
snn

be an inte-
ger.
Assume that and put
 
1
0,
sn n
uH L
1
00 0
:.
s
H
L Then there is a small positive con-
stant such that if 00
Eu u

01
,E
ux
then the problem
(2.3) has a unique global solution with
,t



21
0, ;,0, ;,
sns n
uC HuLH
 



12 1
0, ;0, ;.
sn sn
qC HLH
  
Moreover, if
26,sn and for any multi-indexes
with
21,sn
  there exists some constant
2
n
r such that


2
00
1
r
x
Du xCEx
, then for
any
24,sn
 the solution to Equation (2.3)
has the following decay estimate,
 
2
2
0
,11. We also have
1
r
n
x
x
Duxt CEtt


 


the following corollary by using Theorem 2.1.
Corollary 2.2. Under the same assumptions in Theo-
rem 2.1, the solution satisfies the following decay esti-
mates:
 
42
201,
nk
k
xL
utCE t
 with satisfying
k
04
2
n
ks
 ;
 


 
2
1
42
01,
nk
k
xL
qtCEt


with satisfying
k05
2
n
ks
 .



Remark. In Theorem 2.1, we do not need to assume
that
1
0,
n
uL if The results in Corollary
2.2 is similar to those in [7].
2.n
3. The Global Existence of Solution
This section is devoted to prove the global existence re-
sult stated in Theorem 2.1. In [7], the global existence of
solutions to the problem (2.3) is obtained, but for the
completeness of this paper, here we give the sketch of the
proof.
Since a local existence result can be obtained by the
standard method based on the successive approximation
sequence, we omit its details and only derive the desired
a priori estimates of solutions.
From

11
2.32.3div 2.3, 2
we get that
11
0.
ii
nn
ttx x
ii
u u uuuuu


 


 (3.4)
Now we make energy estimates by using (3.4) under
the following a priori estimate:

1,
xL
ut
 (3.5)
here 11
is a given constant.
Multiplying (3.4) by and integrating with respect
to
u
, by integration by parts we have that

 
12
22
d.
dx2
2
H
LL
ututC utut
t
L
(3.6)
Multiplying
3.4
l
x
by and integrating
with respect to
1
l
xul
, by integration by parts we have that


 
1
1
22
2
d
d
,1
ll
xx
HL
l
xx
LH
ut ut
t
Cutut l

2
.
 
1
(3.7)
We add up (3.7) with 1ls
 and get that
Copyright © 2013 SciRes. IJMNTA
S. K. MAO, Y. Q. LIU
188


 
12
1
2
22
2
d
d
.
ss
s
xx
HH
xx
LH
ut ut
t
Cut ut


 
(3.8)
Combining (3.6) with (3.8) we have that



 
1
1
22
2
d
d
.
s
s
x
HH
xx
LH
ut ut
t
Cut ut


 
s
(3.9)
In view of (3.5), (3.9) yields that
 
1
22
2
0
0d.
s
ss
t
xH
HH
ut uCu

 
(3.10)
From 2 we have that thus
(3.10) yields that

2.3

1
1,qu
 
 
11
22
2
0
0d.
s
ss
t
xH
HH
qt qCu


 
(3.11)
By the continuity argument, we have the following
result.
Theorem 3.1. Let 2
2
n
s


 be an integer. Assume
that and put

0,
sn
uH00
:.
s
H
Eu Then there is
a small positive constant
01
such that if 00
,E
then the problem (2.3) has a unique global solution
with
ux
,t



21
0, ;,0,;,
sns n
uC HuLH
 




12 1
0, ;0, ;.
sn sn
qC HLH

 
4. Estimates on Green Function
In order to study the problem (2.3), we start with the
Green function (or the fundamental solution) to the linear
problem corresponding to the Equation (3.4), which sat-
isfies
 
0,, 0,
,0 ,.
n
tt
n
GGGx t
Gxx x
 

(4.12)
By Fourier transform we get that,

ˆˆˆ
0,,0,
ˆ,0 1,.
n
tt
n
GGG t
G



(4.13)
By direct calculation we have that

2
2
1
ˆ,e
t
Gt
.
Let
 
13
1, ,1,,
0,2 ,0,1,
R
R
 
 
 







be
the smooth cut-off functions, where
and are any
fixed positive numbers satisfying
R
01R.
 Set

213
1,

  and
ˆˆ
,,Gt Gt

,
1,2, 3.i
ii We are going to
study
,
i
Gxt, which is the inverse Fourier transform of
ˆ,.
i
Gt
First we give a lemma which is important for us to
make estimates on the low frequency part.
ˆ,
2
N
n
Cft
Lemma 4.1. If has compact sup-
port in the variables
, is a positive integer, and
there exists a constant such that
N
0,b
ˆ,
f
t
satis-
fies


 

2
||
2
22
,
11
1e
,
k
kk
bt
t
Cttt
 






2
ˆ
m
Df
t




for any multi-indexes ,
with 2N
, then


2
,,
nk
xNN
DfxtCtB xt

,
where and are any fixed integers,
mk
x 0,,aa
and

2
,1
1.
N
N
x
Bxt t





ma
Proof. If ,k
 by direct calculation we have
that
 
 


2
1
|| ||||
2
22
22
2
ˆ
,e ,d
11
1ed1 .
n
n
x
k
kk
nk
mbt
xD fxtCDft
Ctt
tCtt
 

 



 






 
t
If ,k
 we also have
 



2
1
2
22
22
2
ˆ
,e,
11 1
1ed1
n
n
x
k
k
nk
mbt
xD fxtCDft
Cttt
tCt
 

d
.
t







 
Let 0
when 21,
x
t
and 2N
when
21,
x
t we obtain from the above estimates that

2
2
1
,min1,
N
nk t
Dfxt Ct
x










.
Since
2
2
2
2
1,1 ,
12
1,1
1,
x
t
x
x
t
x
t
t



, we have
Copyright © 2013 SciRes. IJMNTA
S. K. MAO, Y. Q. LIU 189
large such that 2
1
1m

22
12
min 1,2,.
11
NN
N
N
N
tBxt
xx
t









 







.
 Since
2
supp ;,G
 
R
we have that
Thus we complete the proof.
By using Lemma 4.1 we can get the following propo-
sition about the estimates on

1,.Gxt
Proposition 4.2. For sufficiently small
, there exists
constant such that
0C


2
1,,
n
xNN
DGxtCtBx t
.
Proof. For
being sufficiently small, by noticing
that is a smooth function of

,
ˆt
G
near 0
and
using Taylor expansion we have that

2
24
2
1
,
ˆee
ttO t
t
G


 .
It yields that



 

2
2
22
22
ˆ,
11
1e.
t
DGt
Ctt
t


 








t
Since

 

12
12
1
1
12
,
!ˆ,,
!!
DGt
D
DGt





 





and

1,
 

we have that



 

2
1
||
2
22
22
,
11
1e.
t
DGt
Ctt
t


 








t
From Lemma 4.1 we obtain that


2
1,,
n
xNN
DGxtCtBx t
.
Thus we complete the proof of Proposition 4.2.
As for we have the following estimates.

2,Gxt
Proposition 4.3. For fixed
and , there exist
positive numbers and such that
R
mC


2
2,e ,
t
m
xN
DGxtCBx t

2
ˆ,e
t
m
GtC
.
(4.14)
It yields that

2,e
t
m
x
DG xtC
. (4.15)
Now we shall give an estimate on
2,
x
Gxt
by
induction on .
Assume that, if 1,l
 then
 
2
ˆ,1e
t
m
DGt Ct
 ,
(4.16)
which is true as 0
by (4.14)
By using (4.13), we have the following problem for
 

22
22
0
1,
ˆˆˆ
,,,
11
ˆ,0.
t
DG tDG tDG t
DGt











,,
(4.17)
By multiplying (4.17), whose variables are now
changed to
,
s
by
ˆ,Gts
and integrating over
the region
0,
s
t, we have that
 
2
2
0
ˆˆˆ
,,, ,
1
t
DG tG tsDGss


 


 

d.
In view of (4.16) for 1,l
it yields that
  
1
0
ˆ,e1ed1
ts st
tmm
R
DGtCts Ct



  
e,
m
which shows that (4.16) is valid as .l
This implies
that, for 1,l
 




2
·
2
{}
,
ˆ,ed
e11
1e.
n
h
xt
ix
t
m
R
t
m
xDG xt
CD Gt
Ct
Ct

 

d
 



(4.18)
By using (4.15) when 21
x
t and (4.18) with
2N
when 21
x
t, as well as the fact that
2
2
2
2
2,1 ,
12
1,1
1,
x
t
x
x
t
x
t
t


we get that
.


2
2,e ,
t
m
xN
DGxtCBx t
.
Proof. For any fixed
, we choose sufficiently m
Copyright © 2013 SciRes. IJMNTA
S. K. MAO, Y. Q. LIU
190
Thus we complete the proof of Proposition 4.3.
Next we will come to consider First we give
a lemmas which is useful in dealing with the high fre-
quency part.

2,Gxt
Lemma 4.4. If


ˆ
supp: ;,
R
f
OR

  and

ˆ
f
satisfies
 
1
ˆˆ
,,fCDfC


 1,
,
then there exist distributions
 
12
,
f
xfx and con-
stant such that
0
C


,
120
f
xf Cx
xfx
where

x
is the Dirac function. Furthermore, for
positive integer 2,Nn



2
11,
N
x
Df xCx



1
22
,supp;2 ,
L
fCfxxx
0

with 0
being sufficiently small.
The proof of Lemma 4.4 can be seen in [9].
Choose sufficiently large such that
R
2
2
1,
2
1
if 1.R
 By Taylor expansion, we
have that
 
24
11
1()
33
ˆ,e
Ot
Gt




 


.
It is obvious that

2
3
ˆ,e
t
GtC
.
Since
 
24
12
12
11
1
33
ˆ,e
Ot
DGtCDD








 




,
by direct calculation, we have that for 1,

1
2
3
ˆ,e
t
DGtC


.
By using Lemma 4.4 we have the following result.
Proposition 4.5. For being sufficiently large,
there exist distributions and con-
stant such that
R
31
Gx

,,t

32 ,,Gxt
0
C
  

2
331320
,e ,,
t
GxtGxt GxtCx
,
where
x
is the Dirac function. Furthermore, for
positive integer 2Nn,
 the following estimates
hold:


2
31 ,1,
N
x
DG xtCx
 and


1
3232 0
.,,supp.,;2 ,
L
GtCGtxx

here 0
is sufficiently small.
Combining Proposition 4.2, Proposition 4.3, and
Proposition 4.5, we have the following theorem on the
Green function.
Theorem 4.6. For any multi-indexes
, there exists a
distribution


2
32 0
,e ,
t
,
K
xtG xtCx
 such
that the following estimate holds:


2
,1 ,
n
xN
DGKxtCtB xt
,
here, 2
n
N
is an arbitrary positive integer.
5. Pointwise Estimates
In this section, we focus on the pointwise estimates of
solutions to the problem (2.3).
By Duhamel principle, the solution to the Equation
(3.4) with initial datum can be ex-
pressed as following,
 
0
,0uxu x
 

001
,., 1
:, ,.
i
n
t
x
n
uxtGtuxGtuu
uxtuxt
d



Now we give a lemma which will be used in the fol-
lowing analysis.
Lemma 5.1. When 12
,nnn2, and
31
min ,,nnn2
we have that

13
2
22
2
11d1
11
n
nn
n
xy x
yyC
tt

 
 

 

 
.
The proof of Lemma 5.1 can be seen in [9].
Since
12
,,,:uxtG KxtKxtII,
by using Lemma 5.1 and Theorem 4.6 with , we
have that
Nr


2
10
1,.
n
xr
DICEtB xt

Noticing that if 0
2,xy
 then
11
2
11yCx


2
,
we have that




2
2320 0
2
0
e,
e,.
n
t
xx
t
r
DIGxytCx yDu y y
CEBx t


d
Thus we obtain that
 

2
0
,1
,, 1.
2
n
x
r
Du xtCEt
n
Bxt s


 


(5.19)
Next we come to make estimates on To this
end, we will use the following lemma.

,.uxt
Lemma 5.2. Assume , then the following ine-
qualities hold,
1n
1) If
0,t
, and 2
A
t, then
Copyright © 2013 SciRes. IJMNTA
S. K. MAO, Y. Q. LIU
Copyright © 2013 SciRes. IJMNTA
191
22
1
121
111
nn
n
n
AA
tt

 

 
 



 Denote







1
2
,0,,
4
2
,:1,,
:sup, ,
n
n
r
x
xT
n
s
xttBx t
MTDux x
.



 



;
2
t, then
2
12 1
1
n
nA
t




.
A
2) If
Now we come to make estimates to by using

,uxt
Theorem 4.6 with and Lemma 5.2. We decompose
Nr
,
x
Du xt
as following,

 



 



 


2
2
0;2 1
2
2
0;2 1
2
;2 1
2
2
;2 1
2
0
,
1,,
1,,d
11,
2
11,
2
i
i
i
i
n
x
tn
xx
yxy i
tn
xx
yxy i
n
t
txy
yxy i
n
t
txy
yxy i
t
R
Du xt
DGKxytuyy
DGKxytuyy
GKxytDu yy
GKxytDu yy
K


dd
d
,dd
,dd














1
31 3241425
,1 ,dd
:.
i
n
yx
i
xytDuuyy
IIIII

 

Next we estimate respectively by using Theorem 4.6. By using Lemma 5.2 (1), we have that
1, 2, 3, 4, 5
i
Ii
 




 







1
222
31 0;2
1
22
2
0;2
||
22
1,1
1,1,dd
,1
,dd
nn
Nr
yx y
t
nn
Nr
yx y
n
r
t
CM TtBxy tByy
CM TtBxtByy
CMTBx tt
I
 










Now we estimate 32 .
I
in two cases.
Case 1. 2
x
t. By using Lemma 5.2 (1), we have that
 




 







1
222
32 0;2
12
22
2
0;2
22
1,1
1
1,1,
1
,1 .
tnn
Nr
yx y
n
t
nn
Nr
yx y
n
r
ICMTtBxytBy y
CM TtBxy tBxty
t
CM TBxtt
,dd
dd
 






 




Case 2. 2
x
t. By using Lemma 5.2 (2), we have that





 




1
222
32 0;2
1
22
2
0
22
1
,1 ,dd
11
,d,1.
tn
yx y
n
Nr
t
nn
n
rr
ICMT t
BxytBy y
CM Tt
ByCMTBxt t







 
 

Combining the two cases, we have that
S. K. MAO, Y. Q. LIU
192



22
32 ,1
n
r
ICMTBxt t.

As for 41,
I
we also need to divide it into two cases.
Case 1. 2
x
t. By using Lemma 5.2 (1), we have that
 



 

 



1
22
41 ;2
2
2
2
1
2
22
2
;2
2
2
22
1,
1,dd
1(1)
1
,,dd
1
,1.
n
t
tN
yx y
n
r
n
nt
tyx y
n
Nr
n
r
ICMTtBxyt
By y
CM Ttt
t
Bxt Byty
t
CM TBxtt



 








Case 2. 2
x
t. By using Lemma 5.2 (2), we have
that
 



 





1
22
41 ;2
2
2
2
1
2
22
2
;2
2
22
1
1,dd
1(1)
,dd
,1.
n
t
tN
yx y
n
r
n
nt
tyx y
N
n
r
ICMTtBxyt
By y
CM Ttt
Bxyty
CM TBxtt
,




 




Combining the two cases, we have that



22
41 ,1
n
r
ICMTBxtt
.
As for 42,
I
by direct calculation, we have that
 




 






1
22
42 ;2
2
2
2
2
22
1
2
;2
2
22
1,
1,dd
1,
(1 ),
1.
dd
,
n
t
tN
yx y
n
r
n
r
n
t
tN
yx y
n
r
ICMTt Bxyt
By y
CM TBxt
tBxyty
CM TBxtt









To estimate 5,
I
we will use the following result,
which is obtained in [7].
Lemma 5.3. 1) If
3, 1,
2, 2,
2
n
snn



and
 
1
0,
sn n
uH L put 1
00 0
:
s
H
Eu u,
L
then
the following estimate holds:
 
42
01,
sk
nk
k
xH
utCE t


with 01 .ks
2) If 2,
2
n
s


 and put

0,2
sn
uH n,
00
:,
s
H
Eu then the following estimate holds:
 
2
01
sk
k
k
xH
utCE t
,
with 0.ks
We estimate 5
I
as following,
 





2
532
01
2
01
e1
e1 ,d.
i
i
tn
t
xx
i
tn
t
xx
i
ICGtDuu
CD uux
d




Notice that consists of terms of

2
1i
n
xx
i
D

u
12
121 2
,, 0,3.
kk
xx
uukk kk
 Without loss of
generality, we assume that then
12
,kk
1max,2.k
Since 6
2
n
s


 and
4,
2
n
s



 we have that
  

1
12
,1
nk
k
xuyMTBy,.


By using Gagliardo-Nirenberg inequality and Lemma
5.3, we have that
 
2
22
02
1, 2
k
k
xL
n
uCE ks

1.
 


Thus we have that
  
12
12 2
0
,,1 .
nk k
kk
xx
uyuy CEMT
 

 
Combined with Proposition 4.5 and the fact that
11
22
11,
11
yx
C
tt







if 1
2,xy

it yields that
 



1
22
50 0
2
0
e1 d
,1.
tn
t
n
r
ICEMT
CE MTBxtt



Combining 313241 42
,,,
I
III and 5,
I
we have that
 

1
2
0,
,,
xh
DuxtCEMTMTxt
.

(5.20)
Proof of Theorem 2.1. In view of (5.19) and (5.20),
we get that
 


1
2
00 ,
,,
xh
Du xtC EEM TM Txt
 .
It yields that
Copyright © 2013 SciRes. IJMNTA
S. K. MAO, Y. Q. LIU
Copyright © 2013 SciRes. IJMNTA
193
 
2
00 .MTCE EMTMT 
Thus if 0 is suitably small, we obtain E

0
M
TCE
by the continuous dependence on the initial data. In view
of Theorem 3.1, the proof of Theorem 2.1 is completed.
6. Acknowledgements
The first author is partially supported by the National
Natural Science Foundation of China (Grant No.
11201142) and by the Fundamental Research Funds for
the Central Universities (Grant No. 11QL40). The second
author is partially supported by the National Natural
Science Foundation of China (Grant No. 11201144).
REFERENCES
[1] R. Courant and K. O. Friedrichs, “Supersonic Flow and
ShockWaves,” Interscience Publishers, Inc., New York,
1948.
[2] W. G. Vincenti and C. H. Kruger, “Introduction to Physi-
cal Gas Dynamics,” Wiley, New York, 1965.
[3] K. Hamer, “Nonlinear Effects on the Propogation of
Sound Waves in a Radiating Gas,” Quarterly Journal of
Mechanics & Applied Mathematics, Vol. 24, No. 2, 1971,
pp. 155-168.
http://dx.doi.org/10.1093/qjmam/24.2.155
[4] W. L. Gao and C. J. Zhu, “Asymptotic Decay toward the
Planar Rarefaction Waves for a Model System of the Ra-
diating Gas in Two Dimensions,” Mathematical Models
and Methods in Applied Sciences, Vol. 18, No. 4, 2008,
pp. 511-541.
http://dx.doi.org/10.1142/S0218202508002760
[5] S. Kawashima and S. Nishibata, “Weak Solutions with a
Shock to a Model System of the Radiating Gas,” Science
Bulletin of Josai University, Vol. 5, Special Issue, 1998,
pp. 119-130.
[6] S. Kawashima and S. Nishibata, “Cauchy Problem for a
Model System of the Radiating Gas: Weak Solution with
a Jump and Classical Solutions,” Mathematical Models
and Methods in Applied Sciences, Vol. 9, No. 1, 1999, pp.
69-91. http://dx.doi.org/10.1142/S0218202599000063
[7] M. D. Francesco, “Initial Value Problem and Relaxation
Limits of the Hamer Model for Radiating Gases in Several
Space Variables,” Nonlinear Differential Equations and
Applications NoDEA, Vol. 13, No. 5-6, 2007, pp. 531-562.
http://dx.doi.org/10.1007/s00030-006-4023-y
[8] Y. Liu and S. Kawashima, “Asymptotic Behavior of So-
lutions to a Model System of a Radiating Gas,” Commu-
nications on Pure and Applied Analysis, Vol. 10, No. 1,
2011, pp. 209-223.
http://dx.doi.org/10.3934/cpaa.2011.10.209
[9] W.-K. Wang and T. Yang, “The Pointwise Estimates of
Solutions for Euler Equations with Damping in Multi-
Dimensions,” Journal of Differential Equations, Vol. 173,
No. 2, 2001, pp. 410-450.
http://dx.doi.org/10.1006/jdeq.2000.3937