Decay Property for Solutions to Plate Type Equations with Variable Coefficients

International Journal of Computer Applications Technology and Research
Volume 7–Issue 07, 259-269, 2018, ISSN: 2319–8656
www.ijcat.com 259
Decay Property for Solutions to Plate Type Equations
with Variable Coefficients
Shikuan Mao
School of Mathematics and Physics,
North China Electric Power University,
Beijing 102206, China
Xiaolu Li
School of Mathematics and Physics,
North China Electric Power University,
Beijing 102206, China
Abstract: In this paper we consider the initial value problem for a plate type equation with variable coefficients and memory in
 1n
R n  ), which is of regularity-loss property. By using spectrally resolution, we study the pointwise estimates in the spectral
space of the fundamental solution to the corresponding linear problem. Appealing to this pointwise estimates, we obtain the global
existence and the decay estimates of solutions to the semilinear problem by employing the fixed point theorem.
Keywords: plate equation, memory, decay, regularity-loss property.
1. INTRODUCTION
In this paper we consider the following initial value problem
for a plate type equation in  1n
R n  :
       
       
2
0 1
1 1 , ,
,0 , ,0
p
g tt g t g t t
t
u u u k u f u u u
u x u x u x u x
           

 
(1.1)
Here 0 2p  , 0, 0   are real numbers ， the
subscript t in tu and ttu denotes the time derivative (i.e.,
t tu u  ,
2
tt tu u  ),
1
1
i j
n ij
g x xij
Gg
G 
    is the Laplace(-Beltrami)
operator associated with the Riemannian metric
 
1
n
ij i j
ij
g g x dx dx

  ,  det ijG g
and    
1ij
ijg g

 ,  ,u u x t is the unknown function
of
n
x R and 0t  , and represents the transversal
displacement of the plate at the point x and t , g ttu
corresponds to the rotational inertial. The term tu represents a
frictional dissipation to the plate. The term
      0
:
tp p
g t g tk u k t u d      
corresponds to the memory term, and  k t satisfies the
following assumptions:
Assumption [A].  2
k C R
 ,   0k s  , and the
derivatives of k satisfy the following conditions
     0 1 ,C k s k s C k s   
       2 3 , .C k s k s C k s s R
   
Where  0,1,2,3iC i  are positive constants.
We suppose the metric g satisfies the following conditions:
Assumption [B]. The matrix
ij
g is symmetric for
each
n
x R , and there exists 0C  and 0C  such that
 i    ,ij n ij
xg C R g x C


   ,
,n n
x R Z    .
 ii  
2 2
1 21
n ij
i jij
C g x C   
  ,
,n n
x R R    .
Assumption [C].  2n
f C R 
 and there exists
Z 
 satisfying 1  such that    n
f U O U ,
as 0U  .
It is well known that under the above assumptions, the
Laplace operator g is essentially self-adjoint on the Hilbert
gt
e

space  2
,n
gH L R d with domain  0
n
C R
,
here gd Gdx  . We denote the unique self-adjoint
extension (to the Sobolev space  2 n
H R ) by the same
symbol g . The spectrum of g is 0  ， , and it
generates a contraction semi-group
gt
e

on  p n
L R

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 1 p   , We note by our assumptions of the metric g ,
the measure gd is equivalent to dx, and by the (functional)
calculus for pseudodifferential operators, we have the
following classical equivalence (cf. [8, 5]): For s R , there
exists 0sC  , such that
 
   
2
1 2, 1
1.2
s s
s
s g sH H
L
n
C u u u C u
u S R
  
    
 
 
where  n
S R is the class of Schwartz functions.
The main purpose of this paper is to study the global
existence and decay estimates of solutions to the initial value
problem (1.1). For our problem, it is difficult to obtain
explicitly the solution operators or their Fourier transform due
to the presence of the memory term and variable coefficients.
However, we can obtain the pointwise estimate in the spectral
space of the fundamental solution operators to the
corresponding linear equation
   
   
2
1 1
0 1.3
g tt g t
p
g t
u u u
k u


     
   
from which the global existence and the decay estimates of
solutions to the semilinear problem can be obtained. The
following are our main theorems.
Theorem 1.1 (energy estimate for linear problem). Let 0s 
be a real number. Assume that
 
 max 1,2 2
0
s p n
u H R
 

and  1
s n
u H R , and put
 max 1,2 20 0 1 .s p s
H H
I u u  
Then the solution to the problem (1.3) with initial
condition   00u u and   10tu u satisfies
     0 1 1
0, ; 0, ;s n s n
u C H R C H R
     
and the following energy estimate:
   
    
1
1
2 2
2 2 2
0
0
.
s s
s s
tH H
t
tH H
u t u t
u u d CI  



  
The second one is about the decay estimates for the
solution to (1.3), which is stated as follows:
Theorem 1.2 (decay estimates for linear problem). Under the
same assumptions as in Theorem 1.1, then the solution to (1.3)
satisfies the following decay estimates:
   1
2
0 1 ,s
H
u t CI t

 

 
for 0 1,s   and
    2
0 1 ,st H
u t CI t



 
for 0 s  .
Theorem 1.3 (existence and decay estimates for semilinear
problem). Let
2
n
s  and 0 2p  be real numbers.
Assume that
 
 max 1,2 2
0
S p n
u H R
 
 ,  1
s n
u H R ,
and put
 max 1,2 20 0 1s p s
H H
I u u  
then there exists a small 0  , such that when 0I  ,
there exists a unique solution to (1.1) in
     0 1 1
0, ; 0, ;s n s n
u C H R C H R
     
satisfying the following decay estimates:
     1
2
0 1 , 1.4s
H
u t CI t

 

 
for 0 1,s   and
     2
0 1 , 1.5st H
u t CI t



 
for 0 s  .
Remark 1. If the semilinear term is the form of  f u , then
we may assume  0 1s n  and  1 2
2
n
s n   in
Theorem 1.3.
For the study of plate type equations, there are many
results in the literatures. In [4], da Luz–Charão studied a
semilinear damped plate equation :
   2
. 1.6tt tt tu u u u f u     
They proved the global existence of solutions and a
polynomial decay of the energy by exploiting an energy
method. However the result was restricted to
dimension 1 5n  , This restriction on the space
dimension was removed by Sugitani–Kawashima (see [23])
by the fundamental method of energy estimates in the Fourier
(or frequency) space and some sharp decay estimates. Since
the method of energy estimates in Fourier space is relatively
simple and effective, it has been adapted to study some related
problems (see [18, 19, 20, 24]).
For the case of dissipative plate equations of memory
type, Liu–Kawashima (see [15, 12]) studied the following
equation
 2
,ttu u u k u f u     

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as well as the equation with rotational term
 2
, ,tt tt tu u u u k u f u u u         ,
and obtained the global existence and decay estimates of
solutions by the energy method in the Fourier space. The
results in these papers and the general dissipative plate
equation (see [13, 14, 16, 23]) show that they are of
regularity-loss property.
A similar decay structure of the regularity-loss type was also
observed for the dissipative Timoshenko system (see [10])
and a hyperbolic-elliptic system related to a radiating gas (see
[9]). For more studies on various aspects of dissipation of
plate equations, we refer to [1, 2, 3, 7]. And for the study of
decay properties for hyperbolic systems of memory-type
dissipation, we refer to [6, 11, 22].
The results in [12] are further studied and generalized to
higher order equations in [16] and to the equations with
variable coefficients in [17]. The main purpose of this paper is
to study the decay estimates and regularity-loss property for
solutions to the initial value problem (1.1) in the spirit of [12,
15, 16, 17]. And we generalize these results to the case of
variable coefficients and semilinear equations.
The paper is arranged as follows: We study the pointwise
estimates of solutions to the problem (2.2) and (2.3) in the
spectral space in Section 2. And in Section 3, we prove the
energy estimates and the decay estimates for solutions to the
linear equation (1.3) by virtue of the estimates in Section 2. In
Section 4, we prove the global existence and decay estimates
for the semilinear problems (1.1).
For the reader’s convenience, we give some notations
which will be used below. Let  F f denote the Fourier
transform of f defined by
   
 
 
2
1ˆ :
2 n
ix
n
R
F f f e f x dx



  
and we denote its inverse transform as
1
F
.
Let  L f denote the Laplace transform of f defined
by
 
   2
2
2 ˆ1s
n
n x
x
s
s
H L R
L R
f f f   
here  
1
2 2
1   denotes the Japanese bracket.
2. Pointwise estimates in the spectral space.
We observe that the equation (1.1) ( respectively (1.3) ) is
equivalent to the following in-homeogeneous equation
      
     
2
1 1 0
, 2.1
p
g tt g g
p
t g
u u k u
u k u F t x

 
      
    
with         0, , ,
p
g tF t x k t u x f u u u   
(respectively,       0,
p
gF t x k t u x  ).
In order to study the solutions to (2.1), we study the
pointwise estimates for solutions to the following ODEs with
parameter R 
 , respectively:
        
 
   
2 4 2
2
1 , 1 0 ,
, 0 2.2 ,
0, 1, 0, 0,
p
tt
p
t
t
G t k G t
G t k G
G G
     
  
 
    

   
  

（）
and
        
 
   
2 4 2
2
1 , 1 0 ,
2.3 ,, 0
0, 0, 0, 1,
p
tt
p
t
t
H t k H t
H t k H
H H
     
  
 
    

    

 
（）
We note that      
2
, , ,tG t H t H t   

  , and
apply the Laplace transform to (2.2) and (2.3) (which is
guaranteed by Proposition 1 given at the end of this section),
then we have formally that
 
 
       
 
2
1
2 2 4 2 2
1
,
1 1 0
t p p
G t L t
k L k

  

       


  
  
       
 
 
       
 
2
1
2 2 4 2 2
1
,
1 1 0
t p p
H t L t
k L k



       


 
  
       
Now by virtue of the solutions to (2.2) and (2.3), the
solution to (2.1) can be expressed as
     
      
0 1
1
0
, ,
, 1 2.4
t
g
u t G t u H t u
H t F d  

   
    
where  ,G t  and  ,H t  ) are defined by the
measurable functional calculus (cf. [21]):
      
      
 
2
2
, , ,
2.5
, , ,
LR
LR
G t G t d P
H t H t d P


    
    
  

 


，
，
for , in the domain of  ,G t  and  ,H t 

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respectively, here P is the family of spectral projections
for the positive self-adjoint operator  
1
2
g   . We
note that  ,G t  and  ,H t  are the solutions (formally)
to the following operator equations:
     
 
   
 
2
1 1 0
0 2.6
0 , 0
p
g tt g g
p
t g
t
G k G
G k G
G I G O

 
       


    

 

and
     
 
   
 
2
1 1 0
0 2.7
0 , 0
p
g tt g g
p
t g
t
H k H
H k H
H O H I

 
       


    

 

respectively, here I stands for the identity operator, and O
denotes the zero operator.
Thus estimates for  u t can be reduced to estimates for
 ,G t  and  ,H t  in terms of (2.4).
First, let us introduce some notations. For any reasonable
complex-valued function  f t ,  0,t   , we define
      
         
        
0
0
2
0
: ,
: ,
: .
t
t
t
k f t k t f d
k f t k t f f t d
k f t k t f t f d
  
  
  
  
   
  



Then direct computations imply the following lemma
Lemma 2.1. For any functions  1
k C R
 and
 1
H R 
 , it holds that
            
           
           
         
0
2
2
0
2
0
1 . ,
1
2 . Re
2
1 1
2 2
3 . .
t
t
t
t
k t k t k d t
k t t k t t
d
k t k t k d t
dt
k t k d k t
    
  
    
   
   
  
  
 



Remark 2. From Lemma 2.1 1), we have
     0 .t
k k k k t k           
Now we come to get the pointwise estimates of  ,G t 
and  ,H t  in the spectral space, and we have the
following proposition.
Proposition 1 (pointwise estimates in the spectral space).
Assume  ,G t  and  ,H t  are the solutions of (2.2) and
(2.3) respectively, then they satisfy the following estimates:
   
    
22 2
2 4
, ,
,
t
p c t
G t G t
k G t Ce
 
   
   
    
    ，
And
   
    
22 2
2 4
, ,
, ,
t
p c t
H t H t
k H t Ce
 
   
   
    
   
here k Gand k H are defined as in (2.8), and
  2
   
   with  
1
2 21 .    
Proof. We only prove the estimate for  ,G t  , and the case
for  ,H t  can be proved in a similar way. To simplify the
notation in the following, we write G for  ,G t  .
Step 1. By multiplying (2.2) by tG and taking the real part,
we have that
  
    
2 22 4 2
2 2
1 1
1
2 2
Re 0 2.9
p
t
t
p
t t
G k t G
G k G G
  
 
   
       
   
   
Apply Lemma 2.1 2) to the term   Re tk G G in (2.9),
and denote
      
 
2 22 4 2
1
2
1 1
, : 1 1
2 2
,
2
p
t
p
E t G k t G
k G
   

    

and

www.ijcat.com 263
     
2 2
2 22
2 , 0 ,
2 2
p p
tR t C G k G k G
 
 
 
     
 
then we obtain that
     1 1, , 0. 2.10E t F t
t
 

 

Step 2. By multiplying (2.2) by G and taking the real part, we
have that
    
  
    
22 2
2 24 2
2
Re 1 1
1
1 0
2
Re 0. 2.11
t t
t
p
t
p
G G G
k G G
k G G
 
   

  
 
     
 
  
In view of Lemma 2.1 1), we have that
        
       
0
2
Re Re
Re 0
t
k G G k G G k d GG
k G G k t k G
      
   

Denote
    
    
      
22
2
24 2
2
22 2
2
, : Re 1 ,
2
, : 1 ,
, : 1 Re ,
t
p
p
t
E t G G G
F t k t G
R t G k G G

 
   
  
  
  
   
then (2.11) yields that
       2 2 2, , , . 2.12E t F t R t
t
  

 

Step 3. Define   2
   
   , and set
       
       
     
1 2
1 2
2
, : , , ,
, : , , ,
, : , ,
E t E t E t
F t F t F t
R t R t
    
    
   
 
 

Here  is a positive constant and will be determined later.
Then (2.10) and (2.12) yields that
       , , , . 2.13E t F t R t
t
  

 

We introduce the following Lyapunov functions:
       
     
2 22 4 2
0
2 2
2 2
0
1 1 1
, : 1 1 ,
2 2 2
, : .
2 2
p
t
p p
t
E t G G k G
F t G k t G k G
   
 

    
  
From the definition of  1 ,E t  and  1 ,F t  , we know
that there exist positive constants  1,2,3iC i  such that
the following estimates hold:
     
     
1 0 1 2 0
1 3 0
, , , ,
, , . 2.14
C E t E t C E t
F t C F t
  
 
 

On the other hand, since
   2 24
2 , ,tE t C G G    
we know that
      
   
2 22 4
2
2
4 0
, 1
, .
2
t
p
E t C G G
C k G C E t
     

  
    
 
  
 
Choosing suitably small such that 1 2
4 min ,
2 2
C C
C
 
  
 
,
and by virtue of (2.14), we have that
       1 2
0 0
3
, , , . 2.15
2 2
C C
E t E t E t   
In view of (2.14), it is easy to verify that
          
24 2
3 0 3, , 1 2.16p
F t C F t C k t G         
Since
   
 
2
2 22
2
2
, 0
2
.
2
p
t
p
R t C G k G
C k G

 

 
    
 

We have
   
   
 
2
2 2
2
5
, 0
2
2
, .
p
t
p
R t C G k G
C k G
C F t

   

  
 
 
    
 


Taking sufficiently small such that

www.ijcat.com 264
1 1 2
4
5
1
, min ,
2 2 2
C C
C
C
   
   
  
, we have that
     
1
, , . 2.17
2
R t F t 
In view of (2.13), the relation (2.17) yields that
     
1
, , 0. 2.18
2
E t F t
t
 

 

On the other hand, (2.15) and (2.16) yield that
       , , 2.19F t c E t   
Then (2.18) and (2.19) yield that
   
   , 0, 2.20
c
E t e E
 
 

By virtue of (2.15) and (2.20), we get the desired results.
3. Decay estimates of solutions to the linear
problem.
In this section we shall use the functional calculus of  and
the pointwise estimates in spectral space obtained in
Proposition 1 to prove the energy estimate in Theorem 1.1 and
the decay estimates in Theorem 1.2.
Proof of Theorem 1.1. From (2.18) and (2.19) we have that
     , , 0.E t C E t
t
   

 

Integrate the previous inequality with respect to t and appeal
to (2.15), then we obtain
         0 0 00
, , , . 3.1
t
E t E CE t      
Multiply (3.1) by
 2 1s
 
  and integrate the resulting
inequality with respect to the measure  0 0,d P u u , as
well as by the definition of  ,G t  in (2.5) and the
equivalence (1.2), then we obtain the following estimate
for   0,G t u :
   
    
 
1
1
1
2 2
0 0
2 2
0 00
2
0
, ,
, ,
. 3.2
s s
s s
s
tH H
t
tH H
H
G t u G t u
G u G u d
C u
  



  
   


Similarly, we have
   
    
 
1
1
1
2 2
1 1
2 2
1 10
2
1
, ,
, ,
. 3.3
s s
s s
s
tH H
t
tH H
H
H t u H t u
H u H u d
C u
  



  
   


From (3.3), we know that
     11 1 1, , , 0, 3.4ss
n
HH
H t u C u u S R t    
which implies
      
 
 
    
 
1
1
2 2
1
0
0
1
0
0 0,
0
, 1
sup , 1
. 3.5
s
s
s p
t p
g g
H
t p
g g
Ht
H
k H u d
k d H u
C u

  
  


 



   
    



Similarly, we have
      
 
 
    
 2 2
1
0
0
1
0
0 0,
0
, 1
sup , 1
. 3.6
s
s
s p
t p
t g g
H
t p
t g g
Ht
H
k H u d
k d H u
C u

  
  
 



   
    



Again from (3.3), we know that
   
2 2
1 1 1, , , 0.ss
n
t HH
H t u C u u S R t    
which implies that
      
      
    
 
1
1
2 2
2
1
0
0 0
21
0
0
212
0
0
2
0
, 1
, 1
, 1
3.7
s
s
s
s p
t p
g g
H
t p
g gL
H
t p
g gL
H
H
k H u d d
k k H u d d
k H u d
C u

    
    
  
 



    
 
      
 
     

 


where in the first inequality, we used the Jensen’s inequality,
while in the second inequality, we used the
1
L -estimates for
the convolution operation with respect to time (or changing
the order of integration).
In a similar way, by (3.3), we have

www.ijcat.com 265
      
      
    
 
1
1
1
1
1
2 2
2
1
0
0 0
21
0
0
212
0
0
2
0
, 1
, 1
, 1
3.8
s
s
s
s p
t p
t g g
H
t p
t g gL
H
t p
t g gL
H
H
k H u d d
k k H u d d
k H u d
C u

    
    
  



 



    
 
      
 
     

 


Thus, in term of (2.4) and the estimates of (3.2)–(3.8), as
well as the fact that
     
      
0 1
1
0
, ,
, 1 . 3.9
t t t
t
t g
u t G t u H t u
H t F d  

   
    
with       0,
p
gF t x k t u x  defined in (2.1), we
have
   
    
1
1
2 2
2 2 2
0
0
.
s s
s s
tH H
t
tH H
u t u t
u u d CI  



  
That is the conclusion of the theorem.
In order to prove Theorem 1.2, we need the following lemma
which is a direct result of Proposition 1.
Lemma 3.1. With   2
   
   introduced in Proposition
1,  ,G t  and  ,H t  satisfy the following estimates:
     
     
     
     
1
1 . ,
2 . , ,
3 . , ,
1 . , .
c t
c t
t
c t
c t
t
G t Ce
G t Ce
H t Ce
H t Ce
 
 
 
 

 
 



 


  
  

By the above lemma, we have the following estimates:
Lemma 3.2. Let 0, 0r   be real numbers, then the
following estimates hold:
       
       
       
       
1
1
2
2
2
2
1 . , 1 , ,
2 . , 1 , ,
3 . , 1 , ,
4 . , 1 , .
rr
rr
rr
rr
n
HH
n
t HH
n
HH
n
t HH
G t C t S R
G t C t S R
H t C t S R
H t C t S R








  
  
  
  

 
 





    
    
    
    
Proof. We only prove the case 1), but the other cases can be
deduced similarly. In view of Lemma 3.1 1) and the
functional calculus (2.5) as well as the equivalence (1.2), we
have that
2
2
2
2
2 2
0
2 ( )
0
1
2 ( )
0
2 ( )
0
1 2
( , ) ( , ) ( , )
( , )
( , )
( , )
: .
r
r
LH
r cp t
L
r cp t
L
r cp t
L
G t C G t d P
C e d P
C e d P
C e d P
I I







    
  
  
  






   
  
  
  
 




It is obvious that
2
2
1 .ct
L
I Ce 

On the other hand,
 
2
2
2
0
2
(1 ) ( , )
(1 ) .r v
r v
v
L
v
H
I C t d P
C t
  
 



   
 

Here
}.22,1max{,0,0  psvrvr
Thus the result for the case 1) is proved.
Proof of Theorem 1.2. Let 0r , then from (2.4) we have
that
0 1
1
0
0
( ) ( , ) ( , )
( ) ( , )( ) (1 )
.
r rr
r
H HH
t
p
g g
H
u t G t u H t u
k H t u d
I II III
  
   
     
  

By Lemma 3.2, we know that
1 2
11 2
2 2
0 1(1 ) (1 ) .r v r v
v v
H H
I II C t u C t u  
 
    
And
3
13
13
3
2 33
12
00
1
0
2
1 22
0 0
1
0
2
0
( ) ( , )( ) (1 )
( ) ( , )( ) (1 )
(1 ) ( ) (1 ) ( )
( ) (1 )
(1 ) ,
r
r
r v
r v
r v p
t
p
g g H
t
p
t g g H
v t
p
g g H
ct p
g g H
v
H
III k H t u d
k H t u d
C t u k d
Ce u
C t u
  
  
  
 
  




 

     
     
    
   
 



where in the second step, we used the exponentially decay

www.ijcat.com 266
property of )(tk , which is a direct result from the
Assumption [A]. Thus, we have
1
( max{1,2 2} 1) 1 max{1,2 2}1
2
12
3
( 2 2 max{1,2 2} 1) 1 max{1,2 2}
3
2
0
2
1
2
0
( ) (1 )
(1 )
(1 ) .
r v p pr
r v
r v p p p
v
H H
v
H
v
H
u t C t u
C t u
t u
      
 
        



 
 
 
Here 0,0,0 321  vvv satisfy
 
1
2
3
max{1,2 2},
1 , 3.10
2 3 max{1,2 2}.
r v s p
r v s
r v p s p
   

  
      
Choose the smallest real numbers 1 2 3, ,v v v such that








.}22,1max{22
,
,1}22,1max{
3
2
1



ppv
v
pv
It gives that








).22(}22,1max{
,1}22,1max{
,
23
21
2
ppvv
pvv
v 
Thus, the inequality (3.10) holds with r satisfying
0 1 .r s    
Taking the maximal r , i.e., 1r s    , we obtain
1
2
0( ) (1 ) .s
H
u t CI t

 

 
That is the result for )(tu .
The estimate for )(tut can be proved in a similar way by
just using the fact (3.9) and Lemma 3.2, and we omit the
details.
4. Global existence and decay estimates of
solutions to the semilinear problem.
In this section, by virtue of the properties of solution operators,
we prove the global existence and optimal decay estimates of
solutions to the semilinear problem by employing the
contraction mapping theorem.
From (2.1), we know that the solution to (1.1) can be
expressed as
0 1
1
00
1
0
( ) ( , ) ( , )
( , ) ( )(1 ) ( )
( , )(1 ) ( ( ), ( ), ( )) .
t
p
g g
t
g t
u t G t u H t u
H t k u d
H t f u u u d
  
    


   
     
     


Lemma 4.1 (Moser estimates). Assume that 0r be a real
number, then
( ).r r r
H L H L H
uv C u v v u  
By the previous lemma and an inductive argument, we have
the following estimates:
Lemma 4.2. Assume that 1,1   be integers,
and 0r be a real number, then
1 1
( (
).
rr
r
L L L HH
L H
u v C u v u v
v u
  
  

 


Define
  
  
0 1
1
: { 0, ,
0, , ; }
s n
s n
X
X u C H R
C H R u

   
  
here
1
2
0 1 0
2
0 1 0
: sup sup{(1 ) ( ) }
sup sup{(1 ) ( ) }.
s
s
X H
s t
t H
s t
u t u t
t u t






 

   
   
 
 
Proposition 2. There exists 0C such that
 ( ) . 4.1L X
U t C u 
Proof. By the Sobolev imbedding theorems, we have
( ) ( ) .S
L H
U t C U t 
By the definition of X
u and ,1 sS
HH
uCu we
obtained the result.
Denote
1
0
0
0 0 1
1
00
: { ; },
: ( , , ),
[ ]( ) : ( ) ( , )(1 ) ( ) ,
( ) : ( , ) ( , )
( , ) ( )(1 ) ( ) .
R X
t
t
g
t
p
g g
B u X u R
U u u u
u t t H t f U d
t G t u H t u
H t k u d
 
  


  
 
       
    
     


We will prove that )(uu  is a contraction mapping
on RB for some 0R .

www.ijcat.com 267
Proof of Theorem 1.3. We denote
),,(: vvvV t  , : ( , , )tW w w w  , then
1
0
[ ]( ) [ ]( )
( , )(1 ) ( ( ) ( ))( ) .
t
g
v t w t
H t f V f W d  
   
    
Step 1: We prove:
 
1
1
2
[ ]( ) [ ]( )
(1 ) ( , ) 4.2
s
H
X X
v t w t
C t v w v w



 
 
  
 
with 0 1s  
Indeed,
 
1
1
1 2
2
0
2
1
2
0
2
1
1 2
( [ ] [ ])( )
( )
( , )(1 ) ( ( ) ( ))( )
( )
( , )(1 ) ( ( ) ( ))( )
: . 4.3
s
s
s
H
t
t
t
g H
t
t
t
g H
v w t
H t f V f W d
H t f V f W d
I I



  
  
 
 
  


  
 
    
 
    
 
 
 
In view of lemma 3.2 3), we have that
 
1 2 1
1 ( 3)
1
2 2
0
2 2
0
(1 ) ( ( ) ( ))( )
(1 ) ( ( ) ( ))( )
4.4
s v
s v
t v
H
t v
H
I
C t f V f W d
C t f V f W d


  
  
    
   



  
   


By Lemma 4.2, we get that
1 [ ( 3)]
1 [ ( 3)]
1 [ ( 3)]
2
( ( ) ( ))( ) ( , )( )
{ ( , )( ) ( )( )
( )( ) ( )( ) }.
s v
s v
s v
H L
L H
H L
f V f W C V W
V W V W
V W V W




 
 
 
    
    
    

 
 
  
In view of (4.1), we have that
 
1 [ ( 3)]
( 3)
1
2
( ( ) ( ))( )
(1 ) ( , ) ( , ) 4.5
s v
H
v
X X
f V f W
C v w v w d




 
   
 
 
 

Let v in (4.5), we have that
 
1
( 3)
1
2 2 2
0
3
1
22 2
0
2
1
2
(1 ) (1 ) ( , ) ( )
(1 ) (1 ) ( , ) ( )
2
(1 ) ( , ) ( ) . 4.6
t v v
X X
t
t
t X X
X X
I
C t v w v w d
t
C v w v w d
C t v w v w






  
 
 
  

  
 

   
   
  

 
Le 0v in (4.5), we have that
 
2
( 3)
1
2 2
2
3
1
2
2
3
1
2 2
2
1
2
(1 ) (1 ) ( , ) ( )
(1 ) ( , ) ( )
(1 ) (1 ) ( , ) ( )
2
(1 ) ( , ) ( ) . 4.7
v v
t
t X X
t
t X X
t
t X X
X X
I
C t v w v w d
C t v w v w d
t
C v w v w d
C t v w v w








  

 
 
  

 
  
 

   
  
   
  



By virtue of (4.6) and (4.7), we obtain the desired results.
Step 2. We prove:
 
1
2
( [ ] [ ])( )
(1 ) ( , ) ( ) . 4.8
st t H
X X
v w t
C t v w v w




 
   
 
Indeed,
2
2
0
2
1
2
0
2
3 4
( [ ] [ ])( )
( )
( , )(1 ) ( ( ) ( ))( )
( )
( , )( ( ) ( ))( )
: .
s
s
s
t t H
t
t
t
t g H
t
t
t
t H
v w t
H t f V f W d
H t f V f W d
I I



  
  


 

  
 
    
 
  
 
 
 
In view of Lemma 3.2 4), we have
2
1 [ ( 3)]
2 2
3
0
2 2
0
(1 ) ( ( ) ( ))( )
(1 ) ( ( ) ( ))( ) .
s v
s v
t v
H
t v
H
I C t f V f W d
C t f V f W d


  
  
  
   


   
   


In a similar way to (4.6) and (4.7), we have

www.ijcat.com 268
 
3
( 3)
1
2 2 2
0
3
1
22 2
0
2
1
2
(1 ) (1 ) ( , ) ( )
(1 ) (1 ) ( , ) ( )
2
(1 ) ( , ) ( ) . 4.9
t v v
X X
t
t
t X X
X X
I
C t v w v w d
t
C v w v w d
C t v w v w






  
 
 
  

  
 

   
   
  

 
and
 
4
( 3)
1
2 2
2
3
1
2
2
3
1
2 2
2
1
2
(1 ) (1 ) ( , ) ( )
(1 ) ( , ) ( )
(1 ) (1 ) ( , ) ( )
2
(1 ) ( , ) ( ) . 4.10
v v
t
t X X
t
t X X
t
t X X
X X
I
C t v w v w d
C t v w v w d
t
C v w v w d
C t v w v w








  

 
 
  

 
  
 

   
  
   
  



By virtue of (4.9) and (4.10), we obtain the desired results.
Combining the estimates (4.2) and (4.8), we obtain that
       
1
, . 4.11XXX
v w C v w v w

   
So far we proved that
    1
1 XX
v w C R v w
    if , Rv w B .
On the other hand,     00 t t   , and from Theorem
1.2 we know that   2 00 X
C I  if 0I is suitably
small.Take 2 02R C I . if 0I is suitably small such that
1
1
1
2
C R
 , then we have that
   
1
.
2 XX
v w v w   
It yields that, for Rv B ,
    2 0
1 1
0
2 2XX X
v v C I R R      
Thus   Rv B  ,  v v  is a contraction
mapping on RB .and by the fixed point theorem
there exists a unique Ru B satisfying  u u  ,
and it is the solution to the semilinear problem (1.1)
satisfying the decay estimates (1.4) and (1.5). So
far we complete the proof of Theorem 1.3.
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