Decay Property for Solutions to Plate Type Equations with Variable Coefficients by ats journals

International Journal of Computer Applications Technology and Research Volume 7–Issue 07, 259-269, 2018, ISSN: 2319–8656

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

R n  n  1 ), 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.

C2k  s   k   s   C3k  s  , s   R   .

1. INTRODUCTION In this paper we consider the following initial value problem for a plate type equation in

R n  n  1 :

Where Ci

 i  0,1, 2, 3 are positive constants.

1    u  1   2  u   u   k     p u  f  u, u , u  (1.1) g tt g t g t t  u  x,0   u0  x  , ut  x,0   u1  x 

Assumption

Here 0  p  2 ,   0,   0 are real numbers ， the

each x  R , and there exists C

subscript

utt

and

denotes the time derivative (i.e.,

ut   t u , utt   t2u ), g 

1 G

ij 1  xi Gg ij  x j is the Laplace(-Beltrami)

We suppose the metric g satisfies the following conditions: The

 ii  C1 

g   g 

xR

u  u  x, t  is the unknown function

t 0

and

, and represents the transversal

displacement of the plate at the point

and

t ,  g utt

corresponds to the rotational inertial. The term u t represents a frictional dissipation to the plate. The term

k    g  ut :  k  t      g  ut   d t

k t 

Assumption

 Z as U

Assumption [A]. derivatives of

k  C 2  R  , k  s   0

satisfy the following conditions

[C].

f  C   Rn 2 

satisfying 

 1 such

that

, and the

and

there

exists

 

f U   O U

Laplace operator  g is essentially self-adjoint on the Hilbert

t g

space

H  L2  Rn , d g  with

d g  Gdx

domain

C0  R n  ,

. We denote the unique self-adjoint

H 2  Rn  )

by the same

  , 0， t g p n generates a contraction semi-group e on L  R  symbol

g

. The spectrum of

 g

and it

C0 k  s   k   s   C1k  s  ,

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 0.

extension (to the Sobolev space

following assumptions:

 0 and C  0 such that

It is well known that under the above assumptions, the

here satisfies the

for

x  R n ,   R n .

corresponds to the memory term, and

symmetric

 ij 1 g ij  x  i j  C2 

ij 1

and

x  R n ,   Z n .

1

g ij

 i  g ij  C   Rn  , x g ij  x   C ,

g   gij  x  dxi  dx j , G  det  gij  ij

matrix

operator associated with the Riemannian metric

[B].

259

International Journal of Computer Applications Technology and Research Volume 7–Issue 07, 259-269, 2018, ISSN: 2319–8656

1  p    , We note by our assumptions of the metric g ,

u t 

the measure d  g is equivalent to dx , and by the (functional)

for 0  

calculus for pseudodifferential operators, we have the following classical equivalence (cf. [8, 5]): For s  R , there

Cs1 u

s     u , 1   g  2 u   Cs u   L2

for 0   H

u  S  R n 

where S

ut  t 

 2

 CI 0 1  t 

H s



 2

s.

Theorem 1.3 (existence and decay estimates for semilinear

n and 0  p  2 be real numbers. 2 S  max1,2 p 2 Assume that u0  H  Rn  , u1  H s  Rn  ,

1.2 

problem). Let

 R  is the class of Schwartz functions.

s

and put

I 0  u0

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

1    u  1    u   u   k     u  0 g

2 g

u t 

1.3

for 0  





u  C 0 0,   ; H s 1  R n   C1 0,   ; H s  R n 



and the following energy estimate: 2 H

s 1

 ut  t  

2 Hs

 u  

2 Hs

 ut  

2 H s 1

d  CI . 2 0

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:

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 CI 0 1  t 

H s 

 CI 0 1  t 



 2



1.4 



1.5

 2

s  0  n  1 and s  1 

f  u  , then

n  n  2 2

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 :

 0   u0 and ut  0   u1 satisfies



s 1

Theorem 1.3.

Then the solution to the problem (1.3) with initial condition u



Remark 1. If the semilinear term is the form of we may assume

 u1

s.

s  max1,2 p  2

I0  

 s  1, and ut  t 

 R  , and put H

such that when





 

I 0  u0

u  C 0 0,   ; H s 1  R n   C1 0,   ; H s  R n 

Theorem 1.1 (energy estimate for linear problem). Let s  0 s  max1,2 p 2 be a real number. Assume that u0  H Rn s

0,

 u1

there exists a unique solution to (1.1) in

from which the global existence and the decay estimates of solutions to the semilinear problem can be obtained. The following are our main theorems.

and u1  H

s  max1,2 p 2

then there exists a small 

satisfying the following decay estimates:

u t 



 s  1, and

 0 , such that

exists Cs

H s 1

 CI 0 1  t 

utt  utt   2u  ut  f  u  .

1.6 

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  n  5 , 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

utt   2u  u  k  u  f  u  ,

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International Journal of Computer Applications Technology and Research Volume 7–Issue 07, 259-269, 2018, ISSN: 2319–8656 as well as the equation with rotational term

utt  utt   2u  u  k  u  f  u , ut , u  ,

1    u  1    u   k  0     u   u   k      u  F  t , x   2.1 g

2 g

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.

with F  t , x   k  t   g

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].

In order to study the solutions to (2.1), we study the pointwise estimates for solutions to the following ODEs with

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

transform of

F  f  denote

defined by

F  f   fˆ   :

 2 

e

 ix

n 2 Rn

and we denote its inverse transform as F Let

the Fourier

L f 

1

f  x  dx

denote the Laplace transform of

 1    2 f

 

L2 Rxn

(respectively, F

parameter



u0  x   f  u, ut , u 

 t , x   k  t    g 

u0  x  ).

  R , respectively:

1   2  Gtt  t ,    1   4   k  0   2 p  G  t ,     Gt  t ,     2 p k   G  0   G  0,    1, G  0,    0, t  and

 1   2  H tt  t ,    1   4   k  0   2 p  H  t ,     （2.3） ,    H t  t ,     2 p k   H  0    H  0,    0, H t  0,    1, We note that G

 t ,    Ht  t ,    

2

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     1   2  G  t ,    L1t   t  2 2 4 2p 2p  1     1     k 0      L k             

  1   2  H  t ,    L1t   t  2 2 4 2p 2p  1     1     k  0        L  k    

defined

 

fˆ

Now by virtue of the solutions to (2.2) and (2.3), the solution to (2.1) can be expressed as

u  t   G  t ,   u0  H  t ,   u1

 

L2 Rxn

  H  t   ,   1   g  F   d t

1

 2.4 

here



  1 



1 2 2

where G 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

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（2.2） ,



 t ,   and H  t ,   ) are defined by the

measurable functional calculus (cf. [21]):

 ，G  t ,     G  t ,   d  P ,  2 R L   ，H  t ,     R H  t ,   d  P ,  L2 for  , in the domain of G  t ,   and H  t ,  

 2.5

261

International Journal of Computer Applications Technology and Research Volume 7–Issue 07, 259-269, 2018, ISSN: 2319–8656 respectively, here

 P  is the family of spectral projections

 t ,   and H  t ,  

 k   t  k  0    k     k  t    k  .

   g  2

for the positive self-adjoint operator  note that G

Remark 2. From Lemma 2.1 1), we have

. We

are the solutions (formally)

Now we come to get the pointwise estimates of

to the following operator equations:



and H



1   g  Gtt  1   2g   k  0    gp  G  p    Gt   k     g  G  0   G  0   I , Gt  0   O 

G t,  

 t ,   in the spectral space, and we have the

following proposition.

 2.6 

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:

and



  Gt  t ,       2 G  t ,   2



1   g  H tt  1     k  0     H  p    H t   k     g  H  0    H  0   O, H t  0   I  2 g

p g

 2.7 

k

G  t ,    Ce

  H t  t,      2 H  t,   2

 t  can be reduced to estimates for

G  t ,   and H  t ,  

 c    t

   4，

And

respectively, here I stands for the identity operator, and O denotes the zero operator. Thus estimates for u

 

  here k

k

H  t ,    Ce

 c    t

  4 ,

G and k H are defined as in (2.8), and 1

       2 with     1   2  2 .

in terms of (2.4).

First, let us introduce some notations. For any reasonable

f  t  , t   0 ,   , we define

complex-valued function

Proof. We only prove the estimate for G

 t ,   , and the case

 t ,   can be proved in a similar way. To simplify the notation in the following, we write G for G  t ,   . for H

 k  f  t  : 0 k  t    f   d , t

 k f  t  : 0 k  t     f    f  t   d , t

k

f  t  :  k  t    f  t   f   d . t

Then direct computations imply the following lemma

Step 1. By multiplying (2.2) by Gt and taking the real part, we have that 2 2 1 1 2 4 2p     Gt    1     k  t   G  2 t  2 

  Gt   2 p Re  k   G  Gt   0

 

  H 1  R  , it holds that

1 .

Apply Lemma 2.1 2) to the term Re





2 1 Re  k    t  t  t    k  t    t  2 t 2 1 1d   k    t    k   t   0 k   d   t  2 2 dt

E1  t ,   :

 2 .



 3 .

 k   t 

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    k   d   k t



  t  .

 k   G  G  in (2.9), t

and denote

 k    t    k   t   0 k   d   t  , t

 2.9 

Lemma 2.1. For any functions k  C1 R and







1 1   2  Gt 2  12 1   4   2 p k t  G 2 2

 2 p 2

k

G,

and

262



International Journal of Computer Applications Technology and Research Volume 7–Issue 07, 259-269, 2018, ISSN: 2319–8656   2 p  2 p 2 2 R2  t ,    C    2 Gt  k  0 G    k  G  , 2 2  

1 1   2  Gt 2  12 1   4  G 2  12  2 p  k G  , 2

F0  t ,   : Gt  2

then we obtain that

 E1  t ,    F1  t ,    0.  2.10 t Step 2. By multiplying (2.2) by G and taking the real part, we have that



E0  t ,   :



Re 1   2  Gt G  1   2  Gt

 k 

G.

E1  t ,   and F1  t ,   , we know

that there exist positive constants Ci

 i  1, 2,3 such that

the following estimates hold:

F1  t ,    C3 F0  t ,   .

1 2   1     k  0    G    G  2 t  

From the definition of

 2.14 

 2 p

k t  G 

C1 E0  t ,    E1  t ,    C2 E0  t ,   ,

 2 p

Re  k   G  G  0.

On the other hand, since

 2.11



E2  t ,    C   4 Gt  G 2

,

In view of Lemma 2.1 1), we have that

Re  k   G  G  Re  k G  G 

we know that

  k    d  GG t

 Re  k G  G   k  t   k  0   G



    E2  t ,     C    2 Gt  1   4  G   C  2

Denote





E2  t ,   : Re 1   2  Gt G 

 2

k

Choosing  suitably small such that

F2  t ,   : 1   4   k  t   2 p  G ,



 G     C4 E0  t ,   . 

G ,

 C1 C2  , ,  2 2

 C4  min 

and by virtue of (2.14), we have that

R2  t ,   : 1   2  Gt   2 p Re  k G  G , 2

then (2.11) yields that

       2

 2.15

In view of (2.14), it is easy to verify that

 E2  t ,    F2  t ,    R2  t ,   . t Step 3. Define

C1 3C E0  t ,    E  t ,    2 E0  t ,   . 2 2

 2.12 

F  t,    C3F0  t,    C3    1   4   k t   2 p  G Since

, and set

E  t ,   : E1  t ,        E2  t ,   ,

  2 p 2 2 R2  t ,    C     2 Gt  k  0 G  2  

F  t ,   : F1  t ,        F2  t ,   , R  t ,   :     R2  t ,   ,

Then (2.10) and (2.12) yields that

We introduce the following Lyapunov functions:

C

 2 p 2

 k 

G.

We have

Here  is a positive constant and will be determined later.

 E t,    F t,    R t,  . t

 2.13

 2  2 p 2 R  t ,     C  Gt     k  0 G  2     C     C5 F  t ,   .

 2 p 2

 k 

G

Taking  sufficiently small such that

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 2.16

International Journal of Computer Applications Technology and Research Volume 7–Issue 07, 259-269, 2018, ISSN: 2319–8656

 1  C C  , C41 min  1 , 2  , we have that  2 2   2C5

Similarly, we have

 

1 R t,    F t,  . 2

H  t ,   u1 

 2.17 

2 H s 1

 H  ,   u

1 Hs

 C u1

In view of (2.13), the relation (2.17) yields that

 H t  t ,   u1

2 H s 1

2 Hs

 H t  ,   u1

 2.18

 3.3

H t,   u1

H s1

 C u1

, u1  S  Rn  , t  0,

 k   H  ,      1    t

F t,    c    E t,  

 2.19 

Then (2.18) and (2.19) yield that

 2.20

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

 E  t ,    C     E  t ,    0. t

E0  t ,         E0  ,    CE0  t ,   .

 3.1

and integrate the resulting

G t,  

d  P u0 , u0  , as in (2.5) and the

equivalence (1.2), then we obtain the following estimate for G

 t ,   u0 :

G  t ,   u0 

2 H s 1

 Gt  t ,   u0

 G  ,   u

 C u0

2 H s 1

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2 0 Hs

  0,t 

 C u0

H s 2 p 2

Hs 2 H s 1

H s 1

 3.5

 k   H  ,      1    t

1

u0d

  k  d sup H t  ,     g  1   g  u0 1

 0,t 

 C u0

H s2 p2

 3.6

Again from (3.3), we know that

Ht  t ,   u1

2 Hs

 C u1

2 Hs

, u1  S  R n  , t  0.



 

k   H    ,     g  1   g  u0d 1

d

p 1  t  k L1   k   H    ,     g  1   g  u0 0 

 k

 H    ,      1    t

L1 0

 C u0

1

2 H

 d  d 

d

 3.7 

2 H s2 p2

where in the first inequality, we used the Jensen’s inequality, 1

while in the second inequality, we used the 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

 Gt  ,   u0

1

which implies that

Integrate the previous inequality with respect to t and appeal to (2.15), then we obtain

well as by the definition of

H s 1

inequality with respect to the measure

u0d

Similarly, we have

By virtue of (2.15) and (2.20), we get the desired results.

2 s 1

3.4

  k  d sup H  ,     g  1   g  u0 0

E  0,  

1

Multiply (3.1) by   

d

which implies

On the other hand, (2.15) and (2.16) yield that

 c   

H s 1

From (3.3), we know that

 1 E  t ,    F  t ,    0. t 2

E t,    e

d  3.2 

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International Journal of Computer Applications Technology and Research Volume 7–Issue 07, 259-269, 2018, ISSN: 2319–8656 t



 

k   H t    ,     g  1   g  u0d 1

2 H s 1

p 1  t  k L1   k   H t    ,     g  1   g  u0 0 

 k



L1 0

 C u0

H t    ,     g  1   g  u0 1

2 H s 1

2 H

s 1

 d  d 

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

G (t ,  )

d



2 H

s2 p2

 C     2 r G (t ,  ) d ( P ,  ) L2 0



 C     2 r e  cp (  ) t d ( P ,  )

 3.8

2 H

d

 C     2 r e  cp (  ) t d ( P ,  ) 0

Thus, in term of (2.4) and the estimates of (3.2)–(3.8), as well as the fact that



 C     2 r e  cp (  ) t d ( P ,  ) 0

ut  t   Gt  t ,   u0  H t  t ,   u1   H t  t   ,   1   g  F   d . 0

with F

 t , x   k  t    g 

 3.9

It is obvious that

have

I1  Ce ct 

u0  x  defined in (2.1), we

: I1  I 2 .

1

On the other hand,

u t 

2 H s 1

 ut  t  



I 2  C (1  t ) v   

2 Hs

 u  

2 Hs

 ut  

2 H s 1

2 r  v 

d  CI .

 C (1  t )  v 

2 0

d ( P ,  ) L2

H r v

Here

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.

      2 introduced in Proposition 1, G  t ,   and H  t ,   satisfy the following estimates: Lemma 3.1. With 

r  0, v  0, r  v  s  max{1,2 p  2}. Thus the result for the case 1) is proved.

u (t )

 G (t ,  )u0

 H (t ,  )u1

 k ( )H (t   , )(



1 . G  t ,    Ce  c   t  2  . Gt  t ,    Ce  c   t  ,  3 . H  t ,    Ce  c   t   1 , 1 . H t  t ,    Ce  c   t .

r  0 , then from (2.4) we have

Proof of Theorem 1.2. Let that

) p (1   g ) 1 u0 d

 I  II  III . By Lemma 3.2, we know that

I  II  C (1  t )



v1 2

H r v1

 C (1  t )



v2 2

H r v2 1

And By the above lemma, we have the following estimates: Lemma 3.2. Let r

III   2 k ( ) H (t   ,  )( g ) p (1   g ) 1 u0

 0,  0 be real numbers, then the

  t k ( ) H (t   ,  )( g ) p (1   g ) 1 u0

following estimates hold:

1 . G  t ,   

 2  . Gt  t ,     3 .  4 .

H t,   Ht t,  

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 C 1  t 





 C 1  t   C 1  t 



2 



 C 1  t 

 2



 



H r 

 2



,   S  R n  ,

H r  1

,   S  R n  ,

H r  1

,   S  R

H r 

,

,   S  R n  .

 C (1  t )



v3 2

( g ) p (1   g ) 1 u0

Ce  ct ( g ) p (1   g ) 1 u0  C (1  t )

v  3 2

H r v3  2 p 3

H r v3 1

t 2 0



d

k ( )d

H r v3 1

where in the second step, we used the exponentially decay

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International Journal of Computer Applications Technology and Research Volume 7–Issue 07, 259-269, 2018, ISSN: 2319–8656 property of k (t ) , which is a direct result from the Assumption [A]. Thus, we have

u (t )

 C (1  t )



 C (1  t )  (1  t ) Here v1



v1 2



v3 2

v2 2

u (t )  G (t , )u0  H (t , )u1 t

  H (t   , )k ( )(1   g ) 1 ( g ) p u0 d 0

u0 u1

H r ( v1max{1,2 p2}1)1max{1,2 p2}

  H (t   , )(1   g ) 1 f (u ( ), ut ( ), u ( ))d .

H r v2 1

Lemma 4.1 (Moser estimates). Assume that r  0 be a real number, then

r ( v  2 p 2max{1,2 p 2}1)1 max{1,2 p 2} 3

 C( u

L

 v

L

By the previous lemma and an inductive argument, we have the following estimates:

 0, v2  0, v3  0 satisfy

r  v1  s  max{1, 2 p  2},  r  v2  1  s, r  v  2 p  3  s  max{1, 2 p  2}. 3 

  1,   1

Lemma 4.2. Assume that

 3.10 

and r  0 be a real number, then

u v 

 C( u

 1 L

 1 L

 v

Choose the smallest real numbers v1 , v2 , v3 such that

v1  max{1,2 p  2}  1   ,  v2   , v  2 p  2  max{1,2 p  2}   .  3

be integers,

L

Define

  C  0,   , H  R   ;

X : {u  C 0 0,   , H s 1  R n   1

 }

It gives that

v2   ,  v1  v2  max{1,2 p  2}  1, v  v  max{1,2 p  2}  (2 p  2). 2  3

here 

Taking the maximal r , i.e., r

u (t )

0   s 1 t  0

 s  1   , we obtain 

 CI 0 (1  t ) 2 .

U (t )

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

L

H s 

 0 such that

C u

 4.1

Proof. By the Sobolev imbedding theorems, we have

U (t )

That is the result for u (t ) . The estimate for ut (t ) can be proved in a similar way by

}

H s 1



Proposition 2. There exists C



H s 1

: sup sup{(1  t ) 2 u (t )  sup sup{(1  t ) 2 ut (t )

Thus, the inequality (3.10) holds with r satisfying

0  r  s 1  .

By the definition of

L

 C U (t ) X

and

u

. C u

H s 1

obtained the result. Denote

BR : {u  X ; u

 R},

U : (u , ut , u ), t

[u ](t ) :  0 (t )   H (t   ,  )(1   g ) 1 f (U ) d , 0

 0 (t ) : G (t ,  )u0  H (t ,  )u1 t

  H (t   ,  ) k ( )(1   g ) 1 ( g ) p u0 d . 0

u  (u ) is on BR for some R  0 . We will prove that

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a contraction mapping

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International Journal of Computer Applications Technology and Research Volume 7–Issue 07, 259-269, 2018, ISSN: 2319–8656

I1  Proof of Theorem 1.3. We denote

t 2 0

H (t   ,  )(1   g ) 1 ( f (V )  f (W ))( ) d .

 C (1  t )

[v](t )  [ w](t )

H s 1

C (1  t )





 2

 1

(v, w)

vw

 4.2 

H s 1

H (t   ,  )(1   g ) 1 ( f (V )  f (W ))( )

d

H s 1

 1

(v, w)



(v  w)

H (t   ,  )(1   g ) 1 ( f (V )  f (W ))( )

H s 1  2

In view of lemma 3.2 3), we have that

I1  0

t 2 0

 C  (1  t   )



  ( v 3) 2

(v, w)

 1

(v  w) X d

(v, w)

 1 X

(v  w) X d



 2

 1

(v, w)

( f (V )  f (W ))( ) v  2

d s 1  2 v 1

( f (V )  f (W ))( )

s 1(  v 3)

d

 4.4 

(v  w)

 4.7 



H s 



 2

(v, w)

 1 X

(v  w)

 4.8 

Indeed,

( t [v]   t [ w])(t )

H s 

 (

( f (V )  f (W ))( )

 (V  W )( )



 3

C (1  t )

t 2 0

 t ) 2

H t (t   ,  )(1   g ) 1 ( f (V )  f (W ))( ) t

H s 

d

 ( 2  t ) 0

H t (t   ,  )( f (V )  f (W ))( )

H s  2

d

: I 3  I 4 .

By Lemma 4.2, we get that

L

 4.6 

Step 2. We prove:

d

 4.3

: I1  I 2 .

v 2

By virtue of (4.6) and (4.7), we obtain the desired results.



(v  w) X d

( t [v]   t [ w])(t )

 ( 2  t )

{ (V , W )( )

C  t (1  t   ) 2 (1   )

 C (1  t )

 1

3 t  t   1  C  t (1  ) 2 (1   ) 2 (v, w) X (v  w) X d 2 2

C  2 (1  t   )

(v, w)

 t )



([v]  [ w])(t )



 C  t (1  t )

Indeed,

 (

I2 

0    s 1

t 2 0

  ( v 3)

Le v  0 in (4.5), we have that

Step 1: We prove:

with



3 t  t  t  1  C  t (1  ) 2  2 (1   ) 2 (v, w) X (v  w) X d 0 2 2

[v](t )  [ w](t )  t

v 2

C  (1  t   ) (1   )

V : (v, vt , v) , W : ( w, wt , w) , then





H s 1[  ( v3)]

(V  W )( ) s 1[  ( v 3)]

 C (V , W )( )

 2 L

H s 1[  ( v3)]

(V  W )( )

In view of Lemma 3.2 4), we have t

I 3  C  2 (1  t   ) 0



 C  2 (1  t   ) 0



v 2



v 2

( f (V )  f (W ))( )

H s v 2

H s 1[  ( v3)]

d d .

In view of (4.1), we have that

( f (V )  f (W ))( ) 

  ( v 3)

s 1[  ( v3)]

C (1   ) 2 (v, w) Let v   in (4.5), we have that

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 1 X

In a similar way to (4.6) and (4.7), we have

 (v, w)

d  4.5 

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International Journal of Computer Applications Technology and Research Volume 7–Issue 07, 259-269, 2018, ISSN: 2319–8656

I3 

4. REFERENCES

t 2 0

v  2

C  (1  t   ) (1   ) 



  ( v 3) 2

t 2 0

t t   C  t (1  ) 2  (1   ) 2 2

 C (1  t )



 2

 1

(v, w)

3  2

(v, w)

(v  w)

 1 X

(v  w) X d

 4.9 

I4  

v 2

C  t (1  t   ) (1   )



[4]

  ( v 3) 2

[2] C. Buriol, Energy decay rates for the Timoshenko system of thermoelastic plates,Nonlinear Analysis, 64 (2006), 92-108. [3] R. C. Charão, E. Bisognin, V. Bisognin and A.F. Pazoto, Asymptotic behavior for a dissipative plate equation in R N with periodic coefficients, Electronic J. Differential Equations, 2008 (2008), 1-23.

and

[1] M. E. Bradley and S. Lenhart, Bilinear spatial control of the velocity term in a Kirchhoff plate equation, Electronic J. Differential Equations, 2001 (2001), 1-15.

(v, w)

 1 X

(v  w) X d

C. R. da Luz and R. C. Charão, Asymptotic properties for a semi-linear plate equation in unbounded domains, J. Hyperbolic Differential Equations, 6 (2009), 269-294.

 C  t (1  t )



 3 2

(v, w)

 1

[5] M. Dimassi and J. Sjöstrand, Spectral Asymptotics in the Semi-Classical Limit, London Mathematical Society Lecture Note Series, 268, Cambridge University Press, (1999).

(v  w) X d

3 t  t   1  C  t (1  ) 2 (1   ) 2 (v, w) X (v  w) X d 2 2

 C (1  t )



 2

 1

(v, w)

(v  w)

 4.10 

By virtue of (4.9) and (4.10), we obtain the desired results. Combining the estimates (4.2) and (4.8), we obtain that

 v     w

 C  v, w

 1 X

 4.11

vw X .

So far we proved that

 v    w

On the other hand, 

 C1R 1 v  w

 0  t    0  t  , and from Theorem

1.2 we know that small.Take R  1

C1 R

if v, w  BR .

 0

 C2 I 0

is suitably

 2C2 I 0 . if I 0 is suitably small such that

1  vw X . 2

It yields that, for v  BR ,

 v

   0



1 v 2

1  C2 I 0  R  R 2

Thus   v   BR , v    v  is a contraction mapping on BR .and by the fixed point theorem there exists a unique u  BR 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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[7] Y. Enomoto, On a thermoelastic plate equation in an exterior domain, Math. Meth.Appl. Sci., 25 (2002), 443472. [8] L. Hörmander, Analysis of Linear Partial Differential Operators, Vol. III, Springer-Verlag, (1983). [9] T. Hosono and S. Kawashima, Decay property of regularity-loss type and application to some nonlinear hyperbolic-elliptic system, Math. Models Meth. Appl. Sci., 16 (2006),1839-1859. [10] K. Ide and S. Kawashima, Decay property of regularityloss type and nonlinear effects for dissipative Timoshenko system, Math. Models Meth. Appl. Sci., 18 (2008), 1001-1025. [11] H. J. Lee, Uniform decay for solution of the plate equation with a boundary condition of memory type, Trends in Math, 9 (2006), 51-55.

1  , then we have that 2   v     w

[6] P. M. N. Dharmawardane, J. E. Muñoz Rivera and S. Kawashima, Decay property for second order hyperbolic systems of viscoelastic materials, J. Math. Anal. Appl., 366(2010), 621-635.

[12] Y. Liu, Decay of solutions to an inertial model for a semilinear plate equation with memory, J. Math. Anal. Appl., 394 (2012), 616-632. [13] Y. Liu and S. Kawashima, Global existence and asymptotic behavior of solutions for quasi-linear dissipative plate equation, Discrete Contin. Dyn. Syst., 29 (2011), 1113-1139. [14] Y. Liu and S. Kawashima, Global existence and decay of solutions for a quasi-linear dissipative plate equation, J. Hyperbolic Differential Equations, 8 (2011), 591-614. [15] Y. Liu and S. Kawashima, Decay property for a plate equation with memory-type dissipation, Kinet. Relat. Mod., 4 (2011), 531-547. [16] S. Mao and Y. Liu, Decay of solutions to generalized plate type equations with memory,Kinet. Relat. Mod., 7 (2014), 121-131. [17] S. Mao and Y. Liu, Decay properties for solutions to plate type equations with variable coefficients, Kinet. Relat. Mod., 10 (2017), 785-797.16

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International Journal of Computer Applications Technology and Research Volume 7–Issue 07, 259-269, 2018, ISSN: 2319–8656 [18] N. Mori and S. Kawashima, Decay property for the Timoshenko system with Fourier’s type heat conduction, J. Hyperbolic Differential Equations, 11 (2014), 135-157. [19] N. Mori and S. Kawashima, Decay property of the Timoshenko-Cattaneo system, Anal.Appl., (2015), 1-21. [20] N. Mori, J. Xu and S. Kawashima, Global existence and optimal decay rates for the Timoshenko system: The case of equal wave speeds, J. Differ. Equations., 258 (2015),1494-1518. [21] M. Reed, and B. Simon, Methods of Modern Mathematical Physics, Vol. I, Academic Press, New York, (1975). [22] J. E. Muñoz Rivera, M.G. Naso and F.M. Vegni, Asymptotic behavior of the energy for a class of weakly dissipative second-order systems with memory, J. Math. Anal. Appl.,286 (2003), 692-704. [23] Y. Sugitani and S. Kawashima, Decay estimates of solutions to a semi-linear dissipative plate equation, J. Hyperbolic Differential Equations, 7 (2010), 471-501. [24] J. Xu, N. Mori and S. Kawashima, Global existence and minimal decay regularity for the Timoshenko system: The case of non-equal wave speeds, J. Differ. Equations., 259(2015), 5533-5553.

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