Energy Decay for a Von Karman Equation of Memory Type with a Delay Term

ISSN Online: 2327-4379 ISSN Print: 2327-4352

DOI: 10.4236/jamp.2017.59152 Sep. 22, 2017 1797 Journal of Applied Mathematics and Physics

Energy Decay for a Von Karman Equation of

Memory Type with a Delay Term

*

Sun-Hye Park

1

, Jong-Yeoul Park

2

, Yong-Han Kang

3#

1_{Center for Education Accreditation, Pusan National University, Busan, South Korea} 2_{Department of Mathematics, Pusan National University, Busan, South Korea} 3_{Institute of Liberal Education, Catholic University of Daegu, Gyeongsan, South Korea}

Abstract

We consider a von Karman equation of memory type with a delay term

(

)

( )

(

)

[ ]

2 2

0 1

0 d , , .

t

t tt tt t t

u uρ − ∆ + ∆ −α u u

∫

g t s− ∆ u s s a u a u x t+ + −τ = u v _By

in-troducing suitable energy and Lyapunov functional, we establish a general decay estimate for the energy, which depends on the behavior of g.

Keywords

Von Karman Equation, Memory Type, Delay Damping Term, General Decay Estimate

1. Introduction

Let _{Ω ⊂}2 _{be a bounded domain with sufficiently smooth boundary Γ = ∂Ω: ,}

0 1

Γ ∪Γ = Γ, Γ0∩Γ ≠ ∅1 , Γ0 and Γ1 have positive measures and

(

1, 2

)

ν

=

ν ν

be the outward unit normal vector on ∂Ω. We denote ut u_t

∂ =

∂ , 2

2 2 1 i

i u u

x =

∂ ∆ =

∂

∑

, where x=

(

x x1, 2

)

∈Ω.

In this paper, we investigate the decay of energy of solutions for a von Karman system with memory and a delay term

(

)

( )

( )

(

)

[ ]

(

)

(

)

(

(

) ( )

)

(

)

(

) ( )

(

)

(

)

( )

( ) ( )

( )

(

)

(

) ( )

( )

2 2

0 1

0

0 1 1 ₀ 1

2 2 ₀ 1

0 1 0

d , , , , in 0, ,

0 on 0, , d 0 on 0, ,

d 0 on 0, ,

,0 , ,0 , , , , , , 0, ,

t

t tt tt t t

t

t tt

t t

u u u u g t s u s s a u x t a u x t u v u

u u g t s u s s

u

u g t s u s s

u x u x u x u x x u x t f x t x t

ρ _{α τ}

ν α

ν

τ τ τ

 _{− ∆ + ∆ − − ∆ + + − = Ω× ∞}

 _∂

 = = Γ × ∞ − − = Γ × ∞

 _∂

 _∂

 _{− − − = Γ × ∞}

 ∂

 = = ∈ Ω − = − ∈ Ω×



∫

∫

∫

 

 

(1)

*This work was support by the national research foundation of Korea (Grant NRF- 2016R1D1A1B03930361).

#Corresponding author. How to cite this paper: Park, S.-H., Park,

J.-Y. and Kang, Y.-H. (2017) Energy Decay for a Von Karman Equation of Memory Type with a Delay Term. Journal of Applied Mathematics and Physics, 5, 1797-1807.

https://doi.org/10.4236/jamp.2017.59152

Received:August 9, 2017 Accepted: September 19, 2017 Published: September 22, 2017 Copyright © 2017 by authors and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).

http://creativecommons.org/licenses/by/4.0/

DOI: 10.4236/jamp.2017.59152 1798 Journal of Applied Mathematics and Physics where ρ is assumed to satisfy 0 2

2

N

ρ

< ≤

− if N≥3 or ρ>0 if N=1,2, 0

α> ; a0 is a positive constant; a1 is a real number; g is the kernel of the

memory term; τ >0 _{represents the time delay;} u u f₀, ,_{1 0} are given functions belonging to suitable spaces; and the Airy stress function v satisfies the following elliptic problem

[ ]

(

)

(

)

2 _{, in 0, ,}

0 on 0, ).

v u u

v v

ν

∆ = − Ω × ∞



 ∂

= = Γ × ∞

 _∂



(2)

The von Karman bracket

[ ]

u,

φ

_{is given by}

[ ]

u,

φ

≡ux x x x1 1

φ

2 2+ux x x x2 2

φ

1 1−2ux x x x1 2

φ

1 2, and

(

)

(

)

1u u 1

µ

B u1 , 2u

ν

u 1

µ

B u2 ,

∂

= ∆ + − = ∆ + −

∂

 

here 0,1 2 µ_∈ _

  is Poisson’s ratio,

1 2 2 2 1 1

2 2

1 2 1 2 x x 1 x x 2 x x,

B u= ν ν u −ν u −ν u

(

)

1 2

(

2 2 1 1

)

2 2

2 1 2 x x 1 2 x x x x .

B u

ν

ν

u

ν ν

u u

τ

∂  

= _ − + − _

∂

From the physical point of view, problem (1) describes small vibrations of a thin homogeneous isotropic plate of uniform thickness of α _; u u x t=

( )

, denotes the transversal displacement of the plate; the Airy stress function

( )

,

v v x t= _{is a vibrating plate.}

When a1=0 and ρ=0, problem (1) was studied by many authors [1]-[8].

The authors in [1] [3] [4] proved uniform decay rates for the von Karman system with frictional dissipative effects in the boundary. The stability for a von Karman system with memory and boundary memory conditions was treated in

[5] [6] [7] [9]. They proved the exponential or polynomial decay rate when the

relaxation function decay is at the same rate. The aim of this work is to prove a general decay result for a nonlinear von Karman equation of memory type with a delay term in the first equation of (1), when the relaxation function does not necessarily decay exponentially or polynomially. As for the works about general decay for viscoelastic system, we refer [10]-[15] and references therein. Considering delay term a u x t1 t

(

, −

τ

)

, the problem is different from existing

literature. Time delays arise in many applications depending not only on the present state but also on some past occurrences. And the presence of delay may be a source of instability (see e.g. [16] [17]). Thus, recently, the control of partial differential equations with time delay effects has become an active area of research (see [18] [17] [19] [20] and references therein). Nicaise and Pignotti [17] examined a wave equation with a time-delay of the form

( )

,

( )

, 0

( )

, 1

(

,

)

0.

tt t t

u x t − ∆u x t a u x t a u x t+ + − =

τ

DOI: 10.4236/jamp.2017.59152 1799 Journal of Applied Mathematics and Physics They proved that the energy of the problem decays exponentially under the condition

1 0

0< <a a ,

and there exists a sequence of delays such that instability occurs in the case

1 0

a a≥ _{. Kirane and Said-Houari [21] considered a viscoelastic wave equation} with a delay

( )

,

( )

, ₀

(

) ( )

, d 0

( )

, 1

(

,

)

0.

t

tt t t

u x t − ∆u x t +

∫

g t s u x s s a u x t− ∆ + +a u x t−

τ

=

(4) The authors proved the existence of a solution and a general decay result under the condition

1 0

0<a a≤ . ₍₅₎ They showed that the energy of solutions is still asymptotically stable even if

1 0

a a= _{owing to the presence of the viscoelastic damping. Recently, Wu [20]} obtained similar decay results as in [21] for problem (1) without von Karman bracket

[ ]

u v, under the condition (5). Motivated by these results, we prove a general decay result for a nonlinear viscoelastic von Karman Equation (1) with a time-delay under the condition

1 0,

a ≤a

(6) which is an extension and improvement of the previous result from [20] to a nonlinear viscoelastic von Karman equation without the assumption a1>0.

The plan of this paper is as follows. In Section 2, we give some notations and materials needed for our work. In Section 3, we derive general decay estimate of the energy.

2. Statement of Main Results

Throughout this paper, we denote

( )

{

3

}

0

: 0 on ,

V= u H∈ Ω u= Γ

( )

2

0

: u 0 on ,

W =_u H∈ Ω u= ∂_ν = Γ _ ∂

 

( )

( ) ( )

( )

( ) ( )

1 1

, d and , d .

uφ u x φ x x uφ _Γ u x φ x

Ω Γ

=

∫

=

∫

Γ

For a Banach space X, ⋅ _{X denotes the norm of X. For simplicity, we denote}

( )

2 L Ω

⋅ _by ⋅ _and 2_{( )} 1 L Γ ⋅ _by

1 Γ

⋅ _{, respectively. We define for all} 1≤ < ∞p

( )

pd .

p p

u =

∫

_Ωu x x

From now on, we shall omit x and t in all functions of x and t if there is no ambiguity, and c denotes a generic positive constant different from line to line or even in the same line.

For 0 1

2 µ

< < , the bilinear form a

( )

⋅ ⋅, is defined by

( )

,

{

x x x x_{1 1 1 1} x x_{2 2} x x_{2 2}

(

x x x x_{1 1 2 2} x x_{2 2} x x_{1 1}

)

2 1

(

)

x x_{1 2} x x_{1 2}

}

d .

a uφ u φ u φ µ u φ u φ µ u φ x

Ω

DOI: 10.4236/jamp.2017.59152 1800 Journal of Applied Mathematics and Physics A simple calculation, based on the integration by parts formula, yields

( )

(

)

2

1 2

d , , , .

uφ a uφ u φ uφ

ν Γ

Ω

Γ

∂

 

∆ Ω = −_{ } +

∂

 

∫

 

Thus, for

(

_u,φ

)

_∈

(

_H4

( )

_{Ω ∩W W}

)

_{× it holds}

( )

(

)

₁

1 2

1 2

d ( , , , .

uφ a uφ u φ uφ

ν Γ

Ω

Γ ∂

 

∆ Ω = −_{ } +

∂

 

∫

 

Since Γ ≠ ∅0 we know (see e.g. [1]) that a u u

( )

, is equivalent to the

( )

2

H Ω _{norm on W, i.e.}

( )

( )

( )

2 2

2 2

1 H , 2 H for some , > 0.1 2

c u _Ω ≤a u u ≤c u _Ω c c

(8) This and Sobolev imbedding theorem imply that for some positive constants p

C , Cp and Cs

( )

₁

( )

( )

2 2 2

, , , and , , .

p p s

u ≤C a u u u_Γ ≤C a u u ∇u ≤C a u u ∀ ∈u W

(9) By (7) and Young’s inequality, we see that

( )

2_{( )} 2_{( )}

2 5 2

, for all 0.

8

H H

a u

φ

δ φ

φ

δ

δ

Ω Ω

≤ + >

From this and (8), it holds that

( )

( )

₂

( )

1

5

, , , for all 0.

8

a u a u u a

c

φ δ φ φ δ

δ

≤ + >

(10) We introduce the relative results of the Airy stress function and von Karman bracket

[ ]

⋅ ⋅, _.

Lemma 2.1. ([4]) If u,φ and ψ belong in _H2

( )

_{Ω and at least one of}

them belongs in 2

( )

0

H Ω _{, then}

(

[ ]

u, ,φ ψ

)

=

(

[

u, ,ψ φ

]

)

.

Lemma 2.2. ([1]) Let _{u H∈} 2

( )

_{Ω and v be the Airy stress function}

satisfying (2). Then, the following relations hold:

[ ]

u v, ∈L2

( )

Ω and

[ ]

u v, ≤C u _H2_{( )Ω} v_W2,∞_{( )Ω} ≤C u _H2_{( )Ω} u_H22_{( )Ω} .

Now, we state the assumptions for problem (1).

(H1) For the relaxation function g, as in [11] [15], we assume that :

g +→+ is a nonincreasing differentiable function satisfying g

( )

0 >0,

( )

0: ₀ d 1₂

l =

∫

∞g s s< and

( )

( ) ( )

for 0,

g t′ ≤ −

ζ

t g t t≥ ₍₁₁₎ where ζ:+→+ is a nonincreasing differentiable function.

Theorem 2.1. Assume that (H1) is hold. Then, for the initial data

(

)

(

4

( )

)

(

3

( )

)

2

(

( )

)

0, ,1 0 0,1 ,

u u f ∈ H Ω ∩W × H Ω ∩V ×L Ω × problem (1) has a unique weak solution u in the class

( )

(

_{0, ;} 4

)

1

(

_{0, ;} 3

( )

)

_.

u C∈ T H Ω ∩W ∩C T H Ω ∩V

DOI: 10.4236/jamp.2017.59152 1801 Journal of Applied Mathematics and Physics

3. General Decay of the Energy

In this section we shall prove a general decay rate of the solution for problem (1). For simplicity of notations, we denote

(

) ( )

0 d ,

t

g u∗ =

∫

g t s u s s−

(

) ( ) ( )

2

0 d ,

t

g u =

∫

g t s u t u s− − s

and

(

)

(

( ) ( ) ( ) ( )

)

2

0 , d .

t

g∂ =u

∫

g t s a u t u s u t u s− − − s

From (9), we see that

2 _.

p

g u C g ≤ ∂ u

₍₁₂₎

From now on, we shall omit t in all functions of t if there is no ambiguity, and

c denotes a generic positive constant different in various occurrences. Multiplying the first equation of (1) by ut, we have

( )

2

(

(

)

)

(

)

0 t 1 t , t , ,t

E t′ = −a u −a u t−

τ

u +a g u u∗

(13) where

( )

2 2

( )

2

2

1 1 _, 1 _.

2 t 2 t 2 4

E t _ρ u ρ_ρ+ α u a u u v +

= + ∇ + + ∆

+

From the symmetry of a

( )

⋅ ⋅, _{, we see that for any u C∈} 1

(

_{0, ;T H}2

( )

_Ω

)

(

)

( ) ( )

2 2

(

( )

)

( )

0

1 1 1 d

, , d , .

2 2 2 d

t t

a g u u g t a u u g u g u g s s a u u

t 

′

∗ = − + ∂ − _ ∂ − _



∫



  (14)

Moreover, (10) gives

(

)

(

)

(

( ) ( ) ( )

)

(

( )

)

( )

( )

(

)

( )

0 0

2 2 0

1

, , d d ,

5

2 d , .

8

t t

t

a g u u g t s a u s u t u t s g s s a u u

g s s a u u g u c

∗ = − − +

≤ + ∂

∫

∫

∫



(15)

Now, we define a modified energy by

( )

(

( )

)

( )

( )

2 2

2 0

2 2

2

1 _{1 1 d ,}

2 2 2

1 1 _{d ,}

2 4 2

t

t t

t t t

t u u g s s a u u

p

g u v u s s

ρ ρ

τ

α ρ

+ +

−

= + ∇ + −

+

+ ∂ + ∆ +

∫

∫





where p is a positive constant satisfying

1 2 0 1.

a ≤ ≤p a − a

(16) It is noted that

( )

( )

0

1 1

E t t

l

≤

−  . Therefore, it is enough to obtain the desired decay for the modified energy 

( )

t which will be done below.

Lemma 3.1. There exist non-negative constants α1 and α2 satisfying

( )

2

(

)

2

( ) ( )

2

1 t 2 t 1₂ , 1₂ .

t α u α u t τ g t a u u g u

′ ≤ − − − − + ′ ∂

DOI: 10.4236/jamp.2017.59152 1802 Journal of Applied Mathematics and Physics Proof. Applying (14) to the last term in the right hand side of (13), we have

( )

(

(

)

)

( ) ( )

(

)

2 2

0 1

2 2

1 1

, ,

2 2

.

2 2

t t t

t t

t a u a u t u g t a u u g u

p _u p _{u t}

τ

τ

′ = − − − − + ′ ∂

+ − −

 

By Young’s inequality,

(

)

(

)

1 2 1

(

)

2

1 t , t ₂ t ₂ t .

a a

a u t

τ

u u u t

τ

− − ≤ + −

Thus, we have

( )

(

)

( ) ( )

2 2

1 1

0

2

2 2 2 2

1 _, 1 _.

2 2

t t

a a

p p

t a u u t

g t a u u g u

τ

   

′ ≤ − − −  − −  −

   

′

− + ∂





Putting 1

1 0 _{2 2}

a p a

α

= − − _, 1

2 _{2 2}

a p

α

= − _{and considering (16), we complete} the proof. 

Now, let us define the perturbed modified energy by

( )

( )

( )

( )

( )

,

L t =N t + Ψ

ε

t + ϒ

θ

t + Φ t

₍₁₇₎

where

( )

1

(

,

)

(

,

)

,

1 t t t

t u u uρ α u u

ρ

Ψ = + ∇ ∇

+

( )

( )

( )

2

e d ,

t _{t s} t t

t _τ − − u s s

−

ϒ =

∫

( )

(

) ( ) ( )

(

)

(

)

(

( )

( )

)

0

0

1 _{, d}

1

, d .

t

t t

t

t

t g t s u t u s u u s

g t s u t u s u s

ρ

ρ α

Φ = − − −

+

− − ∇ − ∇ ∇

∫

∫

Then, it is easily shown that L t

( )

is equivalent with 

( )

t for all t≥0. 

Lemma 3.2. There exist positive constants C C3, 4 and t0>0 satisfying

( )

( )

2

3 4 0

d _{for .}

dtL t ≤ −C t C g+ ∂ u t t≥

(18)

Proof. Poincare’s inequality gives

( )

( )

( )

2

( )

2

(

)

2

e d e

t _{t s}

t t t

t

t u s s u t τ u t

τ

τ

− − −

−

′

ϒ = −

∫

+ − −

(19)

( )

2 ₂

( )

2

(

)

2

1

e−τ _tt_−τ u st ds

λ

u tt e−τ u tt

τ

,

≤ −

∫

+ ∇ − −

(20) where λ1 is the embedding constant from H1

( )

Ω to L2

( )

Ω . Using the

problem (1) and (14), we have

( )

( )

(

)

(

)

1

1 1

2 2

2

1 2

1 _{, ,}

1

, , ,

tt

t t u

t u u u a u u

u

u u u a g u u

ρ

ρ α α

ρ ν

ν + +

Γ Γ

Γ

∂

 

′

Ψ = + ∇ + _{ } −

+ ∂ 

∂

 

+_{ } − + ∗

∂

DOI: 10.4236/jamp.2017.59152 1803 Journal of Applied Mathematics and Physics

(

)

( )

( )

(

( ) ( )

)

(

)

(

(

)

)

(

)

( ) (

)

(

)

(

(

)

)

( )

(

)

( )

(

)

(

(

)

)

1 1 1 2 0 2 0 1 2 2 2 2 0 1 2 2 2 0 2 2 0 1 2 1

, , d

, , ,

1 _{, ,}

1

, ,

1 _{1 2 d ,}

1

5 _{, , .}

8 t t t t t t t t t t t t u t

g t s u s u s u t s

a u u a u t u v v

u u a u u a g u u

a u u a u t u v

u u g s s a u u

g u a u u a u t u v

c ρ ρ ρ ρ ν τ α ρ τ α ρ τ Γ Γ + + + + _{ ∂ }    − −   − ∂ _{ }    − − − − ∆ = + ∇ − + ∗ + − − − − ∆ ≤ + ∇ − − + + ∂ − − − − ∆

∫

∫

   (21)

Young and Poincaré’s inequalities produce

(

)

( )

1 02 2 2

0 , t , ₄ p t ,

a C a u u ηa u u λ u

η

− ≤ + ∇

(

)

(

)

( )

12

(

)

2

1 t , , ₄ p t .

a C

a u t τ u ηa u u u t τ η

− − ≤ + −

Substituting these into (21), we derive

( )

(

) ( )

(

)

2 2

2 1 0 2

0 2 2 2 2 1 2 2 1

1 _{1 2 2 ,}

1 4 5 _. 4 8 p t t p t a C

t u u l a u u

a C

g u u t v

c ρ ρ λ α η ρ η τ η + +   ′

Ψ ≤ +_ + _ ∇ − − −

+ _{ }

+ ∂ + − − ∆

(22)

Similarly, we get from (1) that

( )

(

) (

)

(

( ) ( ) ( )

)

(

)

(

( ) ( ) ( )

)

(

) ( ) ( ) ( )

(

)

(

)

(

(

( )

)

( ) ( )

)

(

[ ]

(

) ( ) ( )

(

)

)

(

) ( ) ( )

(

)

(

)

( )

(

)

(

( )

( )

)

(

)

( )

0 0 0 0 0

1 0 0

2 2

0 0

2

0 0

, d d

, d , d

, d , , d

1 _{, d} 1 _d

1 1

, d d

: t t t t t t t t t t

t t t

t t

t t

t g t s g t a u u t u s s

g t s a u t u t u s s a g t s u t u t u s s

a g t s u t t u t u s s u v g t s u t u s s

u u g t s u t u s s g s s u

u g t s u t u s s g s s u

I

ρ ρ

ρ

τ τ τ

τ ρ ρ α α + + ′

Φ = − − − −

+ − − − − − − − − − − − − ′ − − − − + + ′ − ∇ − ∇ − ∇ − ∇ =

∫

∫

∫

∫

∫

∫

∫

∫

∫

∫

( )

2

( )

2

1 6 0 2 7 0

1 _{d d}

1

t t

t t

I g s s u ρ_ρ I α g s s u

ρ + + + + − + − ∇ +

∫

∫

(23)

In what follows we will estimate the terms in right hand side of (23). By similar arguments given in [8], we have

( )

2 0 0 2

1 0 0 2 2

1 1

5 5

, ,

8 8

l l

I l a u u l g u

c c

δ

δ

  ≤ +_ + + _ ∂   

( )

2

2 0 2

1

5

, ,

8

I l a u u g u

c δ δ ≤ + ∂

( )

(

)

2 2 7 0 _, 4 s t g C

I δα u α g u

δ ′

DOI: 10.4236/jamp.2017.59152 1804 Journal of Applied Mathematics and Physics and

[ ]

(

) ( ) ( )

(

)

( ) ( )

(

)

( )

2 2 2 2 5 ₀ 2

2 0 2

0 2 1 , d 4 4 , . 4 t p H H p

I u v g t s u t u s s

C l

c u u g u

C l

c a u u g u

δ δ δ δ δ δ Ω Ω ≤ + − − ≤ + ∂ ≤ + ∂

∫

 

Using Young inequality and the fact that imbedding _H1

( )

_{Ω L}2(ρ+1) _is continuous, we infer

( ) ( )

(

(

) ( ) ( )

)

( ) ( )

(

( )

)

(

)

( )

( )

( )

( )

(

)

2 2 1

6 2 1 0

2 2 1

2 1

2 _{0 0}

2 1

2 2

2

1 1 _d

1 4

1 1 _{d d}

1 4

0 2 0

,

1 4 1

t t t t t p t

I u g t s u t u s s

u g s s g t s u u s s

g C

u g u

ρ ρ ρ ρ ρ ρ δ ρ δ δλ ρ δ δλ

ρ α δ ρ

+ + + + +  _′  ≤ _ + − − − _ +    _{′ ′}  ≤  ∇ + − −  +     _′ ≤ ₊   ∇ − ₊ ∂  

∫

∫

∫

 

where λ2 is the embedding constant from V to L2(ρ+1)

( )

Ω .

Young’s inequality and (10) give

2 2

2 0 0 1 2

3 t ₄ p

a l C

I δ u λ g u

δ

≤ ∇ + ∂

and

(

)

2 1 02 2

4 t ₄ p .

a l C

I δ u t τ g u

δ

≤ − + ∂

Combining these estimates with (23), we get

( )

(

)

( )

(

( )

)

( )

( )

(

)

(

( )

)

( )

( )

2 1 2 2 2

0 0 ₀

2 2 _{2 2}

2 0 2 0 , d 1 0 0

1 _{d .}

1 4

t

t

t s p

t t

t l l c a u u g s s u

g C g C

u t g s s u cg u g u

ρ ρ ρ ρ δλ α

δ α δ δ

ρ α δ τ ρ δ + + +  _{ }   _{ }      ′

Φ ≤ + + +_ − + + _ ∇

+       + ′ + − − + ∂ − ∂ +

∫

∫

   (24)

Since g is positive, for any t0>0 we have

( )

( )

0

0

0 d 0 d :

t t

g s s≥ g s s g=

∫

∫

for all

0

t t≥ _{. Thus, combining (17), (22) and (24), we arrive}

( )

( )

( )

( )

( )

( )

(

)

(

(

)

(

)

)

( )

2 1

2 2 2

2

1 0 2

0 1

2 2

0 2 0 0 0

d d

2 0

1

1 4

1 _{1 2 2 ,}

1

p

t

t

L t N t t t

t

C a

g u

g u l l l c a u u

ρ ρ ρ ρ λ λ α

α δ α α θλ

ρ η η δ ρ + + + ′ ′ ′

= + Ψ + Φ

DOI: 10.4236/jamp.2017.59152 1805 Journal of Applied Mathematics and Physics

(

)

( )

( )

( )

(

)

2

2 2

2 1

2 2

e e d

4

0 0

.

2 4

t p

t _t t

s p

a C

v u t u s s

g C g C

N _{g u c c g u}

τ τ

τ

θ δ τ

η α

η

− −

−

 

− ∆ −_ − − _ − −

 

+

 

′

+ −  ∂ + + ∂

 

 

∫

 

  

(25)

First, we fix η>0 and θ >0 such that 1 2− l0−2η>0 and

α

g0−

θλ

12 >0,

respectively. Next, we choose N>0 so large that

( )

0

( )

0 0

2 4

s p

g C g C

N α

η +

− > ,

and >0 sufficiently small such that g0− > 0,

2 2

1 0

2

0 1 ₄p 0

C a

g λ

α θλ α

η

 

− − _ + _>

 

 and _e 12 ₀

4 p a C

τ

θ

η

− ₋ _{> . Finally, taking δ >0}

so small that

( )

( )

2 1 2

2 2

1 0

2

0 1

2 0

1 0

4 1

p C a g

ρ ρ

λ

λ α

α θλ α δ α

η ρ

+

 _{ } 

 _{ } 

  _{   }

− − _ + _− _ + + _>

+

   

 

 



 ,

(

)

(

2

)

0 0 0

1 2− l −2η −δ l + +l c >0

 and _e 12 ₀

4 p a C

τ

θ δ

η

− ₋ _{− > , we complete the} proof.

Theorem 3.1. There exist positive constants C0,ω and t0>0 such that

( )

0 ( )d

0e for 0.

t

t s s

E t ≤C −ω ζ∫ t t≥

Proof. Multiplying (18) by

ζ

( )

t , using (11) and (17), we get

( ) ( )

( ) ( )

( )

( ) ( )

( ) ( )

(

( )

)

2

3 4

2

3 4

3 4 2 .

t L t C t t C t g u

C t t C g u

C t t C t

ζ ζ ζ

ζ ζ

′ ≤ − + ∂

′

≤ − − ∂

′

≤ − + −

 

 

 

Since ζ is nonincreasing, we have

( ) ( )

( )

(

ζ t L t +2C4 t

)

′≤ −C3ζ

( ) ( )

t  t fort t≥ 0. Thus, by letting 

( )

t =

ζ

( ) ( )

t L t +2C4

( )

t , we get

( )

t c t

ζ

( ) ( )

t fort t0.

′ ≤ − ≥

 

Since

ζ

( )

t _{is a nonincreasing positive function, we can easily observe that}

( )

t

 is equivalent to 

( )

t . Subsequently, it follows that

( )

t c t

ζ

( ) ( )

t fort t0.

′ ≤ − ≥

 

Integrating this over

( )

t t0, , we conclude that

( )

( )

0 ( )d

1 e for all 0.

t t

c s s

t ≤ t −∫ ζ t t≥

 

Consequently, the equivalent relations of ,L and  yield the desired result. 

4. Conclusion

DOI: 10.4236/jamp.2017.59152 1806 Journal of Applied Mathematics and Physics equation with constant time delay in the velocity by establishing proper Lyapunov functionals corresponding to the delay effect. In the future work, we will consider the equation with time-varying delay effect.

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https://doi.org/10.1137/1.9781611970821

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[7] Park, J.Y. and Park, S.H. (2005) Uniform Decay for a Von Karman Plate Equation with a Boundary Memory Condition. Mathematical Methods in the Applied Sciences, 28, 2225-2240. https://doi.org/10.1002/mma.663

[8] Park, S.H., Park, J.Y. and Kang, Y.H. (2012) General Decay for a Von Karman Equ-ation of Memory Type with Acoustic Boundary Conditions. Zeitschrift für Ange-wandte Mathematik und Physik, 63, 813-823.

https://doi.org/10.1007/s00033-011-0188-2

[9] Lee, M.J., Park, J.Y. and Kang, Y.H. (2015) Exponential Decay Rate for a Quasili-near Von Karman Equation of Memory Type with Acoustic Boundary Conditions. Boundary Value Problems, 2015, 122. https://doi.org/10.1186/s13661-015-0381-x [10] Cavalcanti, M.M., Domingos Cavalcanti, V.N. and Martinez, P. (2008) General

De-cay Rate Estimates for Viscoelastic Dissipative Systems. Nonlinear Analysis, 68, 177-193. https://doi.org/10.1016/j.na.2006.10.040

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