JAMP  Vol.5 No.4 , April 2017
Decay Rate for a Viscoelastic Equation with Strong Damping and Acoustic Boundary Conditions
Author(s) Zhiyong Ma
ABSTRACT
This paper is concerned with a nonlinear viscoelastic equation with strong damping: . The objective of the present paper is to provide some results on the long-time behavior to this equation with acoustic boundary conditions. By using the assumptions on the relaxation function due to Tatar [1], we show an arbitrary rate of decay with not necessary of an exponential or polynomial one and without the assumption condition. The result extends and improves some results given in Cavalcanti [2].

1. Introduction

In this paper, we investigate the following viscoelastic system with acoustic boundary conditons

| u t | ρ u t t Δ u Δ u t t + 0 t g ( t s ) Δ u ( x , s ) d s Δ u t = 0 , ( x , t ) ( 0 , + ) , (1.1)

u t ν ( x , t ) = 0 ( x , t ) Γ × [ 0 , + ) , (1.2)

u ( x , t ) = 0 , ( x , t ) Γ 1 × [ 0 , + ) , (1.3)

u t t ν ( x , t ) + u ν ( x , t ) 0 t g ( t s ) u ν ( x , s ) d s = y t ( x , t ) Γ 0 × [ 0 , + ) , (1.4)

u t ( x , t ) + p ( x ) y t + q ( x ) y ( x , t ) = 0 ( x , t ) Γ 0 × [ 0 , + ) , (1.5)

u ( x , 0 ) = u 0 ( x ) , u t ( x , 0 ) = u 1 ( x ) , x Ω , (1.6)

where Ω n ( n = 1 , 2 ) is a bounded domain with smooth boundary Γ = Γ 0 Γ 1 , ν is the unit outward normal to Γ , the function g represents the kernel of a memory, p and q are specific functions, and ρ is a real number such that

1 < ρ 2 n 2 if n 3 ; ρ > 1 if n = 1 , 2. (1.7)

Our problem is of the form

f ( u t ) u t t Δ u Δ u t t = 0 , (1.8)

which has several modeling features. In the case, f ( u t ) is a constant; Equation (8) has been used to model extensional vibrations of thin rods (see Love [3] , Chapter 20). In the case, f ( u t ) is not a constant; Equation (8) can model materials whose density depends on the velocity u t , for instance, a thin rod which possesses a rigid surface and with an interior which can deform slightly. We refer the reader to Fabrizio and Morro [4] for several other related models.

Recently, Liu [5] considered the following viscoelastic problem with acoustic boundary conditions

u t t Δ u + 0 t g ( t s ) Δ u ( x , s ) d s = 0 , ( x , t ) ( 0 , + ) , (1.9)

u ( x , t ) = 0 , ( x , t ) Γ 1 × [ 0 , + ) , (1.10)

u ν ( x , t ) 0 t g ( t s ) u ν ( x , s ) d s = y t ( x , t ) Γ 0 × [ 0 , + ) , (1.11)

u t ( x , t ) + p ( x ) y t + q ( x ) y ( x , t ) = 0 ( x , t ) Γ 0 × [ 0 , + ) , (1.12)

u ( x , 0 ) = u 0 ( x ) , u t ( x , 0 ) = u 1 ( x ) , x Ω , (1.13)

the authors obtain an arbitrary decay rate of the energy. In the pioneering paper [6] , Beale and Rosencrans considered the acoustic boundary condition (1.12) and the coupled impenetrability boundary condition (1.11) with a general form, which had the presence of y t t in (1.2), in a study of the model for acoustic wave motion of a fluid interacting with a so-called locally reacting surface. Recently, many authors treated wave equations with acoustic boundary conditions, see [7] [8] [9] [10] and references therein. For instance, Rivera and Qin [10] proved the polynomial decay for the wave motion with general acoustic boundary conditions by using the Lyapunov functional technique. Frota and Larkin [8] established global solvability and the exponential decay for problems (1.9)-(1.13) with g 0 . They overcame the difficulties which were arisen due to the absence of y t t in (1.12) by using the degenerated second order equation. Recently, Park and Park [9] investigated problems (1.9)-(1.13) and proved general rates of decay which depended on the behavior of g , under the additional assumption of

that 0 + g ( s ) d s .

Many authors have focused on the viscoelastic problem. In the pioneer work of Dafermos [11] [12] , existence and asymptotic stability for a one-dimensional viscoelastic problem were proved but no rate of decay has been specified. Since then problems related to viscoelasticity have attracted a great deal of attention [13] [14] [15] . It seems all started with kernels of the form g ( t ) = e β t , β > 0 , then with kernels satisfying ξ 1 g ( t ) g ( t ) ξ 2 g ( t ) , for all t 0 , for some constants ξ 1 and ξ 2 and some other conditions on the second derivative, Cavalcanti et al. [2] studied the following equation with Dirichlet boundary conditions

| u t | ρ u t t Δ u Δ u t t + g Δ u γ Δ u t = 0 (1.14)

where g Δ u = 0 t g ( t s ) Δ u ( s ) d s . They established a global existence result for γ 0 and an exponential decay of energy for γ > 0 , and studied the interaction within the | u t | ρ u t t and the memory term g Δ u . Messaoudi and Tatar [16] established, for small initial data, the global existence and uniform stability of solutions to the equation

| u t | ρ u t t Δ u Δ u t t + g Δ u = b | u | p 2 u (1.15)

with Dirichlet boundary condition, where γ 0 , ρ , b > 0 , p > 2 are constants. In the case b = 0 in (15), Messaoudi and Tatar [17] proved the exponential decay of global solutions to (15) without smallness of initial data, considering only the dissipation effect given by the memory.

In [18] [19] , the condition has been replaced by g ( t ) ξ ( t ) g ( t ) , where ξ ( t ) is a positive function. Similarly, Han and Wang [20] proved the energy decay for the viscoelastic equation with nonlinear damping

| u t | ρ u t t Δ u Δ u t t + g Δ u + | u t | m u t = 0 , (1.16)

with Dirichlet boundary condition, where ρ > 0 , m > 0 are constants. Then Park and Park [21] established the general decay for the viscoelastic problem with nonlinear weak damping

| u t | ρ u t t Δ u t t Δ u + g Δ u + h ( u t ) = 0 , (1.17)

with the Dirichlet boundary condition, where ρ > 0 is a constant. We also mention that Fabrizio and Polidoro [22] obtained the exponential decay result under the conditions that g ( t ) 0 and e α t g ( t ) L 1 ( 0 , + ) for some α > 0 . Recently, Tatar [23] improved these results by removing the last condition and established a polynomial asymptotic stability. In fact, he considered the kernels having small flat zones and these zones are not too big (see also [24] for the case of coupled system). More recently, under the assumptions that g ( t ) 0 and g ( t ) γ ( t ) L 1 ( 0 , + ) for some nonnegative function γ ( t ) , Tatar [1] genera- lized these works to an arbitrary decay for wave equation with a viscoelastic damping term. Moreover, we would like to mention some results in [25] - [30] .

The rest of our paper is organized as follows. In Section 2, we give some pre- parations for our consideration and our main result. The statements and the proofs of our main results will be given in Section 3.

For convenience, we denote the norm and scalar product in L 2 ( Ω ) by and ( , ) , respectively. C denotes a general positive constant, which may be different in different estimates.

2. Preliminaries and Main Result

For the memory kernel g we assume that:

( H 1 ) g : + + is a non-increasing differentiable function satisfying that

g ( 0 ) > 0 , l = 1 0 + g ( s ) d s > 0. (2.1)

( H 2 ) suppose that there exists a nondecreasing function γ ( t ) > 0 such

that γ ( t ) γ ( t ) = η ( t ) is a decreasing function and 0 + g ( s ) γ ( s ) d s < + .

For the functions p and q , we assume that p , q C ( Γ 0 ) and p ( x ) > 0 and q ( x ) > 0 for all x Γ 0 . This assumption implies that there exist positive constants p i , q i ( i = 0 , 1 ) such that

p 0 p ( x ) p 1 , q 0 q ( x ) q 1 , x Γ 0 . (2.2)

We use the notation

V = { u H 1 ( Ω ) : u = 0 on Γ 1 } , ( u , v ) = Ω u ( x ) v ( x ) d x , and ( u , v ) Γ 0 = Γ 0 u ( x ) v ( x ) d Γ .

Let λ and λ ˜ be the smallest positive constants such that

u 2 λ u 2 , u Γ 0 2 λ ˜ u 2 . (2.3)

Firstly, we have the following existence and uniqueness results, it can be established by adopting the arguments of [2] [31] .

Theorem 2.1 Let ( u 0 , u 1 ) ( V H 2 ( Ω ) ) × V . Assume that H 1 , H 2 and (2.2) hold. There exists a unique pair of functions ( u , y t ) , which is a solution to the problem (1.1) in the class

u L ( 0 , T , V H 2 ( Ω ) ) , u t L ( 0 , T , V ) , (2.4)

u t t L ( 0 , T , L 2 ( Ω ) ) , y , y t L 2 ( + ; L 2 ( Γ 0 ) ) . (2.5)

We introduce the modified energy functional

E ( t ) = 1 ρ + 2 u t ρ + 2 ρ + 2 + 1 2 ( 1 0 t g ( s ) d s ) u 2 + 1 2 ( g u ) ( t ) + 1 2 u t ( t ) 2 + 1 2 Γ 0 q ( x ) | y ( x , t ) | 2 d Γ , (2.6)

where

( g u ) ( t ) = 0 t g ( t s ) u ( t ) u ( s ) 2 d s .

Clearly

d d t E ( t ) = u t ( t ) 2 1 2 g ( t ) u 2 + 1 2 ( g u ) Γ 0 p y t 2 . (2.7)

To state our main result, we introduce the following notations as in [32] . For every measurable set A + , we define the probability measure g ^ by

g ^ ( A ) = 1 1 l A g ( s ) d s . (2.8)

The flatness set and the flatness rate of g are defined by

F g = { s + : g ( s ) > 0 and g ( s ) = 0 } (2.9)

and

R g = g ^ ( F g ) = 1 1 l F g g ( s ) d s (2.10)

respectively. We denote

G γ ( t ) = γ ( t ) 1 t + g ( s ) γ ( s ) d s . (2.11)

Now, we are in a position to state our main result.

Theorem 2.2 ( [23] ) Let ( u 0 , u 1 ) ( V H 2 ( Ω ) ) × V , Assume that (2.1)-(2.2) hold and R g < 1 2 . If G γ ( 0 ) < ( 1 l ) ( 2 l ) 2 , then there exist positive constants

C and ν such that

E ( t ) C γ ( t ) ν , t 0. (2.12)

3. Arbitrary Rate of Decay

Now we define

Φ ( t ) = 1 ρ + 1 Ω | u t | ρ u t u d x + Ω u t u d x + 1 2 Γ 0 p y 2 d Γ + Γ 0 u y d Γ . (3.1)

Using (1.1) and (3.1), we have

Φ ( t ) = 1 ρ + 1 u t ρ + 2 ρ + 2 u 2 + u t 2 + Ω u 0 t g ( t s ) u ( s ) d s d x + Ω Δ u t u d Γ + 2 Γ 0 u y t d Γ Γ 0 q ( x ) y 2 d Γ . (3.2)

We use here the following identity due to [1] , to give a better estimate for the

term Ω u 0 t g ( t s ) u ( s ) d s d x :

Ω u 0 t g ( t s ) u ( s ) d s d x = 1 2 ( 0 t g ( s ) d s ) u 2 + 1 2 0 t g ( t s ) u ( s ) 2 d s 1 2 ( g u ) ( t ) . (3.3)

From (2.1), (3.2) and (3.3), integration by parts and Young’s inequality, we derive for any δ 0 > 0 ,

Φ ( t ) 1 ρ + 1 u t ρ + 2 ρ + 2 + ( 1 + δ 0 ) u t ( t ) 2 ( 1 + l 2 δ 0 λ ˜ ) u 2 + 1 2 0 t g ( t s ) u ( s ) 2 d s 1 2 ( g u ) ( t ) + 1 δ 0 y t Γ 0 2 Γ 0 q ( x ) y 2 d Γ . (3.4)

As in [5] , we have:

Lemma 3.1 For u H 0 1 ( Ω ) , we have

Ω ( 0 t g ( t s ) ( u ( t ) u ( s ) ) d s ) 2 d x λ ( 1 l ) ( g u ) ( t ) . (3.5)

Now we define the functional

Ψ ( t ) = Ω ( Δ u t 1 ρ + 1 | u t | ρ u t ) 0 t g ( t s ) ( u ( t ) u ( s ) ) d s d x . (3.6)

It follows from (1.1) and (3.6) that

Ψ ( t ) = Ω Δ u t 0 t g ( t s ) ( u ( t ) u ( s ) ) d s d x ( 0 t g ( s ) d s ) u t 2 + ( 1 0 t g ( s ) d s ) Ω u ( t ) ( 0 t g ( t s ) ( u ( t ) u ( s ) ) d s ) d x + Ω ( 0 t g ( t s ) ( u ( t ) u ( s ) ) d s ) 2 d x 0 t g ( s ) d s ρ + 1 u t ρ + 2 ρ + 2 + Ω u t 0 t g ( t s ) ( u ( t ) u ( s ) ) d s d x Ω 1 ρ + 1 | u t | ρ u t 0 t g ( t s ) ( u ( t ) u ( s ) ) d s d x Γ 0 y t ( 0 t g ( t s ) ( u ( t ) u ( s ) ) d s ) d Γ = I 1 I 2 + ( 1 0 t g ( s ) d s ) I 3 + I 4 I 5 + I 6 I 7 I 8 . (3.7)

For any δ > 0 , we have

I 1 δ u t ( t ) 2 g ( 0 ) 4 δ λ ( g ( s ) u ) ( t ) . (3.8)

For all measurable sets A and F such that A = + \ F , I 3 , I 4 and I 6 can be estimated as in [1] :

I 3 δ 1 u 2 + 1 l 4 δ 1 Ω A t g ( t s ) | u ( t ) u ( s ) | 2 d s d x + 3 2 ( 1 l ) g ^ ( F ) u 2 + 1 2 F t g ( t s ) u ( s ) 2 d s , δ 1 > 0 , (3.9)

I 4 ( 1 + 1 δ 2 ) ( 1 l ) Ω A t g ( t s ) | u ( t ) u ( s ) | 2 d s d x + ( 1 + δ 2 ) ( 1 l ) g ^ ( F ) Ω F t g ( t s ) | u ( t ) u ( s ) | 2 d s d x , δ 2 > 0, (3.10)

I 6 δ 1 u t 2 + 1 4 δ 1 Ω A t g ( t s ) | u ( t ) u ( s ) | 2 d s d x + 3 2 g ^ ( F ) u t 2 + 1 2 F t g ( t s ) u ( s ) 2 d s , δ 1 > 0 , (3.11)

where g ^ is defined in (2.8). For any δ > 0 ,

I 7 δ u t ( t ) 2 g ( 0 ) 4 δ λ ( g ( s ) u ) ( t ) . (3.12)

For I 8 , for δ 3 , δ 4 > 0 , we use a different estimate as

I 8 = Γ 0 y t ( A t g ( t s ) ( u ( t ) u ( s ) ) d s ) d Γ + Γ 0 y t ( F t g ( t s ) ( u ( t ) u ( s ) ) d s ) d Γ 1 2 y t Γ 0 2 + λ ˜ ( 1 l ) 2 Ω A t g ( t s ) | u ( t ) u ( s ) | 2 d s d x + 1 4 δ 3 g ^ ( F ) y t Γ 0 2 + δ 3 λ ˜ g ^ ( F ) u 2 + 1 4 δ 4 y t Γ 0 2 + δ 4 λ ˜ ( 1 l ) F t g ( t s ) u ( s ) 2 d s . (3.13)

Taking into account these estimates in (3.6), let t be a number such that

0 t g ( s ) d s = g , we obtain that

Ψ ( t ) ( g 2 + δ 1 ) u t 2 g ρ + 1 u t ρ + 2 ρ + 2 + { ( 1 g ) ( δ 1 + 3 2 ( 1 l ) g ^ ( F ) ) + δ 3 λ ˜ g ^ ( F ) + δ } × u 2 ( 1 l ) ( 1 g 4 δ 1 + 1 + δ 2 δ 2 + λ ˜ 2 ) Ω A t g ( t s ) | u ( t ) u ( s ) | 2 d s d x 3 4 δ g ( 0 ) λ ( g u ) ( t ) + ( 1 + δ 2 ) ( 1 l ) g ^ ( F ) Ω F t g ( t s ) | u ( t ) u ( s ) | 2 d s d x + ( 1 g 2 + δ 4 λ ˜ ( 1 l ) ) F t g ( t s ) u ( s ) 2 + ( 1 2 + g ^ ( F ) 4 δ 3 + 1 4 δ 4 ) y t Γ 0 2 . (3.14)

Let

I ( t ) = Ω 0 t G γ ( t s ) | u ( s ) | 2 d s d x , (3.15)

and G γ ( t ) is given in (2.11), we define the following functional

F ( t ) = M E ( t ) + ε Φ ( t ) + Ψ ( t ) + ϵ I ( t ) , (3.16)

then we know from [1] that

I ( t ) G γ ( 0 ) u 2 η ( t ) 0 t G γ ( t s ) u ( s ) 2 d s 0 t g ( t s ) u ( s ) 2 d s . (3.17)

At the same time, we have the following lemmas.

Lemma 3.2 For M large enough, there exist two positive constants ρ 1 and ρ 2 such that

ρ 1 ( E ( t ) + I ( t ) ) F ( t ) ρ 2 ( E ( t ) + I ( t ) ) . (3.18)

Proof. See, e.g. Liu [5] .

Proof of Theorem 2.2 By using (2.7), (3.4), (3.13)-(3.16), a series of com- putations yields, for t t ,

F ( t ) ( M 2 3 4 δ g ( 0 ) λ ) ( g u ) ( t ) ( g ρ + 1 ε 1 + ρ ) u t ( t ) ρ + 2 ρ + 2 [ M 2 + g 2 δ 1 ε ( 1 + δ 0 ) ] u t ( t ) 2 + ( 1 + δ 2 ) ( 1 l ) g ^ ( F ) Ω F t g ( t s ) | u ( t ) u ( s ) | 2 d s d x + { ( 1 g ) ( δ 1 + 3 2 ( 1 l ) g ^ ( F ) ) + δ 3 λ ˜ g ^ ( F ) + δ + e G γ ( 0 ) ε ( 1 + l 2 δ 0 λ ˜ ) } u 2 + ( 1 l ) ( 1 g 4 δ 1 + 1 + δ 2 δ 2 + λ ˜ 2 ) Ω A t g ( t s ) | u ( t ) u ( s ) | 2 d s d x ( ϵ ε 2 ) 0 t g ( t s ) u ( s ) 2 d s ε 2 ( g u ) ( t ) ( 1 g 2 + δ 4 λ ˜ ( 1 l ) ) F t g ( t s ) u ( s ) 2 d s d x ϵ η ( t ) ε Γ 0 q ( x ) y 2 d Γ [ M p 0 ε δ 0 ( 1 2 + g ^ ( F ) 4 δ 3 + 1 4 δ 4 ) ] y t Γ 0 2 .
(3.19)

For n , as in [32] we introduce the sets

A n = { s + : n g ( s ) + g ( s ) 0 } . (3.20)

It is easy to see that

n A n = + \ { F g N g } , (3.21)

where F g is given in (2.9) and N g is the null set where g is not defined. Additionally, we denote F n = + \ A n , then

lim n g ^ ( F n ) = g ^ ( F g ) , (3.22)

since F n + 1 F n for all n and n F n = F g N g . Then, we take A = A n and F = F n in (3.18), it follows that

F ( t ) ( M 2 3 4 δ g ( 0 ) λ ) ( g u ) ( t ) ( g ρ + 1 ε ρ + 1 ) u t ( t ) ρ + 2 ρ + 2 [ M 2 + g 2 δ 1 ε ( 1 + δ 0 ) ] u t ( t ) 2 ( ϵ 1 + ε g 2 δ 4 λ ˜ ( 1 l ) ) 0 t g ( t s ) u ( s ) 2 d s + { ( 1 g ) ( δ 1 + 3 2 ( 1 l ) g ^ ( F n ) ) + δ 3 λ ˜ g ^ ( F n ) + δ + e G γ ( 0 ) [ σ + ( 1 σ ) ] ε 1 + l 2 + ε δ 0 λ ˜ } u 2 + ( 1 l ) ( 1 g 4 δ 1 + 1 + δ 2 δ 2 + λ ˜ 2 ) Ω A t g ( t s ) | u ( t ) u ( s ) | 2 d s d x ( ε 2 ( 1 + δ 2 ) ( 1 l ) g ^ ( F n ) ) ( g u ) ( t ) ϵ η ( t ) I ( t ) ε Γ 0 q ( x ) y 2 d Γ [ M p 0 ε δ 0 ( 1 2 + g ^ ( F n ) 4 δ 3 + 1 4 δ 4 ) ] y t Γ 0 2 , (3.23)

for some 0 < δ < 1 . Since R g = g ^ ( F g ) < 1 2 , we can choose ε , δ 2 small enough

and n , t large enough such that

ε 2 ( 1 + δ 2 ) ( 1 l ) g ^ ( F n ) 0 (3.24)

and

3 2 ( 1 l ) ( 1 g ) g ^ ( F n ) σ ε 1 + l 2 < 0 (3.25)

with σ = 3 ( 1 l ) ( 1 g ) 2 g ( 1 + l ) . Note that for t large enough. Furthermore, we

require that

1 + ε g 2 + δ 4 λ ˜ ( 1 l ) ϵ 1 G γ ( 0 ) ( ( 1 σ ) ε 1 + l 2 ( 1 g ) δ 1 δ 3 λ ˜ g ^ ( F n ) ε δ 0 λ ˜ + δ ) . (3.26)

Combining (3.24) and (3.25), we obtain

( 1 g ) ( δ 1 + 3 2 ( 1 l ) g ^ ( F n ) ) + δ 3 λ ˜ g ^ ( F n ) + δ + e G γ ( 0 ) ε 1 + l 2 + ε δ 0 λ ˜ < 0 (3.27)

Choose our constants properly so that:

M 2 3 4 δ g ( 0 ) λ M 4 , (3.28)

M p 0 ε δ 0 ( 1 2 + g ^ ( F n ) 4 δ 3 + 1 4 δ 4 ) 0 , (3.29)

( 1 l ) ( 1 g 4 δ 1 + 1 + δ 2 δ 2 + λ ˜ 2 ) M 4 n < 0 (3.30)

together with (3.22) yield

F ( t ) C 1 E ( t ) ϵ η ( t ) I ( t ) , t t . (3.31)

As η ( t ) is decreasing, we have η ( t ) η ( 0 ) for all t t . Then (3.30) becomes

F ( t ) C 1 η ( 0 ) η ( t ) E ( t ) ϵ η ( t ) I ( t ) , t t .

Since F ( t ) is equipped with E ( t ) + I ( t ) , we get

F ( t ) C 2 η ( t ) F ( t ) , (3.32)

integrating (3.31) over [ t , t ] yields

F ( t ) e C 2 t t η ( s ) d s F ( t ) , t t .

Then using the left hand side inequality in (3.17), we get

ρ 1 ( E ( t ) + I ( t ) ) e C 2 t t η ( s ) d s F ( t ) , t t .

By virtue of the continuity and boundedness of E ( t ) in the interval [ 0 , t ] , we conclude that

E ( t ) C γ ν ( t ) , t 0 (3.33)

for some positive constants C and ν .

Acknowledgements

This work was in part supported by Shanghai Second Polytechnical University and the key discipline “Applied Mathematics” of Shanghai Second Polytechnic University with contract number XXKZD1304.

Cite this paper
Ma, Z. (2017) Decay Rate for a Viscoelastic Equation with Strong Damping and Acoustic Boundary Conditions. Journal of Applied Mathematics and Physics, 5, 922-932. doi: 10.4236/jamp.2017.54081.
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