_{1}

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].

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 ⊆ ℝ 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 [

Recently, Liu [

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 [

that ∫ 0 + ∞ g ( s ) d s .

Many authors have focused on the viscoelastic problem. In the pioneer work of Dafermos [

| 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 [

| 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 [

In [

| 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 [

| 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 [

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.

For the memory kernel g we assume that:

( H 1 ) g : ℝ + → ℝ + 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 [

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 ( ℝ + ; 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 [

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 ∈ ℝ + : 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 ( [

C and ν such that

E ( t ) ≤ C γ ( t ) − ν , t ≥ 0. (2.12)

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 [

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 [

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 = ℝ + \ F , I 3 , I 4 and I 6 can be estimated as in [

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 [

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 [

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

For n ∈ ℕ , as in [

A n = { s ∈ ℝ + : n g ′ ( s ) + g ( s ) ≤ 0 } . (3.20)

It is easy to see that

∪ n A n = ℝ + \ { 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 = ℝ + \ 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 .

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.

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