In this paper, we are concerned with a coupled system of nonlinear viscoelastic wave equations with weak damping

{ u t t − Δ u + ∫ 0 t g 1 ( t − τ ) Δ u ( τ ) d τ + u t = f 1 ( u , v ) , in  Ω × ( 0 , + ∞ ) , v t t − Δ v + ∫ 0 t g 2 ( t − τ ) Δ v ( τ ) d τ + v t = f 2 ( u , v ) , in  Ω × ( 0 , + ∞ ) , u = v = 0 , on  ∂ Ω × ( 0 , + ∞ ) , u ( ⋅ , 0 ) = u 0 , u t ( ⋅ , 0 ) = u 1 , v ( ⋅ , 0 ) = v 0 , v t ( ⋅ , 0 ) = v 1 , in  Ω ,

where Ω ⊆ R n (n=1,2,3) is a bounded domain with smooth boundary ∂Ω, u and v represent the transverse displacements of waves. The functions g 1 and g 2 denote the kernel of a memory, f 1 ( u , v ) and f 2 ( u , v ) are the nonlinearities.

In recent years, many mathematicians have paid their attention to the energy decay and dynamic systems of the nonlinear wave equations, hyperbolic systems and viscoelastic equations.

Firstly, we recall some results concerning single viscoelastic wave equation. Kafini and Tatar 1 considered the following Cauchy problem:

{ u t t − Δ u + ∫ 0 t g ( t − s ) Δ u ( x , s ) d s = 0 , x ∈ R n , t > 0 , u ( x , 0 ) = u 0 ( x ) , u t ( x , 0 ) = u 1 ( x ) , x ∈ R n .

They established the polynomial decay of the first-order energy of solutions for compactly supported initial data and for a not necessarily decreasing relaxation function. Later Tatar 2 studied the problem (1.2) with the Dirichlet boundary condition and showed that the decay of solutions was an arbitrary decay not necessarily at exponential or polynomial rate. Cavalcanti et al. 3 studied the following equation with Dirichlet boundary condition:

| u t | ρ u t t − Δ u − Δ u t t + g ∗ Δ u − γ Δ u t = 0 .

The authors established a global existence result for γ ≥ 0 and an exponential decay of energy for γ > 0 . They studied the interaction within the | u t | ρ u t t and the memory term g ∗ Δ u . Later on, several other results were published based on 456. For more results on a single viscoelastic equation, we can refer to 7891011121314.

For a coupled system, Agre and Rammaha 15 investigated the following system:

{ u t t − Δ u + | u t | m − 1 u t = f 1 ( u , v ) , in  Ω × ( 0 , T ) , v t t − Δ v + | v t | r − 1 v t = f 2 ( u , v ) , in  Ω × ( 0 , T ) ,

where Ω⊆Rn (n=1,2,3) is a bounded domain with smooth boundary. They considered the following assumptions on f i (i = 1 , 2 ):

(A₁) Let

F ( u , v ) = a | u + v | p + 1 + 2 b | u v | p + 1 2 , f 1 ( u , v ) = ∂ F ∂ u , f 2 ( u , v ) = ∂ F ∂ v

with a , b > 0 , p ≥ 3 if n = 1 , 2 and p = 3 if n = 3 ; m , r ≥ 1 .

(A₂) There exist two positive constants c 0 , c 1 such that for all u , v ∈ R 2 , F ( u , v ) satisfies

c 0 ( | u | p + 1 + | v | p + 1 ) ≤ F ( u , v ) ≤ c 1 ( | u | p + 1 + | v | p + 1 ) .

Under the assumptions (A₁)-(A₂), they established the global existence of weak solutions and the global existence of small weak solutions with initial and Dirichlet boundary conditions. Moreover, they also obtained the blow up of weak solutions. Mustafa 16 studied the following system:

{ u t t − Δ u + ∫ 0 t g 1 ( t − τ ) Δ u ( τ ) d τ + f 1 ( u , v ) = 0 , v t t − Δ v + ∫ 0 t g 2 ( t − τ ) Δ v ( τ ) d τ + f 2 ( u , v ) = 0 ,

in Ω × ( 0 , + ∞ ) with initial and Dirichlet boundary conditions, proved the existence and uniqueness to the system by using the classical Faedo-Galerkin method and established a stability result by multiplier techniques. But the author considered the following different assumptions on fi (i=1,2) from (A₁)-(A₂):

( A 1 ′ ) f i : R 2 → R (i=1,2) are C 1 functions and there exists a function F such that

f 1 ( x , y ) = ∂ F ∂ x , f 2 ( x , y ) = ∂ F ∂ y , F ≥ 0 , x f 1 ( x , y ) + y f 2 ( x , y ) ≥ F ( x , y ) ,

( A 2 ′ )

| ∂ f i ∂ x ( x , y ) | + | ∂ f i ∂ y ( x , y ) | ≤ d ( 1 + | x | β i 1 − 1 + | y | β i 2 − 1 ) ,

for all ( x , y ) ∈ R 2 , where the constant d > 0 and β i j ≥ 1 , ( n − 2 ) β i j ≤ n for i , j = 1 , 2 .

Han and Wang 17 considered the following coupled nonlinear viscoelastic wave equations with weak damping:

{ u t t − Δ u + ∫ 0 t g 1 ( t − τ ) Δ u ( τ ) d τ + | u t | m − 1 u t = f 1 ( u , v ) , in  Ω × ( 0 , T ) , v t t − Δ v + ∫ 0 t g 2 ( t − τ ) Δ v ( τ ) d τ + | v t | r − 1 v t = f 2 ( u , v ) , in  Ω × ( 0 , T ) , u = v = 0 , on  ∂ Ω × ( 0 , T ) , u ( ⋅ , 0 ) = u 0 , u t ( ⋅ , 0 ) = u 1 , v ( ⋅ , 0 ) = v 0 , v t ( ⋅ , 0 ) = v 1 , in  Ω ,

where Ω⊆Rn is a bounded domain with smooth boundary ∂Ω. Under the assumptions (A₁)-(A₂) on fi (i=1,2), the initial data and the parameters in the equations, they established the local existence, global existence uniqueness and finite time blow up properties. When the weak damping terms | u t | m − 1 u t , | v t | r − 1 v t were replaced by the strong damping terms − Δ u t , − Δ v t , Liang and Gao 18 showed that under certain assumption on initial data in the stable set, the decay rate of the solution energy is exponential when they take

a,b>0 and p > − 1 if n=1,2, − 1 < p ≤ 1 if n=3. Moreover, they obtained that the solutions with positive initial energy blow up in a finite time for certain initial data in the unstable set. For more results on coupled viscoelastic equations, we can refer to 192021.

If we take m = r = 1 in (1.4), the system will be transformed into (1.1). To the best of our knowledge, there is no result on general energy decay for the viscoelastic problem (1.1). Motivated by 1617, in this paper, we shall establish the general energy decay for the problem (1.1) by multiplier techniques, which extends some existing results for a single equation to the case of a coupled system. The rest of our paper is organized as follows. In Section 2, we give some preparations for our consideration and our main result. The statement and the proof of our main result will be given in Section 3.

For the reader’s convenience, we denote the norm and the scalar product in L 2 ( Ω ) by ∥ ⋅ ∥ and ( ⋅ , ⋅ ) , respectively. C 1 denotes a general constant, which may be different in different estimates.

To state our main result, in addition to (A₁)-(A₂), we need the following assumption.

(A₃) g i : R + → R + , i=1,2, are differentiable functions such that

g i ( 0 ) > 0 , 1 − ∫ 0 + ∞ g i ( s ) d s = l i > 0 ,

and there exist nonincreasing functions ξ 1 , ξ 2 : R + → R + satisfying

g i ′ ( t ) ≤ − ξ i ( t ) g i ( t ) , t ≥ 0 .

Now, we define the energy functional

E ( t ) = 1 2 ∫ Ω ( u t 2 + ( 1 − ∫ 0 t g 1 ( s ) d s ) | ∇ u | 2 ) d x + 1 2 ( g 1 ∘ ∇ u ) ( t ) + 1 2 ( g 2 ∘ ∇ v ) ( t ) + 1 2 ∫ Ω ( v t 2 + ( 1 − ∫ 0 t g 2 ( s ) d s ) | ∇ v | 2 ) d x − ∫ Ω F ( u , v ) d x

and the functional

D ( t ) = ( 1 − ∫ 0 t g 1 ( s ) d s ) ∥ ∇ u ( t ) ∥ 2 + ( 1 − ∫ 0 t g 2 ( s ) d s ) ∥ ∇ v ( t ) ∥ 2 + 2 [ ( g 1 ∘ ∇ u ) ( t ) + ( g 2 ∘ ∇ v ) ( t ) ] − 4 ∫ Ω F ( u ( t ) , v ( t ) ) d x ,

where

( g ∘ y ) ( t ) = ∫ 0 t g ( t − s ) ∥ y ( t ) − y ( s ) ∥ 2 d s .

The existence of a global solution to the system (1.1) is established in 17 as follows.

Proposition 17

Let (A₁)-(A₃) hold. Assume that D ( 0 ) = ∥ ∇ u 0 ∥ 2 + ∥ ∇ v 0 ∥ 2 − 4 ∫ Ω F ( u 0 , v 0 ) d x > 0 , 2 p C 0 l ( E ( 0 ) l ) p − 1 2 < 1 and that ( u 0 , u 1 ) ∈ H 0 1 ( Ω ) × L 2 ( Ω ) , ( v 0 , v 1 ) ∈ H 0 1 ( Ω ) × L 2 ( Ω ) , where C 0 is a computable constant and l = min { l 1 , l 2 } . Then the problem (1.1) has a unique global solution ( u ( t ) , v ( t ) ) satisfying

( u ( t ) , u t ( t ) ) ∈ C ( R + ; H 0 1 ( Ω ) × L 2 ( Ω ) ) , ( v ( t ) , v t ( t ) ) ∈ C ( R + ; H 0 1 ( Ω ) × L 2 ( Ω ) ) .

We are now ready to state our main result.

Theorem 2.1 Let (A₁)-(A₃) hold. Assume that D(0)=∥∇u0∥2+∥∇v0∥2−4∫ΩF(u0,v0)dx>0, 2pC0l(E(0)l)p−12<1 and that (u0,u1)∈H01(Ω)×L2(Ω), (v0,v1)∈H01(Ω)×L2(Ω), where C0 is a computable constant and l=min{l1,l2}. Then there exist constants C , η > 0 such that, for t large, the solution of (1.1) satisfies

E ( t ) ≤ C e − η ∫ 0 t ξ ( s ) d s ,

where

ξ ( t ) = min { ξ 1 ( t ) , ξ 2 ( t ) } , t ≥ 0 .

In this section, we carry out the proof of Theorem 2.1. Firstly, we will estimate several lemmas.

Lemma 3.1 Let u ( t ) , v ( t ) be the solution of (1.1). Then the following energy estimate holds for any t ≥ 0 :

E ′ ( t ) = − ( ∥ u t ∥ 2 + ∥ v t ∥ 2 ) + 1 2 [ ( g 1 ′ ∘ ∇ u ) + ( g 2 ′ ∘ ∇ v ) ] − 1 2 [ g 1 ( t ) ∥ ∇ u ( t ) ∥ 2 + g 2 ( t ) ∥ ∇ v ( t ) ∥ 2 ] ≤ 0 .

Proof Multiplying the first equation of (1.1) by u t and the second equation by v t , respectively, integrating the results over Ω, performing integration by parts and noting that F t ( u , v ) = f 1 ( u , v ) u t + f 2 ( u , v ) v t , we can easily get (3.1). The proof is complete. □

Lemma 3.2 Under the assumption (A₃), the following hold:

Proof Using Hölder’s inequality, we get

On the other hand, we repeat the above proof with − g ′ , instead of g, we can get (3.3). The proof is now complete. □

Lemma 3.3 Let (A₁)-(A₃) hold and u(t), v(t) be the solution of (1.1). Then the functional I ( t ) defined by

I ( t ) : = ∫ Ω ( u u t + v v t ) d x

satisfies

I ′ ( t ) ≤ − l 1 2 ∥ ∇ u ( t ) ∥ 2 − l 2 2 ∥ ∇ v ( t ) ∥ 2 + ( 1 + 1 4 δ ) ( ∥ u t ∥ 2 + ∥ v t ∥ 2 ) + C 1 δ ( g 1 ∘ ∇ u ) + C 1 δ ( g 2 ∘ ∇ v ) + C 1 ∫ Ω F ( u , v ) d x

for all δ > 0 .

Proof By (1.1), a direct differentiation gives

I ′ ( t ) = ∥ u t ∥ 2 − ∥ ∇ u ∥ 2 + ∫ Ω ∇ u ∫ 0 t g 1 ( t − τ ) ∇ u ( τ ) d τ d x − ∫ Ω u t u d x + ∫ Ω f 1 u d x + ∥ v t ∥ 2 − ∥ ∇ v ∥ 2 + ∫ Ω ∇ v ∫ 0 t g 2 ( t − τ ) ∇ v ( τ ) d τ d x − ∫ Ω v t v d x + ∫ Ω f 2 v d x .

From the assumptions (A₁)-(A₂), we derive

and

f 1 u + f 2 v = a ( p + 1 ) | u + v | p + 1 + b ( p + 1 ) | u v | p + 1 2 ≤ C 1 F ( u , v ) .

By Young’s inequality and (3.2), we deduce for any δ>0

Similarly, we have

∫ Ω ∇ v ⋅ ∫ 0 t g 2 ( t − τ ) ∇ v ( τ ) d τ d x ≤ ∥ ∇ v ∥ 2 ⋅ ∫ 0 t g 2 ( τ ) d τ + δ ∥ ∇ v ∥ 2 + C 1 4 δ ( g 2 ∘ ∇ v ) .

Using Young’s inequality and Poincaré’s inequality, we obtain for any δ>0

∫ Ω u u t d x ≤ δ ∥ u ∥ 2 + 1 4 δ ∥ u t ∥ 2 ≤ δ λ 2 ∥ ∇ u ∥ 2 + 1 4 δ ∥ u t ∥ 2 ,

where λ is the first eigenvalue of −Δ with the Dirichlet boundary condition. Similarly,

∫ Ω v v t d x ≤ δ ∥ v ∥ 2 + 1 4 δ ∥ v t ∥ 2 ≤ δ λ 2 ∥ ∇ v ∥ 2 + 1 4 δ ∥ v t ∥ 2 ,

which together with (3.5)-(3.9) gives

I ′ ( t ) ≤ − ( l 1 − δ − δ λ 2 ) ∥ ∇ u ∥ 2 − ( l 2 − δ − δ λ 2 ) ∥ ∇ v ∥ 2 + ( 1 + 1 4 δ ) ( ∥ u t ∥ 2 + ∥ v t ∥ 2 ) + C 1 4 δ ( g 1 ∘ ∇ u ) + C 1 4 δ ( g 2 ∘ ∇ v ) + C 1 ∫ Ω F ( u , v ) d x .

Now, we choose δ>0 so small that

l 1 − δ − δ λ 2 ≥ l 1 2 , l 2 − δ − δ λ 2 ≥ l 2 2 ,

which together with (3.10) gives (3.4). The proof is complete. □

Lemma 3.4 Let (A₁)-(A₃) hold and u(t), v(t) be the solution of (1.1). Then the functional J ( t ) defined by

J ( t ) = J 1 ( t ) + J 2 ( t ) ,

with

satisfies

J ′ ( t ) ≤ − ( ∫ 0 t g 1 ( τ ) − 2 δ ) ∥ u t ∥ 2 + δ C 1 ∥ ∇ u ∥ 2 + C 1 δ ( g 1 ∘ ∇ u ) − C 1 δ ( g 1 ′ ∘ ∇ u ) − ( ∫ 0 t g 2 ( τ ) − 2 δ ) ∥ v t ∥ 2 + δ C 1 ∥ ∇ v ∥ 2 + C 1 δ ( g 2 ∘ ∇ v ) − C 1 δ ( g 2 ′ ∘ ∇ v ) .

Proof A direct differentiation for J 1 ( t ) yields

J 1 ′ ( t ) = − ∫ Ω u t t ⋅ ∫ 0 t g 1 ( t − τ ) ( u ( t ) − u ( τ ) ) d τ − ∫ Ω u t ⋅ ∫ 0 t g 1 ′ ( t − τ ) ( u ( t ) − u ( τ ) ) d τ d x − ( ∫ 0 t g 1 ( τ ) d τ ) ∫ Ω u t 2 d x .

Using the first equation of (1.1) and integrating by parts, we obtain

J 1 ′ ( t ) = ( 1 − ∫ 0 t g 1 ( τ ) d τ ) ∫ Ω ∇ u ⋅ ∫ 0 t g 1 ( t − τ ) ( ∇ u ( t ) − ∇ u ( τ ) ) d τ d x + ∫ Ω ( ∫ 0 t g 1 ( t − τ ) | ∇ u ( t ) − ∇ u ( τ ) | d τ ) 2 d x + ∫ Ω u t ⋅ ∫ 0 t g 1 ( t − τ ) ( u ( t ) − u ( τ ) ) d τ d x − ∫ Ω f 1 ( u , v ) ∫ 0 t g 1 ( t − τ ) ( u ( t ) − u ( τ ) ) d τ d x − ∫ Ω u t ⋅ ∫ 0 t g 1 ′ ( t − τ ) ( u ( t ) − u ( τ ) ) d τ d x − ( ∫ 0 t g 1 ( τ ) d τ ) ∫ Ω u t 2 d x .

From Young’s inequality, Poincaré’s inequality and Lemma 3.2, we derive

Now, we estimate the first term on the right-hand side of (3.17). Using the assumptions (A₁)-(A₂) and Young’s inequality, we arrive at

where we used the embedding H 0 1 ( Ω ) ↪ L s ( Ω ) for 2 ≤ s ≤ 2 n / ( n − 2 ) if n=3 or s ≥ 2 if n = 1 , 2 and the fact 1 2 ( ∥ u t ∥ 2 + ∥ v t ∥ 2 ) + 1 4 l 1 ∥ ∇ u ∥ 2 + 1 4 l 2 ∥ ∇ v ∥ 2 ≤ 2 E ( 0 ) proved in Lemma 5.1 in 17. Combining (3.13)-(3.18), we get

J 1 ′ ( t ) ≤ − ( ∫ 0 t g 1 ( τ ) d τ − 2 δ ) ∥ u t ∥ 2 + δ C 1 ∥ ∇ u ∥ 2 + δ C 1 ∥ ∇ v ∥ 2 + C 1 δ ( g 1 ∘ ∇ u ) − C 1 δ ( g 1 ′ ∘ ∇ u ) .

The same estimate to J 2 ( t ) , we can derive

J 2 ′ ( t ) ≤ − ( ∫ 0 t g 2 ( τ ) d τ − 2 δ ) ∥ v t ∥ 2 + δ C 1 ∥ ∇ u ∥ 2 + δ C 1 ∥ ∇ v ∥ 2 + C 1 δ ( g 2 ∘ ∇ v ) − C 1 δ ( g 2 ′ ∘ ∇ v ) ,

which together with (3.19) gives (3.11). The proof is now complete. □

Proof of Theorem 2.1 For N 1 , N 2 > 0 , we define the functional K by

K : = N 1 E ( t ) + N 2 J ( t ) + I ( t ) ,

and let

g 0 = min { ∫ 0 t 0 g 1 ( s ) d s , ∫ 0 t 0 g 2 ( s ) d s }

for some fixed t 0 > 0 .

Using Lemma 3.1 and Lemmas 3.3-3.4, a direct differentiation gives

K ′ ( t ) ≤ − ( l 2 − N 2 δ C 1 ) ( ∥ ∇ u ∥ 2 + ∥ ∇ v ∥ 2 ) + ( C 1 δ + N 2 C 1 δ ) [ ( g 1 ∘ ∇ u ) + ( g 2 ∘ ∇ v ) ] − ( N 1 + N 2 − 2 δ − 1 − 1 4 δ ) ( ∥ u t ∥ 2 + ∥ v t ∥ 2 ) + C 1 ∫ Ω F ( u , v ) d x + ( N 1 2 − N 2 C 1 δ ) [ ( g 1 ′ ∘ ∇ u ) + ( g 2 ′ ∘ ∇ v ) ] ,

where l=min{l1,l2}.

Now, we choose δ = 1 4 C 1 N 2 and N 1 , N 2 large enough so that

Inserting (3.21)-(3.23) into (3.20), we have

K ′ ( t ) ≤ − c 1 ( ∥ ∇ u ∥ 2 + ∥ ∇ v ∥ 2 ) − c 2 ( ∥ u t ∥ 2 + ∥ v t ∥ 2 ) + c 3 [ ( g 1 ′ ∘ ∇ u ) + ( g 2 ′ ∘ ∇ v ) ] + ( 4 C 1 2 N 2 l + 4 C 1 2 N 2 l ) [ ( g 1 ∘ ∇ u ) + ( g 2 ∘ ∇ v ) ] + C 1 ∫ Ω F ( u , v ) d x .

Therefore, for two positive constants ω and C, we obtain

K ′ ( t ) ≤ − ω E ( t ) + C [ ( g 1 ∘ ∇ u ) + ( g 2 ∘ ∇ v ) ] , for all  t ≥ t 0 .

On the other hand, we choose N1 even larger so that K ( t ) is equivalent to E ( t ) , i.e.,

K ( t ) ∼ E ( t ) .

Multiplying (3.25) by ξ ( t ) = min { ξ 1 ( t ) , ξ 2 ( t ) } and using (A₃), we get

ξ ( t ) K ′ ( t ) ≤ − ω ξ ( t ) E ( t ) + C ∫ Ω ∫ 0 t ξ 1 ( t − τ ) g 1 ( t − τ ) | ∇ u ( t ) − ∇ u ( τ ) | 2 d τ d x + C ∫ Ω ∫ 0 t ξ 2 ( t − τ ) g 2 ( t − τ ) | ∇ v ( t ) − ∇ v ( τ ) | 2 d τ d x ≤ − ω ξ ( t ) E ( t ) − C ∫ Ω ∫ 0 t g 1 ′ ( t − τ ) | ∇ u ( t ) − ∇ u ( τ ) | 2 d τ d x − C ∫ Ω ∫ 0 t g 2 ′ ( t − τ ) | ∇ v ( t ) − ∇ v ( τ ) | 2 d τ d x ≤ − ω ξ ( t ) E ( t ) − C E ′ ( t ) , for all  t ≥ t 0 .

By virtue of (A₃) and ξ ( t ) ≤ 0 , we have

d d t ( ξ ( t ) K ( t ) + C E ( t ) ) ≤ − ω ξ ( t ) E ( t ) , for all  t ≥ t 0 .

Using (3.26), we can easily get

L ( t ) : = ξ ( t ) K ( t ) + C E ( t ) ∼ E ( t ) ,

which together with (3.28) yields, for some positive constant η,

L ′ ( t ) ≤ − η ξ ( t ) L ( t ) , for all  t ≥ t 0 .

Integrating (3.30) over ( t 0 , t ) , we arrive at

L ( t ) ≤ L ( t 0 ) e − η ∫ t t 0 ξ ( τ ) d τ ≤ C e − η ∫ t t 0 ξ ( τ ) d τ ,

which together with (3.29) and the boundedness of E and ξ yields (2.3). The proof is now complete. □

The authors declare that they have no competing interests.

The paper is a joint work of all authors who contributed equally to the final version of the paper. All authors read and approved the final manuscript.