It is well known that viscoelastic materials have memory effects. These properties are due to the mechanical response influenced by the history of the materials themselves. As these materials have a wide application in the natural sciences, their dynamics are of great importance and interest. From the mathematical point of view, their memory effects are modeled by an integro-differential equations. Hence, questions related to the behavior of the solutions for the PDE system have attracted considerable attention in recent years. Many authors have focused on this problem for the last two decades and several results concerning existence, decay and blow-up have been obtained, see 12345678910111213141516171819202122232425262728 and the reference therein.

In 3, Cavalcanti et al. studied the following problem

∣ u t ∣ ρ u t t - Δ u - Δ u t t + ∫ 0 t g ( t - s ) Δ u ( s ) d s - γ Δ u t = 0 , in  Ω × ( 0 , ∞ ) , u ( x , 0 ) = u 0 ( x ) , u t ( x , 0 ) = u 1 ( x ) , x ∈ Ω , u ( x , t ) = 0 , x ∈ ∂ Ω , t ≥ 0 ,

where Ω ⊂ R^N, N ≥ 1, is a bounded domain with a smooth boundary ∂Ω, γ ≥ 0, 0 < ρ ≤ 2 N - 2 if N ≥ 3 or ρ > 0 if N = 1, 2, and the function g: R⁺ → R⁺ is a nonincreasing function. This type of equations usually arise in the theory of viscoelasticity when the material density varies according to the velocity. In that paper, they proved a global existence result of weak solutions for γ ≥ 0 and a uniform decay result for γ > 0. Precisely, they showed that the solutions goes to zero in an exponential rate for γ > 0 and g is a positive bounded C¹-function satisfying

1 - ∫ 0 ∞ g ( s ) d s = 1 - l > 0 ,

and

- ξ 1 g ( t ) ≤ g ′ ( t ) ≤ - ξ 2 g ( t ) ,

for all t ≥ 0 and some positive constants ξ₁ and ξ₂. Later, this result was extended by Messaoudi and Tatar 15 to a situation where a nonlinear source term is competing with the dissipation terms induced by both the viscoelasticity and the viscosity. Recently Messaoudi and Tatar 14 studied problem (1.1) for the case of γ = 0, they improved the result in 3 by showing that the solution goes to zero with an exponential or polynomial rate, depending on the decay rate of the relaxation function g.

The assumptions (1.2) and (1.3), on g, are frequently encountered in the linear case (ρ = 0), see 12456132223293031. Lately, these conditions have been weakened by some researchers. For instance, instead of (1.3) Furati and Tatar 8 required the functions e^αt g(t) and e^αtg'(t) to have sufficiently small L¹-norm on (0, ∞) for some α > 0 and they can also have an exponential decay of solutions. In particular, they do not impose a rate of decreasingness for g. Later on Messaoudi and Tatar 21 improved this result further by removing the condition on g'. They established an exponential decay under the conditions g'(t) ≤ 0 and e^αt g(t) ∈ L¹(0, ∞) for some large α > 0. This last condition was shown to be necessary condition for exponential decay 7. More recently Tatar 25 investigated the asymptotic behavior to problem (1.1) with ρ = γ = 0 when h(t)g(t) ∈ L¹(0, ∞) for some nonnegative function h(t). He generalized earlier works to an arbitrary decay not necessary of exponential or polynomial rate.

Motivated by previous works 2125, in this paper, we consider the initial boundary value problem for the following nonlinear viscoelastic equation:

∣ u t ∣ ρ u t t - Δ u - Δ u t t + ∫ 0 t g ( t - s ) Δ u ( s ) d s + ∣ u ∣ p u = 0 , in  Ω × ( 0 , ∞ ) ,

with initial conditions

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

and boundary condition

u ( x , t ) = 0 , x ∈ ∂ Ω , t ≥ 0 ,

where Ω ⊂ R^N, N ≥ 1, is a bounded domain with a smooth boundary ∂Ω. Here ρ, p > 0 and g represents the kernel of the memory term, with conditions to be stated later [see assumption (A1)-(A3)].

We intend to study the arbitrary decay result for problem (1.4)-(1.6) under the weaker assumption on g, which is not necessarily decaying in an exponential or polynomial fashion. Indeed, our result will be established under the conditions g'(t) ≤ 0 and ∫ 0 ∞ ξ ( s ) g ( s ) d s < ∞ for some nonnegative function ξ(t). Therefore, our result allows a larger class of relaxation functions and improves some earlier results concerning the exponential decay or polynomial decay.

The content of this paper is organized as follows. In Section 2, we give some lemmas and assumptions which will be used later, and we mention the local existence result in Theorem 2.2. In Section 3, we establish the statement and proof of our result related to the arbitrary decay.

In this section, we give some assumptions and lemmas which will be used throughout this work. We use the standard Lebesgue space L^p(Ω) and Sobolev space H 0 1 ( Ω ) with their usual inner products and norms.

Lemma 2.1. (Sobolev-Poincaré inequality) Let 2 ≤ p ≤ 2 N N - 2 , the inequality

∥ u ∥ p ≤ c s ∥ ∇ u ∥ 2 for  u ∈ H 0 1 ( Ω ) ,

holds with the optimal positive constant c_s, where || · ||_p denotes the norm of L^p(Ω).

Assume that ρ satisfies

0 < ρ ≤ 2 N - 2 if  N ≥ 3 or ρ > 0 if N = 1 , 2 .

With regards to the relaxation function g(t), we assume that it verifies

(A1) g(t) ≥ 0, for all t ≥ 0, is a continuous function satisfying

0 < ∫ 0 ∞ g ( s ) d s = l < 1 .

(A2) g'(t) ≤ 0 for almost all t > 0.

(A3) There exists a positive nondecreasing function ξ(t): [0, ∞) → (0, ∞) such that ξ ′ ( t ) ξ ( t ) = η ( t ) is a decreasing function and

∫ 0 ∞ ξ ( s ) g ( s ) d s < ∞ .

Now, we state, without a proof, the existence result of the problem (1.4)-(1.6) which can be established by Faedo-Galerkin methods, we refer the reader to 35.

Theorem 2.2. Suppose that (2.1) and (A1) hold, and that ( u 0 , u 1 ) ∈ H 0 1 ( Ω ) × H 0 1 ( Ω ) . Assume 0 < ρ ≤ 2 N-2 , if N ≥ 3, p > 0, if N = 1, 2. Then there exists at least one global solution u of (1.4)-(1.6) satisfying

u ∈ L ∞ ( [ 0 , ∞ ) ; H 0 1 ( Ω ) ) , u t ∈ L ∞ ( [ 0 , ∞ ) ; H 0 1 ( Ω ) ) , u t t ∈ L 2 ( [ 0 , ∞ ) ; L 2 ( Ω ) ) .

Next, we introduce the modified energy functional for problem (1.4)-(1.6)

E ( t ) = 1 ρ + 2 ∥ u t ∥ ρ + 2 ρ + 2 + 1 2 1 - ∫ 0 t g ( s ) d s ∥ ∇ u ( t ) ∥ 2 2 + 1 2 ∥ ∇ u t ( t ) ∥ 2 2 + 1 2 ( g ∘ ∇ u ) ( t ) + 1 p + 2 ∥ u ( t ) ∥ p + 2 p + 2 ,

where

( g ∘ ∇ u ) ( t ) = ∫ 0 t ∫ Ω g ( t - s ) ∣ ∇ u ( t ) - ∇ u ( s ) ∣ 2 d x d s .

Lemma 2.3. Let u be the solution of (1.4)-(1.6), then the modified energy E(t) satisfies

E ′ ( t ) = 1 2 ( g ′ ∘ ∇ u ) ( t ) - 1 2 g ( t ) ∥ ∇ u ( t ) ∥ 2 2 ≤ 1 2 ( g ′ ∘ ∇ u ) ( t ) ≤ 0 .

Proof. Multiplying Eq. (1.4) by u_t and integrating it over Ω, then using integration by parts and the assumption (A1)-(A2), we obtain (2.6).

Remark. It follows from Lemma 2.3 that the energy is uniformly bounded by E(0) and decreasing in t. Besides, from the definition of E(t) and (2, 2), we note that

( 1 - l ) ∥ ∇ u ∥ 2 2 + ∥ ∇ u t ( t ) ∥ 2 2 + ( g ∘ ∇ u ) ( t ) ≤ 2 E ( 0 ) , ∀ t ≥ 0 .

In this section, we shall state and prove our main result. For this purpose, we first define the functional

L ( t ) = E ( t ) + ∑ i = 1 3 λ i Φ i ( t ) ,

where λ_i are positive constants, i = 1, 2, 3 to be specified later and

Φ 1 ( t ) = 1 ρ + 1 ∫ Ω ∣ u t ∣ ρ u t u d x + ∫ Ω ∇ u t ( t ) ∇ u ( t ) d x ,

Φ 2 ( 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 ,

Φ 3 ( t ) = ∫ Ω ∫ 0 t H ( t - s ) ∣ ∇ u ( s ) ∣ 2 d s d x ,

here

H ( t ) = ξ ( t ) - 1 ∫ t ∞ g ( s ) ξ ( s ) d s .

Remark. This functional was first introduced by Tatar 25 for the case of ρ = 0 and without imposing the dispersion term and forcing term as far as (1.4) is concerned.

The following Lemma tells us that L(t) and E(t) + Φ₃(t) are equivalent.

Lemma 3.1. There exists two positive constants β₁ and β₂ such that the relation

β 1 ( E ( t ) + Φ 3 ( t ) ) ≤ L ( t ) ≤ β 2 ( E ( t ) + Φ 3 ( t ) ) ,

holds for all t ≥ 0 and λ_i small, i = 1, 2.

Proof. By Hölder inequality Young's inequality Lemma 2.1, (2.7) and (2.2), we deduce that

∫ Ω ∣ u t ∣ ρ u t u d x ≤ 1 2 ∥ u t ∥ 2 ( ρ + 1 ) 2 ( ρ + 1 ) + 1 2 ∥ u ∥ 2 2 ≤ c s 2 ( ρ + 1 ) 2 ∥ ∇ u t ∥ 2 2 ( ρ + 1 ) + c s 2 2 ∥ ∇ u ∥ 2 2 ≤ α 1 2 ∥ ∇ u t ∥ 2 2 + c s 2 2 ∥ ∇ u ∥ 2 2 , ∫ Ω ∇ u t ( t ) ∇ u ( t ) d x ≤ 1 2 ( ∥ ∇ u t ∥ 2 2 + ∥ ∇ u ∥ 2 2 ) ,

∫ Ω ∇ u t ∫ 0 t g ( t - s ) ( ∇ u ( t ) - ∇ u ( s ) ) d s d x ≤ 1 2 ∥ ∇ u t ∥ 2 2 + 1 2 ∫ Ω ∫ 0 t g ( t - s ) ( ∇ u ( t ) - ∇ u ( s ) ) 2 d s d x ≤ 1 2 ∥ ∇ u t ∥ 2 2 + l 2 ( g ∘ ∇ u ) ( t ) ,

and

∫ Ω ∣ u t ∣ ρ u t ∫ 0 t g ( t - s ) ( u ( t ) - u ( s ) ) d s d x ≤ 1 2 ∥ u t ∥ 2 ( ρ + 1 ) 2 ( ρ + 1 ) + 1 2 ∫ Ω ∫ 0 t g ( t - s ) u ( t ) - u ( s ) 2 d s d x ≤ α 1 2 ∥ ∇ u t ∥ 2 2 + l c s 2 2 ( g ∘ ∇ u ) ( t ) ,

where α 1 = c s 2 ( ρ + 1 ) ( 2 E ( 0 ) ) ρ . Therefore, from above estimates, the definition of E(t) by (2.4) and (2.2), we have

L ( t ) = E ( t ) + ∑ i = 1 3 λ i Φ i ( t ) ≤ E ( t ) + c 1 | | ∇ u | | 2 2 + c 2 | | ∇ u t | | 2 2 + c 3 g ∘ ∇ u ( t ) + λ 3 Φ 3 ( t )

and

L ( t ) ≥ E ( t ) - c 1 ∥ ∇ u ∥ 2 2 - c 2 ∥ ∇ u t ∥ 2 2 - c 3 g ∘ ∇ u ( t ) + λ 3 Φ 3 ( t ) ≥ 1 ρ + 2 ∥ u t ∥ ρ + 2 ρ + 2 + 1 2 ( 1 - l ) - c 1 ∥ ∇ u ( t ) ∥ 2 2 + 1 2 - c 2 ∥ ∇ u t ( t ) ∥ 2 2 + 1 2 - c 3 g ∘ ∇ u ( t ) + 1 p + 2 ∥ u ( t ) ∥ p + 2 p + 2 + λ 3 Φ 3 ( t ) ,

where c 1 = λ 1 ( c s 2 + ρ + 1 ) 2 ( ρ + 1 ) , c 2 = ( λ 1 + λ 2 ) ( α 1 + ρ + 1 ) 2 ( ρ + 1 ) , and c 3 = l ( ρ + 1 + c s 2 ) λ 2 2 ( ρ + 1 ) . Hence, selecting λ_i , i = 1, 2 such that

λ 1 < min ( ρ + 1 ) ( 1 - l ) c s 2 + ρ + 1 , ρ + 1 α 1 + ρ + 1 , λ 2 < min ρ + 1 l ( c s 2 + ρ + 1 ) , ρ + 1 α 1 + ρ + 1 - λ 1 ,

and again from the definition of E(t), there exist two positive constants β₁ and β₂ such that

β 1 ( E ( t ) + Φ 3 ( t ) ) ≤ L ( t ) ≤ β 2 ( E ( t ) + Φ 3 ( t ) ) , t ≥ 0 .

To obtain a better estimate for ∫ Ω ∇ u ∫ 0 t g ( t - s ) ∇ u ( s ) d s d x , we need the following Lemma which repeats Lemma 2 in 25.

Lemma 3.2. For t ≥ 0, we have

∫ Ω ∇ u ∫ 0 t g ( t - s ) ∇ u ( s ) d s d x = 1 2 ∫ 0 t g ( s ) d s ∥ ∇ u ∥ 2 2 + 1 2 ∫ 0 t g ( t - s ) ∥ ∇ u ( s ) ∥ 2 2 d s - 1 2 g ∘ ∇ u ( t ) .

Proof. Straightforward computations yield this identity.

Now, we are ready to state and prove our result. First, we introduce the following notations as in 2425. For every measurable set A ⊂ R⁺, we define the probability measure g ^ by

g ^ ( A ) = 1 l ∫ A g ( s ) d s .

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

F g = { s ∈ R + ∣ g ( s ) > 0 and  g ′ ( s ) = 0 }

and

R g = g ^ ( F g ) = 1 l ∫ F g g ( s ) d s .

Before proceeding, we note that there exists t₀ > 0 such that

∫ 0 t g ( s ) d s ≥ ∫ 0 t 0 g ( s ) d s = g * > 0 , ∀ t ≥ t 0 ,

since g is nonnegative and continuous.

Theorem 3.3. Let (u0,u1) ∈ H01(Ω) × H01(Ω) be given. Suppose that (A1)-(A3), (2, 1) and the hypothesis on p hold. Assume further that R g < g * α l ( 2 α + 2 c s 2 ) , H ( 0 ) < g * ( 4 + l ) - 3 l 4 and g * > 3 l 4 + l with

α = ( g * ( 4 + l ) - 3 l ) 1 - l p 8 c s 2 ( p + 1 ) ( 2 E ( 0 ) ) p .

Then the solution energy of (1.4)-(1.6) satisfies

E ( t ) ≤ K ξ ( t ) - μ , t ≥ 0 ,

where μ and K are positive constants.

Proof. In order to obtain the decay result of E(t), it suffices to prove that of L(t). To this end, we need to estimate the derivative of L(t). It follows from (3.2) and Eq. (1.4) that

Φ 1 ′ ( t ) = 1 ρ + 1 ∥ u t ∥ ρ + 2 ρ + 2 + ∥ ∇ u t ∥ 2 2 - ∥ ∇ u ∥ 2 2 + ∫ Ω ∇ u ∫ 0 t g ( t - s ) ∇ u ( s ) d s d x - ∥ u ∥ p + 2 p + 2 ,

which together with the identity (3.6) and (2.2) gives

Φ 1 ′ ( t ) ≤ 1 ρ + 1 ∥ u t ∥ ρ + 2 ρ + 2 + ∥ ∇ u t ∥ 2 2 - 1 - l 2 ∥ ∇ u ∥ 2 2 + 1 2 ∫ 0 t g ( t - s ) ∥ ∇ u ( s ) ∥ 2 2 d s - 1 2 ( g ∘ ∇ u ) ( t ) - ∥ u ∥ p + 2 p + 2 .

Next, we would like to estimate Φ 2 ′ ( t ) . Taking a derivative of Φ₂ in (3.3) and using Eq. (1.4) to get

Φ 2 ′ ( t ) = 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 - 1 ρ + 1 ∫ 0 t g ( s ) d s ∥ u t ∥ ρ + 2 ρ + 2 - ∫ 0 t g ( s ) d s ∥ ∇ u t ∥ 2 2 - ∫ Ω ∇ u t ( t ) ∫ 0 t g ′ ( t - s ) ( ∇ u ( t ) - ∇ u ( s ) ) d s d x - 1 ρ + 1 ∫ Ω ∣ u t ∣ ρ u t ∫ 0 t g ′ ( t - s ) ( u ( t ) - u ( s ) ) d s d x + ∫ Ω ∣ u ∣ p u ∫ 0 t g ( t - s ) ( u ( t ) - u ( s ) ) d s d x .

We now estimate the first two terms on the right-hand side of (3.11) as in 25.

Indeed, for all measure set A and F such that A = R⁺ - F, we have

∫ Ω ∇ u ( t ) ∫ 0 t g ( t - s ) ( ∇ u ( t ) - ∇ u ( s ) ) d s d x = ∫ Ω ∇ u ( t ) ∫ A ∩ [ 0 , t ] g ( t - s ) ( ∇ u ( t ) - ∇ u ( s ) ) d s d x + ∫ Ω ∇ u ( t ) ∫ F ∩ [ 0 , t ] g ( t - s ) ( ∇ u ( t ) - ∇ u ( s ) ) d s d x ≤ ∫ Ω ∇ u ( t ) ∫ A ∩ [ 0 , t ] g ( t - s ) ( ∇ u ( t ) - ∇ u ( s ) ) d s d x + ∫ F ∩ [ 0 , t ] g ( s ) d s ∥ ∇ u ∥ 2 2 - ∫ Ω ∇ u ( t ) ∫ F ∩ [ 0 , t ] g ( t - s ) ∇ u ( s ) d s d x .

To simplify notations, we denote

A t = A ∩ [ 0 , t ] and  F t = F ∩ [ 0 , t ] .

Using Hölder inequality Young's inequality and (2.2), we see that, for δ₁ > 0,

∫ Ω ∇ u ( t ) ∫ A t g ( t - s ) ( ∇ u ( t ) - ∇ u ( s ) ) d s d x ≤ δ 1 ∥ ∇ u ∥ 2 2 + l 4 δ 1 ∫ Ω ∫ A t g ( t - s ) ∣ ∇ u ( t ) - ∇ u ( s ) ∣ 2 d s d x

and

∫ Ω ∇ u ( t ) ∫ F t g ( t - s ) ∇ u ( s ) d s d x ≤ 1 2 ∫ F t g ( s ) d s ∥ ∇ u ∥ 2 2 + 1 2 ∫ F t g ( t - s ) ∥ ∇ u ( s ) ∥ 2 2 d s .

Thus, from the definition of g ^ ( F ) by (3.8), (3.12) becomes

∫ Ω ∇ u ( t ) ∫ 0 t g ( t - s ) ∇ u ( t ) - ∇ u ( s ) d s d x ≤ δ 1 + 3 2 l g ^ ( F ) ∥ ∇ u ∥ 2 2 + l 4 δ 1 ∫ Ω ∫ A t g ( t - s ) ∣ ∇ u ( t ) - ∇ u ( s ) ∣ 2 d s d x + 1 2 ∫ F t g ( t - s ) ∥ ∇ u ( s ) ∥ 2 2 d s .

The second term on the right-hand side of (3.11) can be estimated as follows (see 25), for δ₂ > 0,

∫ Ω ∫ 0 t g ( t - s ) ∇ u ( t ) - ∇ u ( s ) d s 2 d x = ∫ Ω ∫ A t g ( t - s ) ∇ u ( t ) - ∇ u ( s ) d s + ∫ F t g ( t - s ) ∇ u ( t ) - ∇ u ( s ) d s 2 d x ≤ 1 + 1 δ 2 l ∫ Ω ∫ A t g ( t - s ) | ∇ u ( t ) - ∇ u ( s ) | 2 d s d x + ( 1 + δ 2 ) l g ^ ( F ) ∫ Ω ∫ F t g ( t - s ) | ∇ u ( t ) - ∇ u ( s ) | 2 d s d x .

Using Hölder inequality Young's inequality and (A2) to deal with the fifth term, for δ₃ > 0,

∫ Ω ∇ u t ( t ) ∫ 0 t g ′ ( t - s ) ( ∇ u ( t ) - ∇ u ( s ) ) d s d x ≤ δ 3 ∥ ∇ u t ∥ 2 2 + 1 4 δ 3 ∫ Ω ∫ 0 t g ′ ( t - s ) ( ∇ u ( t ) - ∇ u ( s ) ) d s 2 d x ≤ δ 3 ∥ ∇ u t ∥ 2 2 - g ( 0 ) 4 δ 3 ( g ′ ∘ ∇ u ) ( t ) .

Exploiting Hölder inequality Young's inequality Lemma 2.1 and (A2) to estimate the sixth term, for δ₄ > 0,

1 ρ + 1 ∫ Ω ∣ u t ∣ ρ u t ∫ 0 t g ′ ( t - s ) ( u ( t ) - u ( s ) ) d s d x ≤ 1 ρ + 1 δ 4 ∥ u t ∥ 2 ( ρ + 1 ) 2 ( ρ + 1 ) + 1 4 δ 4 ∫ Ω ∫ 0 t g ′ ( t - s ) ( u ( t ) - u ( s ) ) d s 2 d x ≤ 1 ρ + 1 δ 4 c s 2 ( ρ + 1 ) ∥ ∇ u t ∥ 2 2 ( ρ + 1 ) - g ( 0 ) c s 2 4 δ 4 g ′ ∘ ∇ u ( t ) ≤ 1 ρ + 1 α 1 δ 4 ∥ ∇ u t ∥ 2 2 - g ( 0 ) c s 2 4 δ 4 g ′ ∘ ∇ u ( t ) .

For the last term, thanks to Hölder inequality Young's inequality Lemma 2.1, (2.7), (2.2) and (3.8), we have, for δ₅ > 0,

∫ Ω ∣ u ∣ p u ∫ 0 t g ( t - s ) ( u ( t ) - u ( s ) ) d s d x ≤ δ 5 ∫ Ω ∣ u ∣ 2 ( p + 1 ) d x + 1 4 δ 5 ∫ Ω ∫ 0 t g ( t - s ) ( u ( t ) - u ( s ) ) d s 2 d x ≤ δ 5 α 2 ∥ ∇ u ∥ 2 2 + l c s 2 2 δ 5 ∫ Ω ∫ A t g ( t - s ) ∣ ∇ u ( t ) - ∇ u ( s ) ∣ 2 d s d x + l c s 2 2 δ 5 g ^ ( F ) ∫ Ω ∫ F t g ( t - s ) ∣ ∇ u ( t ) - ∇ u ( s ) ∣ 2 d s d x ,

where α 2 = c s 2 ( p + 1 ) 2 E ( 0 ) 1 - l p . Thus, gathering these estimates (3.13)-(3.17) and using (3.9), we obtain, for t ≥ t₀,

Φ 2 ′ ( t ) ≤ ( 1 - g * ) δ 1 + 3 2 l g ^ ( F ) + δ 5 α 2 ∥ ∇ u ∥ 2 2 - g ( 0 ) 4 δ 3 + g ( 0 ) c s 4 ( ρ + 1 ) δ 4 ( g ′ ∘ ∇ u ) ( t ) + 1 - g * 2 ∫ F t g ( t - s ) ∥ ∇ u ( s ) ∥ 2 2 d s + δ 3 + α 1 δ 4 ρ + 1 - g * ∥ ∇ u t ∥ 2 2 - g * ρ + 1 ∥ u t ∥ ρ + 2 ρ + 2 + l 1 + 1 δ 2 + c s 2 2 δ 5 + 1 - g * 4 δ 1 ∫ Ω ∫ A t g ( t - s ) ∣ ∇ u ( t ) - ∇ u ( s ) ∣ 2 d s d x + 1 + δ 2 + c s 2 2 δ 5 l g ^ ( F ) ∫ Ω ∫ F t g ( t - s ) ∣ ∇ u ( t ) - ∇ u ( s ) ∣ 2 d s d x .

Further, taking a derivative of Φ₃(t), using the fact that ξ′(t) ξ(t) = η(t) is a decreasing function and the definition of Φ₃(t) by (3.4), we derive that (see 25)

Φ ′ 3 ( t ) = H ( 0 ) ∥ ∇ u ∥ 2 2 - ∫ 0 t ξ ′ ( t - s ) ξ ( t - s ) H ( t - s ) ∥ ∇ u ( s ) ∥ 2 2 d s - ∫ 0 t g ( t - s ) ∥ ∇ u ( s ) ∥ 2 2 d s ≤ H ( 0 ) ∥ ∇ u ∥ 2 2 - η ( t ) ∫ 0 t H ( t - s ) ∥ ∇ u ( s ) ∥ 2 2 d s - ∫ 0 t g ( t - s ) ∥ ∇ u ( s ) ∥ 2 2 d s = H ( 0 ) ∥ ∇ u ∥ 2 2 - η ( t ) Φ 3 ( t ) - ∫ 0 t g ( t - s ) ∥ ∇ u ( s ) ∥ 2 2 d s .

Hence, we conclude from (2.6), (3.10), (3.18) and (3.19) that for any t ≥ t₀ > 0,

L ′ ( t ) = E ′ ( t ) + λ 1 Φ ′ 1 ( t ) + λ 2 Φ ′ 2 ( t ) + λ 3 Φ ′ 3 ( t ) ≤ 1 2 - λ 2 g ( 0 ) 4 δ 3 + g ( 0 ) c s 2 4 ( ρ + 1 ) δ 4 ( g ′ ∘ ∇ u ) ( t ) + 1 ρ + 1 ( λ 1 - λ 2 g * ) ∥ u t ∥ ρ + 2 ρ + 2 - λ 1 ∥ u ∥ p + 2 p + 2 - λ 1 2 ( g ∘ ∇ u ) ( t ) + λ 1 + λ 2 δ 3 + α 1 δ 4 ρ + 1 - g * ∥ ∇ u t ∥ 2 2 + λ 1 2 - λ 3 ∫ 0 t g ( t - s ) ∥ ∇ u ( s ) ∥ 2 2 d s + λ 2 ( 1 - g * ) δ 1 + 3 2 l g ^ ( F ) + λ 2 δ 5 α 2 - λ 1 ( 1 - l 2 ) + λ 3 H ( 0 ) ∥ ∇ u ∥ 2 2 + λ 2 l 1 + 1 δ 2 + c s 2 2 δ 5 + 1 - g * 4 δ 1 ∫ Ω ∫ A t g ( t - s ) ∣ ∇ u ( t ) - ∇ u ( s ) ∣ 2 d s d x + λ 2 1 + δ 2 + c s 2 2 δ 5 l g ^ ( F ) ∫ Ω ∫ F t g ( t - s ) ∣ ∇ u ( t ) - ∇ u ( s ) ∣ 2 d s d x + λ 2 1 - g * 2 ∫ F t g ( t - s ) ∥ ∇ u ( s ) ∥ 2 2 d s - λ 3 η ( t ) Φ 3 ( t ) .