In this article, we study the asymptotic behavior of the energy function related to a nonlinear wave equation of Kirchhoff type subject to memory condition at the boundary as follows:

u t t - M | | ∇ u | | 2 2 Δ u + l ( t ) h ( ∇ u ) - Δ u t + a ( x ) f ( u ) = 0 in  Ω × ( 0 , ∞ ) ,

u = 0 on  Γ 0 × ( 0 , ∞ ) ,

u + ∫ 0 t g ( t - s ) M | | ∇ u ( s ) | | 2 2 ∂ u ∂ ν ( s ) + ∂ u t ∂ ν ( s ) d s = 0 on  Γ 1 × ( 0 , ∞ ) ,

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

where Ω is a bounded domain with smooth boundary ∂Ω = Γ₀ ∪ Γ₁. The partition Γ₀ and Γ₁ are closed and disjoint, with meas(Γ₀) > 0, ν represents the unit normal vector directed towards the exterior of Ω, u is the transverse displacement, and g is the relaxation function considered positive and nonincreasing belonging to W^1,2 (Ω).

From the physical point of view, we know that the memory effect described in integral equation (1.3) can be caused by the interaction with another viscoelastic element. In fact, the boundary condition (1.3) signifies that Ω is composed of a material which is clamped in a rigid body in the portion Γ₀ of its boundary and is clamped in a body with viscoelastic properties in the portion of Γ₁.

When Γ₁ = ϕ, problem (1.1) has its origin in describing the nonlinear vibrations of an elastic string. More precisely, we have

ρ h ∂ 2 u ∂ t 2 = p 0 + E h 2 L ∫ 0 L ∂ u ∂ x 2 d x ∂ 2 u ∂ x 2 + f

for 0 < × < L, t ≥ 0; where u is the lateral deflection, x the space coordinate, t the time, E the Young modulus, ρ the mass density, h the cross section area, L the length, p₀ the initial axial tension and f the external force. Kirchhoff 1 was the first one who introduced (1.5) to study the oscillations of stretched strings and plates, so that (1.5) is called the wave equation of Kirchhoff type after him. In this direction, problem (1.1) with ∂Ω = Γ₀ and l(t) = 0 has been investigated by many authors in recent years, and many results concerning existence, nonexistence and asymptotic behavior have been established, see 2345678910111213.

On the other hand, regarding the viscoelastic wave equations with memory term acting in the boundary or in the domain, there are numerous results related to asymptotic behavior of solutions. For example, in the case where M(s) = 1, Cavalcanti et al. 14 investigated the existence and uniform decay of strong solutions of wave equation (1.1) with a nonlinear boundary damping of memory type and a nonlinear boundary source when l(t) = 0. Cavalcanti and Guesmia 15 considered the following system:

u t t - Δ u + F ( x , t , u , ∇ u ) = 0 in  Ω × ( 0 , ∞ ) ,

u = 0 on  Γ 0 × ( 0 , ∞ ) ,

u + ∫ 0 t g ( t - s ) ∂ u ∂ ν ( s ) d s = 0 on  Γ 1 × ( 0 , ∞ ) ,

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

where Ω is a bounded domain with smooth boundary ∂Ω = Γ₀ ∪ Γ₁. They obtained the general decay result which depends on the relaxation function g. In particular, if the relaxation function g decays exponentially (or polynomially), then the solution also decays exponentially (or polynomially) and with the same decay rate. Moreover, when u₀ = 0 on Γ₁, they obtained exponential or polynomial decay of solutions, even if the relaxation function g does not converge to 0 at ∞. Later, Messaoudi and Soufyane 16 generalized this result to the case of a system of Timoshenko type. They established general decay rate results, from which the usual exponential and polynomial decay rates are only special cases. Recently, Messaoudi and Soufyane 17 studied the following problem:

u t t - Δ u + f ( u ) = 0

in a bounded domain with boundary conditions (1.7)-(1.9). They improved the results of 15 by applying the multiplier techniques. Indeed, they obtained not only a general decay result, but their works also allowed certain relaxation functions which are not necessarily of exponential or polynomial decay. For other related works, we refer the reader to 181920 and references therein.

Conversely, in the case where M is not a constant function, Santos 21 considered

u t t - μ ( t ) u x x = 0 , ( x , t ) ∈ ( 0 , 1 ) × R + , u ( 0 , t ) = 0 , u ( 1 , t ) = - ∫ 0 t g ( t - s ) μ ( s ) u x ( 1 , s ) d s , ∀ t > 0 , u ( 0 ) = u 0 , u t ( 0 ) = u 1 , x ∈ ( 0 , 1 ) ,

where μ(t) is a nonincreasing function satisfying μ(t) ≥ μ₀ > 0. By denoting k the resolvent kernel of g', he showed that the solution decays exponentially (or polynomially) to zero provided k decays exponentially (or polynomially) to zero. Later on, Santos et al. 22 generalized this result to a nonlinear n-dimensional equation of Kirchhoff type of the form

u t t - M | | ∇ u | | 2 2 Δ u - Δ u t + f ( u ) = 0

in a bounded domain with boundary conditions (1.2)-(1.3). In that article, they proved that the energy decays with the same rate of decay of the relaxation function. This latter result improved an earlier one by Park et al. 23, where the authors considered (1.10) in a bounded domain with nonlinear boundary damping and memory term and M(s) = 1 + s and f = 0.

We note that stability of problems with the nonlinear term h(∇u) requires a careful treatment because we do not have any information about the influence of the integral ∫ Ω h ( ∇ u ) u t d x about the sign of the derivative E'(t). Although the subject is important, there are few mathematical results in the presence of the nonlinearity given by h(∇u), see 242526. In light of this and previous articles 1722, it is interesting to investigate whether we still have the similar general decay result as in 17 for nondissipative distributed system (1.1) with the memory-type damping acting on a part of the boundary. Hence, the main purpose of this article is to answer the above question for system (1.1)-(1.4). Consequently, by following the arguments close to the one in 17 with necessary modification required the nature of our problem, we establish a general decay result which includes the usual exponential and polynomial decay rates. Furthermore, our results allow a larger class of relax functions which are not necessarily of exponential and polynomial decay. Therefore, this improves earlier results in the literature 2227.

In order to obtain our results, we consider system (1.1)-(1.4), under some assumptions on a(x), l(t), M and f. Precisely, we state the general assumptions:

(A1) a(x): Ω → R⁺ is a function.

(A2) f ∈ C¹(R) is a function and satisfies

u f ( u ) ≥ β F ( u ) ≥ 0 f o r β > 2 ,

where F ( u ) = ∫ 0 u f ( s ) d s with

F ( u ) ≤ d | u | p for all  u ∈ R ,

d > 0 and 1 ≤ p ≤ n n - 2 .

(A3) M is a C¹ function on [0, ∞) satisfying

M ( λ ) ≥ m 0 > 0 and M ( λ ) λ ≥ M ^ ( λ ) for all λ ≥ 0 ,

Where M ^ ( λ ) = ∫ 0 λ M ( s ) d s .

(A4) h : Rⁿ → R is a C¹ function such that ∇h is bounded and there exists β₁ > 0 such that

| h ( ξ ) | ≤ β 1 | ξ | for all  ξ ∈ R n ,

and l(t) is a positive and nonincreasing function.

The remainder of this article is organized as follows. In Section 2, we introduce some notations, present Lemma 2.1 to describe more general relations between the relaxation function g and the corresponding resolvent kernel k and state the existence result to system (1.1)-(1.4). In Section 3, we give the proof of our main result Theorem 3.5.

In this section, we introduce some notations and establish the existence of solutions of the problem (1.1)-(1.4). In what follows, let ||·||_p denote the usual L^p (Ω) norm | | ⋅ | | L p ( Ω ) , for 1 ≤ p ≤ ∞. We define the convolution product operator by

( g * u ) ( t ) = ∫ 0 t g ( t - s ) u ( s ) d s ,

and set

( g ∘ ϕ ) ( t ) = ∫ 0 t g ( t - s ) | | ϕ ( t ) - ϕ ( s ) | | 2 2 d s ,

( g ♢ ϕ ) ( t ) = ∫ 0 t g ( t - s ) ( ϕ ( t ) - ϕ ( s ) ) d s .

Using Hölder's inequality, we observe that

| g ♢ ϕ ( t ) | 2 ≤ ∫ 0 t | g ( s ) | d s ( | g | ∘ ϕ ) ( t ) .

Next, we shall use Equation 1.3 to estimate the boundary term M | | ∇ u ( s ) | | 2 2 ∂ u ∂ ν + ∂ u t ∂ ν . Differentiating (1.3), we obtain

M | | ∇ u ( t ) | | 2 2 ∂ u ∂ ν ( t ) + ∂ u t ∂ ν ( t ) + 1 g ( 0 ) ∫ 0 t g ′ ( t - s ) M | | ∇ u ( s ) | | 2 2 ∂ u ∂ ν ( s ) + ∂ u t ∂ ν ( s ) d s = - 1 g ( 0 ) u t .

Assume the function k is the resolvent kernel of the relaxation function g, then

k + 1 g ( 0 ) k * g ′ = - 1 g ( 0 ) g ′ .

Applying Volterra's inverse operator yields

M | | ∇ u ( t ) | | 2 2 ∂ u ∂ ν ( t ) + ∂ u t ∂ ν ( t ) = - 1 g ( 0 ) ( u t + k * u t ) ,

which implies that

M | | ∇ u ( t ) | | 2 2 ∂ u ∂ ν ( t ) + ∂ u t ∂ ν ( t ) = - τ { u t + k ( 0 ) u - k ( t ) u 0 + k ′ * u } on  Γ 1 × ( 0 , ∞ ) ,

where τ = 1 g ( 0 ) . Reciprocally, taking u₀ = 0 on Γ₁, identity (2.5) implies (1.3). As we are interested in relaxation functions of more general decay and the resolvent k appeared in Equation 2.5, we want to know if the resolvent k has the same property with the relaxation function g involved in (1.3). The following lemma answers this question. Let h be a relaxation function and k its resolvent kernel, that is

k ( t ) = h ( t ) + ( k * h ) ( t ) .

Lemma 2.1. 151722 If h : [0, ∞) → R⁺ is continuous, then k is also a positive continuous function. Moreover,

(1) If there exists a positive constant c₀ such that

h ( t ) ≤ c 0 e - ∫ 0 t γ ( s ) d s ,

where γ : [0, ∞) → R⁺, is a nonincreasing function satisfying, for some positive constant ε < 1,

c 1 = ∫ 0 ∞ e - ∫ 0 t ( 1 - ∈ ) γ ( s ) d s d t < 1 c 0 .

Then, k satisfies

k ( t ) ≤ c 0 1 - c 0 c 1 e - ∫ 0 t ε γ ( s ) d s .

(2) Suppose that

h ( t ) ≤ c 0 ( 1 + t ) p

for c₀ < p - 1. Then, there exists a positive constant ε < 1 such that

k ( t ) ≤ β ( 1 + t ) ε p ,

where β > 0 is a constant.

Based on this lemma, we will use (2.5) instead of (1.3), i.e., we can consider system (1.1)-(1.4) as follows:

u t t - M | | ∇ u | | 2 2 Δ u + l ( t ) h ( ∇ u ) - Δ u t + a ( x ) f ( u ) = 0 in  Ω × ( 0 , ∞ ) , u = 0 on  Γ 0 × ( 0 , ∞ ) , M | | ∇ u ( t ) | | 2 2 ∂ u ∂ ν ( t ) + ∂ u t ∂ ν ( t ) = - τ { u t + k ( 0 ) u - k ( t ) u 0 + k ′ * u } on  Γ 1 × ( 0 , ∞ ) , u ( x , 0 ) = u 0 ( x ) , u t ( x , 0 ) = u 1 ( x ) in  Ω .

We notice that, due to the condition (1.2), the solution of system (1.1)-(1.4) must belong to the following space:

V = { v ∈ H 1 ( Ω ) ; v = 0 on  Γ 0 } ,

which endowed with the norm ||∇·||₂ is a Hilbert space. Now, we are ready to give the well-posedness of system (1.1)-(1.4).

Theorem 2.2. Let k ∈ W^2,1 (R⁺) ∩ W^1,∞ (R⁺), (u₀, u₁) ∈ (H² (Ω) ∩ V)² and satisfy the compatibility condition

M | | ∇ u 0 | | 2 2 ∂ u 0 ∂ ν + ∂ u 1 ∂ ν + τ u 1 = 0 on  Γ 1 .

Assume further that (A1)-(A4) hold. Then, there exists a unique solution u of system (1.1)-(1.4) such that

u ∈ L ∞ ( 0 , ∞ ; H 2 ( Ω ) ∩ V ) , u t ∈ L ∞ ( 0 , ∞ ; V ) , u t t ∈ L ∞ ( 0 , ∞ ; L 2 ( Ω ) ) .

Proof. Using the Galerkin method and procedures similar to that of 2228, we can obtain the result.    □

In this section, we study the asymptotic behavior of the solutions of system (1.1)-(1.4) when the resolvent kernel k satisfies

k ( 0 ) > 0 , k ( t ) ≥ 0 , k ′ ( t ) ≤ 0 , k ″ ( t ) ≥ - γ ( t ) k ′ ( t ) ,

where γ : [0, ∞) → R⁺ is a function satisfying the following condition:

γ ( t ) > 0 , γ ′ ( t ) ≤ 0 and ∫ 0 ∞ γ ( s ) d s = ∞ .

To get our result, we further assume that

0 < l ( t ) ≤ γ ( t ) for all  t ≥ 0 .

Let x₀ be a fixed point in Rⁿ. Set

m = m ( x ) = x - x 0 , R ( x 0 ) = max | | m ( x ) | | 2 ; x ∈ Ω ̄

and partition the boundary ∂Ω into two sets

Γ 0 = { x ∈ ∂ Ω ; m ( x ) ⋅ ν ≤ 0 } , Γ 1 = { x ∈ ∂ Ω ; m ( x ) ⋅ ν > 0 } .

Define the first-order energy function of system (1.1)-(1.4) by

E ( t ) = 1 2 | | u t | | 2 2 + M ^ | | ∇ u | | 2 2 + ∫ Ω a ( x ) F ( u ) d x + τ 2 k ( t ) ∫ Γ 1 | u | 2 d Γ - τ 2 ∫ Γ 1 k ′ ∘ u d Γ .

The following lemma is associated with the property of the convolution operator, which is used to estimate the energy identity.

Lemma 3.1. If g, ϕ ∈ C¹(R⁺), then

( g * ϕ ) ϕ t = - 1 2 g ( t ) | ϕ ( t ) | 2 + 1 2 g ′ ∘ ϕ - 1 2 d d t g ∘ ϕ - ∫ 0 t g ( s ) d s | ϕ ( t ) | 2 .

Proof. Our conclusion is followed by differentiating the term g ○ ϕ.   □

Lemma 3.2. Under the assumptions of (A1)-(A4), the energy function E(t) satisfies

d d t E ( t ) ≤ - τ 2 ∫ Γ 1 | u t | 2 d Γ + τ 2 k 2 ( t ) ∫ Γ 1 | u 0 | 2 d Γ + τ 2 k ′ ( t ) ∫ Γ 1 | u | 2 d Γ - τ 2 ∫ Γ 1 k ″ ∘ u d Γ - ∫ Ω | ∇ u t | 2 d x - ∫ Ω l ( t ) h ( ∇ u ) u t d x .

Proof. Multiplying Equation 1.1 by u_t , and integrating by parts over Ω, we get

d d t 1 2 | | u t | | 2 2 + M ^ | | ∇ u | | 2 2 + ∫ Ω a ( x ) F ( u ) d x = ∫ Γ 1 M | | ∇ u | | 2 2 ∂ u ∂ ν + ∂ u t ∂ ν u t d Γ - ∫ Ω | ∇ u t | 2 d x - ∫ Ω Δ l ( t ) h ( ∇ u ) u t d x .

Exploiting (2.5), (3.6) and the definition of E(t) by (3.5), we have

d d t E ( t ) ≤ - τ ∫ Γ 1 | u t | 2 d Γ + τ ∫ Γ 1 k ( t ) u 0 u t d Γ + τ 2 k ′ ( t ) ∫ Γ 1 | u | 2 d Γ - τ 2 ∫ Γ 1 k ″ ∘ u d Γ - ∫ Ω | ∇ u t | 2 d x - ∫ Ω l ( t ) h ( ∇ u ) u t d x .

Then, using Hölder's inequality and Young's inequality, the inequality (3.7) is obtained.    □

Next, we construct a Lyapunov functional which is equivalent to E(t). To do so, for N > 0 large enough, let

L ( t ) = N E ( t ) + ψ ( t ) ,

where

ψ ( t ) = ∫ Ω m ⋅ ∇ u ( t ) + n 2 - θ u u t d x

for 0 < θ < 1.

For the purpose of achieving our main result, we need the following lemmas.

Lemma 3.3. There exist two positive constants α₁ and α₂ such that the relation

α 1 E ( t ) ≤ L ( t ) ≤ α 2 E ( t )

holds for all t ≥ 0.

Proof. From (3.9) and using Young's inequality, we get

| ψ ( t ) | ≤ R ( x 0 ) + B 1 n 2 - θ | | u t | | 2 | | ∇ u | | 2 ≤ 2 m 0 R ( x 0 ) + B 1 n 2 - θ E ( t ) ,

where we have used the fact that | | u t | | 2 2 ≤ 2 E ( t ) by (3.5) and

| | ∇ u | | 2 2 ≤ 2 m 0 E ( t )

due to M ^ ( λ ) ≥ m 0 λ > 0 by (A3) and (3.5). Here B₁ > 0 is the smallest constant such that

| | u | | 2 ≤ B 1 | | ∇ u | | 2 , ∀ u ∈ V .

Thus, from (3.8), we deduce that

| L ( t ) - N E ( t ) | = | ψ ( t ) | ≤ 2 m 0 R ( x 0 ) + B 1 n 2 - θ E ( t ) .

Hence, selecting

N > 2 m 0 R ( x 0 ) + B 1 n 2 - θ ,

there exist two positive constants α₁ and α₂ such that the relation

α 1 E ( t ) ≤ L ( t ) ≤ α 2 E ( t )

holds.    □

Lemma 3.4. Let (A1)-(A4) and (3.1)-(3.3) hold, with β₁ (given by (A4)) small enough and

lim t → ∞ k ( t ) = 0 .

Then, for some t₀ large enough, the functional L(t) verifies, along the solution u of (1.1)-(1.4) ,

L ′ ( t ) ≤ - α 2 E ( t ) + c 4 k 2 ( t ) ∫ Γ 1 | u 0 | 2 d Γ - c 5 ∫ Γ 1 k ′ ∘ u d Γ + ∫ Ω n + α - n 2 - θ β a ( x ) + m ⋅ ∇ a F ( u ) d x

for all t ≥ t₀, where α = min {2θ, 1 - θ} and c_i are positive constants given in the proof, i = 4, 5.

Proof. First, we are going to estimate the derivative of ψ(t). From (3.9) and using Equation 1.1, we have

d d t ψ ( t ) = 1 2 ∫ Γ 1 ( m ⋅ ν ) | u t | 2 d Γ - θ ∫ Ω | u t | 2 d x + ∫ Ω m ⋅ ∇ u ( t ) + n 2 - θ u M | | ∇ u | | 2 2 Δ u d x + ∫ Ω m ⋅ ∇ u ( t ) + n 2 - θ u Δ u t d x - ∫ Ω l ( t ) h ( ∇ u ) m ⋅ ∇ u ( t ) + n 2 - θ u d x - ∫ Ω m ⋅ ∇ u ( t ) + n 2 - θ u a ( x ) f ( u ) d x .

Performing integration by parts and using Young's inequality, we obtain

d d t ψ ( t ) ≤ 1 2 ∫ Γ 1 ( m ⋅ ν ) | u t | 2 d Γ - θ ∫ Ω | u t | 2 d x + ∫ Γ 1 M | | ∇ u | | 2 2 ∂ u ∂ ν + ∂ u t ∂ ν m ⋅ ∇ u ( t ) + n 2 - θ u d Γ - M | | ∇ u | | 2 2 2 ∫ Γ 1 ( m ⋅ ν ) | ∇ u | 2 d Γ - ( 1 - θ ) M ( | | ∇ u | | 2 2 ) | | ∇ u | | 2 2 + ε c 0 M | | ∇ u | | 2 2 | | ∇ u | | 2 2 + C ε ∫ Ω | ∇ u t | 2 d x - ∫ Ω l ( t ) h ( ∇ u ) m ⋅ ∇ u ( t ) + n 2 - θ u d x - ∫ Ω m ⋅ ∇ u ( t ) + n 2 - θ u a ( x ) f ( u ) d x ,

where ε > 0, c_ε and c₀ are some positive constants. In the following, we will estimate the last two terms on the right-hand side of (3.15). It follows from (1.13), Hölder's inequality, (3.11), (3.3) and (3.10) that

∫ Ω l ( t ) h ( ∇ u ) m ⋅ ∇ u ( t ) + n 2 - θ u d x ≤ γ ( 0 ) β 1 R ( x 0 ) + B 1 n 2 - θ | | ∇ u | | 2 2 ≤ 2 γ ( 0 ) β 1 c 1 m 0 E ( t ) ,

where c 1 = R ( x 0 ) + B 1 ( n 2 - θ ) . Taking (1.11) and (3.4) into account, we have

- ∫ Ω m ⋅ ∇ u ( t ) + n 2 - θ u a ( x ) f ( u ) d x = - ∫ Ω a ( x ) m ⋅ ∇ F ( u ) d x - n 2 - θ ∫ Ω a ( x ) u f ( u ) d x ≤ ∫ Ω ( n a ( x ) + m ⋅ ∇ a ) F ( u ) d x - ∫ Γ 1 a ( x ) ( m ⋅ ν ) F ( u ) d Γ - n 2 - θ β ∫ Ω a ( x ) F ( u ) d x ≤ ∫ Ω n - n 2 - θ β a ( x ) + m ⋅ ∇ a F ( u ) d x .

A substitution of (3.16)-(3.17) into (3.15), we obtain

d d t ψ ( t ) ≤ 1 2 ∫ Γ 1 ( m ⋅ ν ) | u t | 2 d Γ - θ | | u t | | 2 2 - ( 1 - θ - ε c 0 ) M | | ∇ u | | 2 2 | | ∇ u | | 2 2 + ∫ Γ 1 M | | ∇ u | | 2 2 ∂ u ∂ ν + ∂ u t ∂ ν m ⋅ ∇ u ( t ) + n 2 - θ u d Γ + C ε ∫ Ω | ∇ u t | 2 d x - M | | ∇ u | | 2 2 2 ∫ Γ 1 ( m ⋅ ν ) | ∇ u | 2 d Γ + 2 γ ( 0 ) β 1 c 1 m 0 E ( t ) + ∫ Ω ( n - n 2 - θ β ) a ( x ) + m ⋅ ∇ a F ( u ) d x .

Now, we analyze the boundary term on the right-hand side of (3.18). Applying Young's inequality and M(λ) ≥ m₀ > 0 by (1.12), we have, for ε₁ > 0,

∫ Γ 1 M | | ∇ u | | 2 2 ∂ u ∂ ν + ∂ u t ∂ ν ( m ⋅ ∇ u ( t ) + n 2 - θ u ) d Γ ≤ ε 1 ∫ Γ 1 | m ⋅ ∇ u ( t ) | 2 + n 2 - θ 2 | u | 2 d Γ + C ε 1 ∫ Γ 1 M | | ∇ u | | 2 2 ∂ u ∂ ν + ∂ u t ∂ ν 2 d Γ ≤ ε 1 ∫ Γ 1 ( m ⋅ ν ) | ∇ u | 2 d Γ + n 2 - θ 2 B * ε 1 | | ∇ u | | 2 2 + C ε 1 ∫ Γ 1 M | | ∇ u | | 2 2 ∂ u ∂ ν + ∂ u t ∂ ν 2 d Γ ≤ ε 1 ∫ Γ 1 ( m ⋅ ν ) | ∇ u | 2 d Γ + n 2 - θ 2 B * ε 1 m 0 M | | ∇ u | | 2 2 | | ∇ u | | 2 2 + C ε 1 ∫ Γ 1 M | | ∇ u | | 2 2 ∂ u ∂ ν + ∂ u t ∂ ν 2 d Γ ,

where C ε 1 is a positive constant and B_* > 0 is the constant such that

∫ Γ 1 | u | 2 d Γ ≤ B * | | ∇ u | | 2 2 , ∀ u ∈ V .

Thus, (3.18) becomes

d d t ψ ( t ) ≤ 1 2 ∫ Γ 1 ( m ⋅ ν ) | u t | 2 d Γ - θ | | u t | | 2 2 - 1 - θ - ε c 0 - n 2 - θ 2 B * ε 1 m 0 M ( | | ∇ u | | 2 2 ) | | ∇ u | | 2 2 - M ( | | ∇ u | | 2 2 ) 2 - ε 1 ∫ Γ 1 ( m ⋅ ν ) | ∇ u | 2 d Γ + C ε ∫ Ω | ∇ u t | 2 d x + C ε ∫ Γ 1 M | | ∇ u | | 2 2 ∂ u ∂ ν + ∂ u t ∂ ν 2 d Γ + 2 γ ( 0 ) β 1 c 1 m 0 E ( t ) + ∫ Ω n - n 2 - θ β a ( x ) + m ⋅ ∇ a F ( u ) d x .

By rewriting the boundary condition (2.5) as

M | | ∇ u | | 2 2 ∂ u ∂ ν + ∂ u t ∂ ν = - τ { u t + k ( t ) u ( t ) - k ( t ) u 0 - k ′ ♢ u } ,

and, then, combining (3.7) and (3.20), we deduce that

L ′ ( t ) = N E ′ ( t ) + ψ ′ ( t ) ≤ - N τ 2 - ( m ⋅ ν ) 2 - 8 τ 2 c ε 1 ∫ Γ 1 | u t | 2 d Γ - ( N - c ε ) | | ∇ u t | | 2 2 - θ | | u t | | 2 2 - 1 - θ - ε c 0 - n 2 - θ 2 B * ε 1 m 0 M ( | | ∇ u | | 2 2 ) | | ∇ u | | 2 2 + 8 τ 2 c ε 1 k 2 ( t ) ∫ Γ 1 | u | 2 d Γ + N τ 2 + 8 τ 2 c ε 1 k 2 ( t ) ∫ Γ 1 | u 0 | 2 d Γ - N τ 2 ∫ Γ 1 k ″ ∘ u d Γ - M | | ∇ u | | 2 2 2 - ε 1 ∫ Γ 1 ( m ⋅ ν ) | ∇ u | 2 d Γ + 8 τ 2 c ε 1 ∫ Γ 1 | k ′ ♢ u | 2 d Γ + 2 γ ( 0 ) β 1 c 1 m 0 E ( t ) - N l ( t ) ∫ Ω h ( ∇ u ) u t d x + ∫ Ω n - n 2 - θ β a ( x ) + m ⋅ ∇ a F ( u ) d x .

Similarly as in deriving (3.16), we note that

l ( t ) ∫ Ω h ( ∇ u ) u t d x ≤ γ ( t ) β 1 1 2 | | u t | | 2 2 + 1 2 | | ∇ u | | 2 2 ≤ γ ( t ) β 1 c 3 E ( t ) ≤ γ ( 0 ) β 1 c 3 E ( t ) ,

where c 3 = 1 + 1 m 0 . This implies that

L ′ ( t ) ≤ - N τ 2 - ( m ⋅ ν ) 2 - 8 τ 2 c ε 1 ∫ Γ 1 | u t | 2 d Γ - θ | | u t | | 2 2 - ( N - c ε ) | | ∇ u t | | 2 2 - 1 - θ - ε c 0 - n 2 - θ 2 B * ε 1 m 0 M | | ∇ u | | 2 2 | | ∇ u | | 2 2 + 8 τ 2 c ε 1 k 2 ( t ) ∫ Γ 1 | u | 2 d Γ + N τ 2 + 8 τ 2 c ε 1 k 2 ( t ) ∫ Γ 1 | u 0 | 2 d Γ - N τ 2 ∫ Γ 1 k ″ ∘ u d Γ - M | | ∇ u | | 2 2 2 - ε 1 ∫ Γ 1 ( m ⋅ ν ) | ∇ u | 2 d Γ + 8 τ 2 c ε 1 ∫ Γ 1 | k ′ ♢ u | 2 d Γ + β 1 γ ( 0 ) N c 3 + 2 c 1 m 0 E ( t ) + ∫ Ω n - n 2 - θ β a ( x ) + m ⋅ ∇ a F ( u ) d x .