General Decay for the Degenerate Equation with a Memory Condition at the Boundary

and Applied Analysis 3 The proof of this lemma follows by differentiating the term g ⬦ V. Lemma 2 (see [26]). Suppose that f ∈ L(Ω), g ∈ H1/2(Γ 1 ) and h ∈ H3/2(Γ 1 ); then, any solution of a (V, w) = ∫ Ω fwdx + ∫ Γ 1 gwdΓ + ∫ Γ 1 h ∂w ∂] dΓ, ∀w ∈ W, (20) satisfies V ∈ H4(Ω) and also Δ 2 V = f, V = ∂V ∂] = 0 on Γ 0 , B 1 V = h, B 2 V = g on Γ 1 . (21) We formulate the following assumptions. (A1) Let f ∈ C1(R) satisfy f (s) s ≥ 0, ∀s ∈ R. (22) Additionally, we suppose that f is superlinear; that is, f (s) s ≥ (2 + η) F (s) , F (z) = ∫ z 0 f (s) ds, ∀s ∈ R, (23) for some η > 0 with the following growth condition: 󵄨󵄨󵄨󵄨f (x) − f (y) 󵄨󵄨󵄨󵄨 ≤ c (1 + |x| ρ−1 + 󵄨󵄨󵄨󵄨y 󵄨󵄨󵄨󵄨 ρ−1 ) 󵄨󵄨󵄨󵄨x − y 󵄨󵄨󵄨󵄨 , ∀x, y ∈ R (24) for some c > 0 and ρ ≥ 1 such that (n − 2)ρ ≤ n. (A2) K ∈ C(Ω); H2 0 (Ω) ∩ L ∞ (Ω) with K(x) ≥ 0, for all x ∈ Ω, and satisfy the following condition ∇K ⋅ m ≥ 0 in Ω. (25) The well-posedness of system (1)–(5) is given by the following theorem. Theorem3 (see [27]). Consider assumptions (A1)-(A2) and let k i ∈ C 2 (R+) be such that k i , −k 󸀠 i , k 󸀠󸀠 i ≥ 0 (i = 1, 2) . (26) If u 0 ∈ W ∩ H 4 (Ω), u 1 ∈ W, satisfying the compatibility condition

From the physical point of view, we know that the memory effect described in integral equations ( 3) and ( 4) can be caused by the interaction with another viscoelastic element.In fact, the boundary conditions (3) and (4) mean that Ω is composed of a material which is clamped in a rigid body in Γ 0 and is clamped in a body with viscoelastic properties in the complementary part of its boundary named Γ 1 .Problems related to (1)-( 5) are interesting not only from the point of view of PDE general theory, but also due to its applications in mechanics.
The existence of global solutions and exponential decay to the degenerate equation with Ω = Γ 0 has been investigated by several authors.See Cavalcanti et al. [1] and Menezes et al. [2].For instance, when () is equal to 1, (1) describes the transverse deflection (, ) of beams.There exists a large body of literature regarding viscoelastic problems with the memory term acting in the domain or at the boundary

Preliminaries
In this section, we introduce some notations and establish the existence of solutions of the problem (1)- (5).
Note that, because of condition (2), the solution of system (1)-( 5) must belong to the following space: Let us define the bilinear form (⋅, ⋅) as follows: Since Γ 0 ̸ = 0, we know that (, ) is equivalent to the  2 (Ω) norm on ; that is, and here and in the sequel, we denote by  0 and  0 generic positive constants.Simple calculation, based on the integration by parts formula, yields We assume that there exists  0 ∈ R  such that If we denote the compactness of Γ 1 by () =  −  0 , condition (12) implies that there exists a small positive constant  0 such that 0 <  0 ≤ () ⋅ ](), for all  ∈ Γ 1 .
Next, we will use ( 3) and ( 4) to estimate the values B 1 and B 2 on Γ 1 .Denoting by the convolution product operator and differentiating (3) and (4), we arrive at the following Volterra equations: Applying the Volterra inverse operator, we get where the resolvent kernels satisfy Denoting that  1 = 1/ 1 (0) and  2 = 1/ 2 (0), we have Therefore, we use (17) instead of the boundary conditions ( 3) and ( 4).Let us denote that The following lemma states an important property of the convolution operator.
Lemma 1.For , V ∈  1 ([0, ∞) : R), one has The proof of this lemma follows by differentiating the term  ⬦ V.
Additionally, we suppose that  is superlinear; that is, for some  > 0 with the following growth condition: for some  > 0 and  ≥ 1 such that ( − 2) ≤ .

General Decay
In this section, we show that the solution of system (1)-( 5) may have a general decay not necessarily of exponential or polynomial type.For this we consider that the resolvent kernels satisfy the following hypothesis.(H)   : R + → R + is twice differentiable function such that and there exists a nonincreasing continuous function   : The following identity will be used later.
Lemma 4 (see [26]).For every V ∈  4 (Ω) and for every  ∈ R, one has Let us introduce the energy function Now, we establish some inequalities for the strong solution of system (1)-( 5).
Lemma 5.The energy functional  satisfies, along the solution of (1)-( 5), the estimate Substituting the boundary terms by (17) and using Lemma 1 and the Young inequality, our conclusion follows.
Let us consider the following binary operator: Then applying the Holder inequality for 0 ≤  ≤ 1 we have Let us define the functional The following lemma plays an important role in the construction of the desired functional.
Let us introduce the Lyapunov functional with  > 0. Now, we are in a position to show the main result of this paper.