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

Lemma 1. For g, v ∈ C¹([0, ∞) : ℝ), one has

Lemma 2 (see [26].)Suppose that f ∈ L²(Ω),  g ∈ H^1/2(Γ₁) and h ∈ H^3/2(Γ₁); then, any solution of

satisfies v ∈ H⁴(Ω) and also

Theorem 3 (see [27].)Consider assumptions (A1)-(A2) and let k_i ∈ C²(ℝ⁺) be such that

If u₀ ∈ W∩H⁴(Ω), u₁ ∈ W, satisfying the compatibility condition then there is only one solution u of the system (1)–(5) satisfying

Lemma 4 (see [26].)For every v ∈ H⁴(Ω) and for every μ ∈ ℝ, one has

Lemma 5. The energy functional E satisfies, along the solution of (1)–(5), the estimate

Proof. Multiplying (1) by u^′, integrating over Ω, and using (10), we get

Substituting the boundary terms by (17) and using Lemma 1 and the Young inequality, our conclusion follows.

Lemma 6. Suppose that the initial data (u₀, u₁)∈(H⁴(Ω)∩W) × W, satisfying the compatibility condition (27). Then, the solution of system (1)–(5) satisfies

Proof. Differentiating ψ using (1) and Lemma 4, we get

Let us next examine the integrals over Γ₀ in (39). Since u = ∂u/∂ν = 0 on Γ₀, we have B₁u = B₂u = 0 on Γ₀ and since Therefore, from (39) and (40), we have Using the Young inequality, we get where ϵ is a positive constant. Since the bilinear form a(u, u) is strictly coercive on W, using the trace theory, we obtain where λ₀ is a constant depending on Ω,  μ,  θ and n. Substituting inequalities (43)–(45) into (42) and taking into account that m · ν ≤ 0 on Γ₀, as well as (23) and (25), we have Since the boundary conditions (17) can be written as our conclusion follows.

Theorem 7. Let (u₀, u₁) ∈ W × L²(Ω). Suppose that the resolvent kernels k₁,  k₂ satisfy the condition (H). Then, there exist constants ω, C > 0 such that, for some t₀ large enough, the solution of (1)–(5) satisfies

Otherwise, for all t ≥ t₀, where

Proof. Applying inequality (36) with α = 1/2 in Lemma 6 and from Lemma 5, we obtain

We take θ and ϵ so small such that Since K ∈ L^∞(Ω) and then choosing N large enough, we obtain On the other hand, we can choose N even larger so that If ξ(t) = min {ξ₁(t), ξ₂(t)},  t ≥ t₀, then, using (30) and (33), we have which gives Using the fact that ξ is a nonincreasing continuous function as ξ₁ and ξ₂ are nonincreasing, and so ξ is differentiable, with ξ^′(t) ≤ 0, for a.e. t, then we infer that Since using (55), we obtain, for some positive constant ω,

Case  1. If u₀ = ∂u₀/∂ν = 0 on Γ₁, then (60) reduces to

A simple integration over (t₀, t) yields By using (33) and (59), we then obtain for some positive constant C Thus, estimate (49) is proved.

Case  2. If (u₀, (∂u₀/∂ν))≠(0,0) on Γ₁, then (60) gives

where In this case, we introduce A simple differentiation of G, using (64), leads to Again, a simple integration over (t₀, t) yields which implies, for all t ≥ t₀, By using (59), we deduce that Consequently, by the boundedness of ξ, (50) is established.

Remark 8. Note that the exponential and polynomial decay estimates are only particular cases of (49) and (50). More precisely, we have exponential decay for ξ₁(t) ≡ c₁ and ξ₂(t) ≡ c₂ and polynomial decay for ξ₁(t) = c₁(1 + t) ⁻¹ and ξ₂(t) ≡ c₂, where c₁ and c₂ are positive constants.

Example 9. As in [24], we give some examples to illustrate the energy decay rates given by (49).

• (1)

  If $$,  0 < p ≤ 1, then, for $$, where ξ(t) = bp(1 + t) ^p−1. For suitably chosen positive constants a and b, k_i satisfies (H) and (49) gives

  $$ ()

• (2)

  If k₁(t) = a₁/(1 + t) ^q,  q > 0, and $$,  0 < p ≤ 1, then, for $$, where ξ(t) = q(1 + t) ⁻¹. Then

  $$ ()
  The aforementioned two examples are included in the following more general one.

• (3)

  For any nonincreasing functions k_i(t),  i = 1,2, which satisfy (H), ξ_i = −k^′/k are also nonincreasing differentiable functions, and cξ₁(t) ≤ ξ₂(t), for some 0 < c ≤ 1, and (49) gives

  $$ ()