Well-posedness and general decay for a Lamé viscoelastic system with logarithmic damping

Khoukhi, Alae Nore; Meflah, Mabrouk

doi:10.1108/AJMS-09-2025-0144

Purpose

We study a Lamé-type viscoelastic system with logarithmic velocity damping and quantify its stabilizing effect. The dissipation is weak near zero (σ(v) ⋅ v ∼|v|⁴ as |v| → 0) yet becomes stronger than linear for large speeds. Our goal is to prove global well-posedness and quantify the long-time decay of the natural energy.

Design/methodology/approach

We combine the Faedo–Galerkin scheme, Aubin–Lions compactness, and Minty's monotonicity to construct weak solutions and derive an energy identity. A Lyapunov function adapted to the logarithmic dissipation links the decay of the relaxation kernel g to the mechanical energy, assuming g ≥ 0 is nonincreasing and −g′(t) ≥ ξ(t)g(t) with nonincreasing ξ.

Findings

We obtain global existence and uniqueness of weak solutions and a general decay estimate $E (t) \leq C \exp (- κ \int_{0}^{t} ξ (s) d s)$ ⁠. Hence, exponential decay holds when inf_t≥0 ξ(t) > 0, whereas polynomial rates follow if ξ(t) ∼ c(1 + t)⁻¹. The results clarify the combined role of Lamé ellipticity, hereditary memory, and logarithmic damping in stabilization.

Originality/value

Prior Lamé–viscoelastic studies used logarithmic terms mainly as sources; here the damping itself is logarithmic. This reveals a distinct stabilization mechanism and unifies exponential and polynomial regimes.

1. Introduction

Let Ω be a bounded domain in $R^{n} (n = 2,3)$ with smooth boundary ∂Ω. This paper investigates the well-posedness and asymptotic behavior of a viscoelastic Lamé system with logarithmic damping, governed by the following initial-boundary value problem:

\{\begin{cases} u_{t t} - μ Δ u - (λ + μ) ▿ (div u) + σ (u_{t}) + \int_{0}^{t} g (t - s) Δ u (s) d s = 0, & x \in Ω, t > 0, \\ u (x, 0) = u_{0} (x), u_{t} (x, 0) = u_{1} (x), & x \in Ω, \\ u (x, t) = 0, & x \in \partial Ω, t > 0, \end{cases}

(1.1)

where the unknown $u = (u_{1}, \dots, u_{n}) : Ω \times R^{+} \to R^{n}$ is the displacement vector field, μ and λ are the Lamé constants satisfying μ > 0 and λ + μ > 0, $g : R^{+} \to R^{+}$ is a relaxation kernel modeling the viscoelastic memory, and $σ : R^{n} \to R^{n}$ is the logarithmic damping operator defined by $σ (v) = v \log (1 + | v |^{2})$ for $v \in R^{n}$ ⁠.

The Lamé system (1.1) constitutes the fundamental linear model of isotropic, homogeneous elasticity. It describes wave propagation in elastic solids where the stress tensor depends linearly on both the strain (via the instantaneous Lamé operator (μΔu + (λ + μ)▿(div u)) and its history (through the Volterra integral term). From a physical viewpoint, this system governs the dynamics of numerous engineering and geophysical materials; we refer to the classical monograph [1] for a comprehensive derivation and discussion of the Lamé equations. The mathematical treatment of such vector-valued boundary value problems is covered in detail in Ref. [2].

The logarithmic damping term σ(u_t) introduces a nonlinear frictional resistance that exhibits distinctive qualitative features: it is weaker than linear damping near the origin (σ(v) ⋅ v ∼|v|⁴ as |v| → 0) yet becomes slightly stronger at large velocities (σ(v) ⋅ v ∼|v|² log(1 + |v|²) as |v| → ∞). Such behavior is relevant for modeling complex viscoelastic materials, including certain polymers and biological tissues, where the dissipative mechanism deviates from classical linear or polynomial laws.

Mathematically, the analysis of Lamé systems presents unique challenges compared to scalar wave equations due to the vectorial nature of the displacement field and the coupling of components through the term ▿(div u). These challenges are compounded when memory effects and non-standard damping mechanisms are introduced. While the stability of scalar viscoelastic wave equations has been extensively studied (see, e.g. Ref. [3] for polynomial decay rates), the literature on vectorial Lamé systems with memory is less developed but growing rapidly.

Recent contributions on Lamé systems include [4], who examined a purely viscoelastic Lamé model, and [5], which studies attractors for coupled Lamé systems. The work of [6] investigates synchronization phenomena in coupled Lamé systems, establishing conditions for exponential synchronization. Various damping mechanisms have been considered in the context of Lamé systems, including frictional damping [7], viscoelastic damping with infinite memories [8], and nonlinear damping with source terms [9]. Logarithmic nonlinearities have been investigated mainly as source terms in scalar wave and plate models [10–12]. However, their role as damping mechanisms in vector-valued elasticity systems governed by the Lamé operator has not yet been addressed in the literature. This distinction is essential, since the Lamé system involves component coupling through the operator ▿(▿⋅u), which fundamentally alters both the functional framework and the energy analysis.

The present work aims to bridge this significant gap by providing a rigorous analysis of the Lamé system (1.1) with logarithmic damping and viscoelastic memory. Our main contributions are twofold. First, we prove the existence and uniqueness of global weak solutions using Faedo–Galerkin approximation combined with monotonicity arguments tailored to the logarithmic damping. Second, by constructing a suitable Lyapunov functional,we establish a general decay result of the form

E (t) \leq C \exp (- κ \int_{0}^{t} ξ (s) d s), t \geq 0,

where E(t) is the total energy and ξ(t) is a function determined by the relaxation kernel g. This result recovers exponential and polynomial decay rates as special cases, depending on the behavior of g. Our analysis handles the interplay between three challenging features: the vectorial structure of the Lamé operator, the hereditary memory term, and the non-standard logarithmic damping.

The paper is organized as follows. Section 2 introduces the functional framework and assumptions. Section 3 proves the well-posedness of weak solutions. Section 4 establishes the general decay result via a Lyapunov functional. Finally, Section 5 offers concluding remarks.

2. Preliminaries

In this section, we collect some materials and assumptions needed to establish our main results. Throughout the paper, C denotes a positive constant which may change from line to line.

2.1 Function spaces

Let $Ω \subset R^{n}$ (n = 2, 3) be a bounded Lipschitz domain. We define

H ≔ L^{2} {(Ω)}^{n}, V ≔ H_{0}^{1} {(Ω)}^{n}, V^{*} ≔ dual of V .

For u = (u₁, …, u_n) and v = (v₁, …, v_n) in H, we define the inner product and norm by

〈 u, {v 〉}_{H} ≔ \sum_{i = 1}^{n} \int_{Ω} u_{i} v_{i} d x, ‖ u ‖_{H} ≔ \sqrt{〈 u, {u 〉}_{H}} .

Similarly, in $V = H_{0}^{1} {(Ω)}^{n}$ ⁠, we use

〈 ▿ u, ▿ v 〉 ≔ \sum_{i = 1}^{n} \int_{Ω} ▿ u_{i} \cdot ▿ v_{i} d x,

and define the H¹-seminorm

‖ ▿ u ‖ ≔ \sqrt{〈 ▿ u, ▿ u 〉} .

For vector fields $z : Ω \to R^{n}$ ⁠, we also use the notation

‖ z ‖ ≔ ‖ z ‖_{H} = {(\int_{Ω} | z (x) |^{2} d x)}^{1 / 2} .

By Poincaré’s inequality, ‖▿u‖ is equivalent to the usual H¹-norm on V.

2.2 Problem setting

The unknown $u (x, t) \in R^{n}$ represents the displacement vector field of an elastic body occupying the domain Ω.

We study the following initial–boundary value problem:

\{\begin{cases} u_{t t} - μ Δ u - (λ + μ) ▿ (div u) + σ (u_{t}) + \int_{0}^{t} g (t - s) Δ u (s) d s = 0, & x \in Ω, t > 0, \\ u (x, 0) = u_{0} (x), u_{t} (x, 0) = u_{1} (x), & x \in Ω, \\ u (x, t) = 0, & x \in \partial Ω, t > 0 . \end{cases}

(2.1)

Here $σ : R^{n} \to R^{n}$ denotes the logarithmic damping operator,

σ (v) = v \log (1 + | v |^{2}), v \in R^{n},

and $g : [0, \infty) \to R_{+}$ is a relaxation kernel.

2.3 Lamé operator and associated bilinear form

We consider the vector Lamé operator

L u ≔ μ Δ u + (λ + μ) ▿ (div u), u : Ω \to R^{n},

with Lamé parameters μ > 0, $λ \in R$ ⁠.

We use the notation

▿ u : ▿ v ≔ \sum_{i = 1}^{n} ▿ u_{i} \cdot ▿ v_{i} .

Define, for u, v ∈ V,

a (u, v) ≔ μ \int_{Ω} ▿ u : ▿ v d x + (λ + μ) \int_{Ω} (div u) (div v) d x .

(2.2)

In particular,

a (u, u) = μ ‖ ▿ u ‖^{2} + (λ + μ) ‖ div u ‖^{2} .

Ellipticity condition. We impose the following assumption on the Lamé parameters: (A1) The Lamé parameters satisfy

μ > 0 and λ + μ \geq 0 .

This condition guarantees that the bilinear form a(⋅, ⋅) is coercive on V.

Lemma 2.1.

(Coercivity and boundedness of a). Under Assumption (A1), there exist constants K₁, K₂ > 0 (depending on Ω, n, λ, μ) such that for all u ∈ V,

K_{1} ‖ ▿ u ‖^{2} \leq a (u, u) \leq K_{2} ‖ ▿ u ‖^{2} .

(2.3)

Proof. Upper bound. For a.e. x ∈ Ω, write $div u (x) = \sum_{i = 1}^{n} \partial_{i} u_{i} (x)$ ⁠. By Cauchy–Schwarz,

{(div u (x))}^{2} = {(\sum_{i = 1}^{n} \partial_{i} u_{i} (x))}^{2} \leq n \sum_{i = 1}^{n} {(\partial_{i} u_{i} (x))}^{2} \leq n | ▿ u (x) |^{2} .

Integrating gives ∫_Ω|div u|²dx ≤ n∫_Ω|▿u|²dx. Hence

a (u, u) = μ \int_{Ω} | ▿ u |^{2} d x + (λ + μ) \int_{Ω} | div u |^{2} d x \leq (μ + n (λ + μ)) \int_{Ω} | ▿ u |^{2} d x,

which yields the right inequality in (2.3) with K₂ = μ + n(λ + μ).

Lower bound. From Assumption (A1), μ > 0 and (λ + μ) ≥ 0. Therefore both terms in a(u, u) are nonnegative and

a (u, u) \geq μ \int_{Ω} | ▿ u |^{2} d x = μ ‖ ▿ u ‖^{2},

which gives the left inequality in (2.3) with K₁ = μ.

Finally, since $V = H_{0}^{1} {(Ω)}^{n}$ satisfies Poincaré’s inequality ‖u‖ ≤ C_P‖▿u‖, the bounds (2.3) imply that a(⋅, ⋅) is equivalent to the H¹-seminorm (and to the H¹-norm) on V.

Consequently, we equip V with the norm

‖ u ‖_{V} ≔ \sqrt{a (u, u)},

which is equivalent to the standard $H_{0}^{1} {(Ω)}^{n}$ norm. □

2.4 Phase space

The natural phase space for problem (2.1) is

H ≔ V \times H .

For U = (u, v) and $\tilde{U} = (\tilde{u}, \tilde{v})$ in $H$ ⁠, we define the inner product

〈 U, {\tilde{U} 〉}_{H} ≔ a (u, \tilde{u}) + 〈 v, {\tilde{v} 〉}_{H},

with associated norm

‖ U ‖_{H}^{2} = a (u, u) + ‖ v ‖_{H}^{2} .

2.5 Relaxation kernel and memory functional

Kernel hypotheses. We impose the following assumptions on the relaxation kernel g:

(A2) (Kernel regularity) $g \in C^{1} ([0, \infty))$ ⁠, g(t) ≥ 0, g′(t) ≤ 0 for all t ≥ 0, with g(0) > 0, and

μ - \int_{0}^{\infty} g (s) d s ≕ ℓ > 0 .

This ensures positivity of the elastic part and well-posedness of the associated energy. (A3) There exists a nonincreasing differentiable function $ξ : R_{+} \to R_{+}$ such that

g^{'} (t) \leq - ξ (t) g (t), t \geq 0 .

Moreover, ξ satisfies, for some constant L > 0,

| \frac{ξ^{'} (t)}{ξ (t)} | \leq L, ξ^{'} (t) \leq 0, \int_{0}^{\infty} ξ (s) d s = \infty .

These conditions guarantee that the decay of g transfers to the energy, and allow explicit decay rates.

For w: [0, T] → V, define the memory functional

(g ◦ ▿ w) (t) ≔ \frac{1}{2} \int_{0}^{t} g (t - s) ‖ ▿ w (t) - ▿ w (s) ‖^{2} d s .

(2.4)

2.6 Logarithmic damping: definition and lemmas

(A4) (Logarithmic damping) The damping function $σ : R^{n} \to R^{n}$ is defined by

σ (v) = v \log (1 + | v |^{2}), v \in R^{n} .

It satisfies the following properties:

Monotonicity: (σ(v) − σ(w)) ⋅ (v − w) ≥ 0, for all $v, w \in R^{n}$ ⁠.
Dissipativity: σ(v) ⋅ v = |v|² log(1 + |v|²) ≥ c₀ min{|v|⁴, |v|²}, for some universal c₀ > 0.
Local Lipschitz continuity: For every R > 0, there exists C_R > 0 such that |σ(v) − σ(w)| ≤ C_R|v − w| whenever |v|, |w| ≤ R.

Proof.

Monotonicity. Since σ = ▿Ψ with the convex potential

Ψ (v) = \frac{1}{2} (1 + | v |^{2}) \log (1 + | v |^{2}) - \frac{1}{2} | v |^{2},

it follows that ▿Ψ is monotone.

Dissipativity. By direct computation,

σ (v) \cdot v = | v |^{2} \log (1 + | v |^{2}) \geq 0 .

For |v| ≤ 1, log(1 + |v|²) ≥|v|²/2, hence $σ (v) \cdot v \geq \frac{1}{2} | v |^{4}$ ⁠. For $| v | \geq \sqrt{e - 1}$ ⁠, log(1 + |v|²) ≥ 1, hence σ(v) ⋅ v ≥|v|². Thus the unified bound σ(v) ⋅ v ≥ c₀ min{|v|⁴, |v|²} holds.

Local Lipschitz continuity. The Jacobian matrix is

D σ (v) = \log (1 + | v |^{2}) I + \frac{2}{1 + | v |^{2}} v \otimes v,

which is bounded on the ball {|v| ≤ R}. Therefore σ is Lipschitz continuous on bounded sets, which yields the stated estimate. □

Assumption (A4) gathers the main properties of the logarithmic damping, which will be used later in the Galerkin approximation and in the decay analysis.

2.7 Energy functional and dissipation

We define, for a (weak) solution u,

E (t) ≔ \frac{1}{2} ‖ u_{t} (t) ‖^{2} + \frac{1}{2} a (u (t), u (t)) + (g ◦ ▿ u) (t),

where (g ◦ ▿u) (t) is given by (2.4).

Lemma 2.2.

(Energy identity and monotonicity). Under (A1)–(A4), and for every weak solution u ∈ L^∞(0, T; V) with u_t ∈ L^∞(0, T; H), the energy satisfies E ∈ W^1,1(0, T) and for a.e. t ∈ (0, T),

\begin{align} E^{'} (t) & = - \int_{Ω} σ (u_{t} (t)) \cdot u_{t} (t) d x - \frac{g (t)}{2} ‖ ▿ u (t) ‖^{2} - \frac{1}{2} \int_{0}^{t} (- g^{'} (t - s)) ‖ ▿ u (t) - ▿ u (s) ‖^{2} d s \\ \leq 0 . \end{align}

(2.5)

In particular, E is nonincreasing on [0, T].

Proof. The derivation is first established for smooth solutions; the extension to weak solutions follows from the Galerkin approximation together with compactness and monotonicity arguments.

Step 1: Test (2.1) with u_t(t) in H and integrate over Ω:

(u_{t t}, u_{t}) - μ (Δ u, u_{t}) - (λ + μ) (▿ (▿ \cdot u), u_{t}) + \int_{Ω} σ (u_{t}) \cdot u_{t} d x + (\int_{0}^{t} g (t - s) Δ u (s) d s, u_{t}) = 0 .

Boundary terms vanish by u|_∂Ω = 0. Using the definition of a(⋅, ⋅) and symmetry (guaranteed by (A1)),

(u_{t t}, u_{t}) = \frac{d}{d t} \frac{1}{2} ‖ u_{t} ‖^{2}, - μ (Δ u, u_{t}) - (λ + μ) (▿ (▿ \cdot u), u_{t}) = a (u, u_{t}) = \frac{d}{d t} \frac{1}{2} a (u, u) .

The damping contribution satisfies, by dissipativity in (A4), ∫_Ω σ(u_t) ⋅ u_t dx ≥ 0. For the memory term, integrate by parts in space:

(\int_{0}^{t} g (t - s) Δ u (s) d s, u_{t} (t)) = - \int_{0}^{t} g (t - s) (▿ u (s), ▿ u_{t} (t)) d s .

Summing up,

\frac{d}{d t} (\frac{1}{2} ‖ u_{t} ‖^{2} + \frac{1}{2} a (u, u)) + \int_{Ω} σ (u_{t}) \cdot u_{t} d x - \int_{0}^{t} g (t - s) (▿ u (s), ▿ u_{t} (t)) d s = 0 .

(2.6)

Step 2: Dafermos identity for (g ◦ ▿u) (t). By (A2)–(A3), (g ◦ ▿u) (t) is absolutely continuous and

\begin{align} \frac{d}{d t} (g ◦ ▿ u) (t) & = - \frac{g (t)}{2} ‖ ▿ u (t) ‖^{2} + \int_{0}^{t} g (t - s) (▿ u_{t} (t), ▿ u (t) - ▿ u (s)) d s \\ - \frac{1}{2} \int_{0}^{t} (- g^{'} (t - s)) ‖ ▿ u (t) - ▿ u (s) ‖^{2} d s . \end{align}

(2.7)

Split the mixed integral:

\int_{0}^{t} g (t - s) (▿ u_{t} (t), ▿ u (t) - ▿ u (s)) d s = \underset{I_{1}}{\underset{︸}{\int_{0}^{t} g (t - s) (▿ u_{t} (t), ▿ u (t)) d s}} - \underset{I_{2}}{\underset{︸}{\int_{0}^{t} g (t - s) (▿ u_{t} (t), ▿ u (s)) d s}} .

Note that the memory term in (2.6) equals $+ I_{2}$ ⁠:

- \int_{0}^{t} g (t - s) (▿ u (s), ▿ u_{t} (t)) d s = + I_{2} .

Step 3: Combining identities. Add (2.7) to (2.6). The terms $\pm I_{2}$ cancel. We obtain

\begin{array}{l} \frac{d}{d t} (\frac{1}{2} ‖ u_{t} ‖^{2} + \frac{1}{2} a (u, u) + (g ◦ ▿ u) (t)) + \int_{Ω} σ (u_{t}) \cdot u_{t} d x \\ = - \frac{g (t)}{2} ‖ ▿ u (t) ‖^{2} + I_{1} - \frac{1}{2} \int_{0}^{t} (- g^{'} (t - s)) ‖ ▿ u (t) - ▿ u (s) ‖^{2} d s . \end{array}

But $I_{1} = \int_{0}^{t} g (t - s) (▿ u_{t} (t), ▿ u (t)) d s$ is already part of $\frac{d}{d t} (g ◦ ▿ u) (t)$ in (2.7), hence the whole right-hand side is exactly the one displayed in (2.5), i.e.

E^{'} (t) = - \int_{Ω} σ (u_{t} (t)) \cdot u_{t} (t) d x - \frac{g (t)}{2} ‖ ▿ u (t) ‖^{2} - \frac{1}{2} \int_{0}^{t} (- g^{'} (t - s)) ‖ ▿ u (t) - ▿ u (s) ‖^{2} d s .

Step 4: Sign of E′(t). We check the sign of each contribution in (2.5):

By dissipativity in (A4):

\int_{Ω} σ (u_{t}) \cdot u_{t} d x \geq 0 .

(2)
By (A2):

g (t) \geq 0 .

(3)
By (A3):

- g^{'} (t - s) \geq 0 for all s \in [0, t] .

Hence every term on the right-hand side of (2.5) is nonpositive, and therefore

E^{'} (t) \leq 0 .

Integrability and absolute continuity. Integrating (2.5) over (0, t) gives

E (0) - E (t) = \int_{0}^{t} \int_{Ω} σ (u_{τ}) \cdot u_{τ} d x d τ + \frac{1}{2} \int_{0}^{t} g (τ) ‖ ▿ u (τ) ‖^{2} d τ + \frac{1}{2} \int_{0}^{t} \int_{0}^{τ} (- g^{'} (τ - s)) ‖ ▿ u (τ) - ▿ u (s) ‖^{2} d s d τ .

The three terms on the right are nonnegative and finite on (0, T), hence the right-hand side of (2.5) belongs to L¹(0, T). Therefore E ∈ W^1,1(0, T) and (2.5) holds for a.e. t ∈ (0, T).

Remark on weak solutions. The above derivation was carried out formally for smooth functions. For the weak solution constructed via the Galerkin scheme in Section 3, all steps can be rigorously justified: uniform a priori bounds from (A1)–(A4) yield compactness (Aubin–Lions), while the monotonicity of σ and the Dafermos identity for (g ◦ ▿u) (t) ensure that (2.5) also holds for the weak solution. □

3. Existence and uniqueness of solutions

3.1 Weak formulation

Let T > 0. A function

u \in L^{\infty} (0, T; V), u_{t} \in L^{\infty} (0, T; H), u_{t t} \in L^{2} (0, T; V^{*})

is called a weak solution to (2.1) on (0, T) if u(0) = u₀, u_t(0) = u₁, and for a.e. t ∈ (0, T),

〈 u_{t t} (t), v 〉 + a (u (t), v) + \int_{Ω} σ (u_{t} (t)) \cdot v d x - \int_{0}^{t} g (t - s) (▿ u (s), ▿ v) d s = 0, \forall v \in V,

(3.1)

where (⋅, ⋅) denotes the H-inner product and ⟨⋅, ⋅⟩ the duality between V* and V.

Theorem 3.1.

(Existence and uniqueness). Assume (A1)–(A4) and let (u₀, u₁) ∈ V × H. Then there exists a unique weak solution

u \in L^{\infty} (0, T; V), u_{t} \in L^{\infty} (0, T; H), u_{t t} \in L^{2} (0, T; V^{*})

to problem (2.1) in the weak sense (3.1). Moreover, the energy identity (2.5) holds on (0, T) and the energy E(t) is nonincreasing.

Proof.

Step 1. Weak formulation. We recall that u is a weak solution if u(0) = u₀, u_t(0) = u₁, and (3.1) holds for all v ∈ V.
Step 2. Galerkin approximation. Let ${w_{j}}_{j \geq 1} \subset V$ be an H-orthonormal basis of eigenfunctions of the Lamé operator, and set V_m ≔ span{w₁, …, w_m}. We seek $u^{m} (t) = \sum_{j = 1}^{m} d_{j}^{m} (t) w_{j}$ solving, for all v ∈ V_m,

(u_{t t}^{m} (t), v) + a (u^{m} (t), v) + \int_{Ω} σ (u_{t}^{m} (t)) \cdot v d x + \int_{0}^{t} g (t - s) (▿ u^{m} (s), ▿ v) d s = 0,

(3.2)

with u^m(0) = P_mu₀, $u_{t}^{m} (0) = P_{m} u_{1}$ ⁠, where P_m: H → V_m is the H-orthogonal projector. By the Carathéodory existence theorem (see, e.g. Amann [13]), the Galerkin system (3.2) admits a local solution.

Step 3. A priori estimates. Testing (3.2) with $v = u_{t}^{m} (t)$ in H and repeating the energy calculation of Lemma 2.2 (which is fully justified here since u^m is smooth in time) yields the discrete energy identity

\frac{d}{d t} E_{m} (t) = - \int_{Ω} σ (u_{t}^{m} (t)) \cdot u_{t}^{m} (t) d x - \frac{g (t)}{2} ‖ ▿ u^{m} (t) ‖^{2} - \frac{1}{2} \int_{0}^{t} (- g^{'} (t - s)) ‖ ▿ u^{m} (t) - ▿ u^{m} (s) ‖^{2} d s \leq 0,

(3.3)

where

E_{m} (t) = \frac{1}{2} ‖ u_{t}^{m} (t) ‖^{2} + \frac{1}{2} a (u^{m} (t), u^{m} (t)) + (g ◦ ▿ u^{m}) (t) .

By the coercivity of a(⋅, ⋅) (A1) together with Poincaré’s inequality, $a (u^{m}, u^{m}) ≃ ‖ u^{m} ‖_{V}^{2}$ ⁠, so (3.3) implies the uniform bound

\sup_{t \in [0, T]} (‖ u_{t}^{m} (t) ‖^{2} + ‖ u^{m} (t) ‖_{V}^{2}) + \sup_{t \in [0, T]} (g ◦ ▿ u^{m}) (t) \leq C,

(3.4)

with a constant C > 0 depending only on the initial energy E_m(0) ≤ E(0) and independent of m.

Step 4. Compactness and passage to the limit. From the uniform bound (3.4), we extract a subsequence (still denoted by u^m) such that

u^{m} ⇀^{*} u in L^{\infty} (0, T; V), u_{t}^{m} ⇀^{*} u_{t} in L^{\infty} (0, T; H) .

In addition, testing (3.2) with arbitrary v ∈ V_m and using assumptions (A1)–(A4) provides a uniform estimate for $u_{t t}^{m}$ in L²(0, T; V*). Together with (3.4), the Aubin–Lions lemma (see Simon [14]) yields the compactness

u^{m} \to u in L^{2} (0, T; H), u_{t}^{m} \to u_{t} in L^{2} (0, T; V^{*}) .

Linear terms. Passing to the limit in all linear contributions of (3.2) is straightforward by weak convergence.

Nonlinear damping. For the damping operator σ, monotonicity (A4) and Minty's method (see Minty [15] and Brézis [16]) give the identification of the weak limit:

\int_{0}^{T} \int_{Ω} (σ (u_{t}^{m}) - σ (v)) \cdot (u_{t}^{m} - v) d x d t \geq 0, \forall v \in L^{2} (0, T; H),

which implies

σ (u_{t}^{m}) ⇀ σ (u_{t}) in L_{loc}^{1} ((0, T) \times Ω) .

Memory term. The Dafermos identity for (g ◦ ▿⋅), valid under (A2)–(A3), combined with the strong convergence of u^m in L²(0, T; H), yields

\int_{0}^{t} g (t - s) (▿ u^{m} (s), ▿ v) d s \to \int_{0}^{t} g (t - s) (▿ u (s), ▿ v) d s, \forall v \in V, a.e. t \in (0, T) .

Limit identification. Hence the limit function u satisfies the weak formulation (3.1). The initial conditions follow from u^m(0) = P_mu₀, $u_{t}^{m} (0) = P_{m} u_{1}$ ⁠, together with weak^* continuity in L^∞(0, T; V × H).

Step 5. Uniqueness. Let u and v be two weak solutions corresponding to the same initial data, and set w = u − v. Then w satisfies (3.1) with homogeneous initial conditions:

〈 w_{t t}, ϕ 〉 + a (w, ϕ) + \int_{Ω} (σ (u_{t}) - σ (v_{t})) \cdot ϕ d x + \int_{0}^{t} g (t - s) (▿ w (s), ▿ ϕ) d s = 0, \forall ϕ \in V .

Choosing ϕ = w_t and repeating the energy calculation of Lemma 2.2, we obtain

\frac{d}{d t} E_{w} (t) = - \int_{Ω} (σ (u_{t}) - σ (v_{t})) \cdot (u_{t} - v_{t}) d x - \frac{g (t)}{2} ‖ ▿ w (t) ‖^{2} - \frac{1}{2} \int_{0}^{t} (- g^{'} (t - s)) ‖ ▿ w (t) - ▿ w (s) ‖^{2} d s,

where

E_{w} (t) = \frac{1}{2} ‖ w_{t} (t) ‖^{2} + \frac{1}{2} a (w (t), w (t)) + (g ◦ ▿ w) (t) .

By monotonicity of σ (A4) and the positivity of the memory terms (A2)–(A3), the right-hand side is nonpositive, hence

\frac{d}{d t} E_{w} (t) \leq 0 .

Since E_w(0) = 0 (zero initial data), it follows that E_w(t) ≡ 0 for all t ≥ 0, and thus w ≡ 0. This proves uniqueness of weak solutions. □

4. Energy decay

In this section we establish the asymptotic stability of the system. Recall from Lemma 2.2 that the total energy

E (t) = \frac{1}{2} ‖ u_{t} (t) ‖^{2} + \frac{1}{2} a (u (t), u (t)) + (g ◦ ▿ u) (t)

is nonincreasing. Our goal is to quantify its decay rate under the structural assumptions (A2)–(A3) on the relaxation kernel g.

4.1 Lyapunov functional

For a parameter ɛ > 0 define

V (t) ≔ E (t) + ε J (t), J (t) ≔ \int_{0}^{t} g (t - s) (▿ u (t), ▿ u (t) - ▿ u (s)) d s .

(4.1)

Lemma 4.1.

(Equivalence with the energy). Assume (A1)–(A2). Then there exist ɛ₀, c₁, c₂ > 0 such that for all $ε \in (0, ε_{0}]$ and all t ≥ 0,

c_{1} E (t) \leq V (t) \leq c_{2} E (t) .

(4.2)

Proof. Recall

(g ◦ ▿ u) (t) = \frac{1}{2} \int_{0}^{t} g (t - s) ‖ ▿ u (t) - ▿ u (s) ‖^{2} d s .

To estimate $J (t)$ ⁠, rewrite it as a weighted inner product. Define

f (s) ≔ \sqrt{g (t - s)} ▿ u (t), h (s) ≔ \sqrt{g (t - s)} (▿ u (t) - ▿ u (s)) .

Then $J (t) = \int_{0}^{t} f (s) \cdot h (s) d s$ ⁠, and by Cauchy–Schwarz,

| J (t) | \leq {(\int_{0}^{t} | f (s) |^{2} d s)}^{1 / 2} {(\int_{0}^{t} | h (s) |^{2} d s)}^{1 / 2} .

A direct computation gives

\int_{0}^{t} | f (s) |^{2} d s = \int_{0}^{t} g (t - s) ‖ ▿ u (t) ‖^{2} d s = ‖ ▿ u (t) ‖^{2} G (t), G (t) ≔ \int_{0}^{t} g (t - s) d s,

and

\int_{0}^{t} | h (s) |^{2} d s = \int_{0}^{t} g (t - s) ‖ ▿ u (t) - ▿ u (s) ‖^{2} d s = 2 (g ◦ ▿ u) (t) .

Therefore

| J (t) | \leq ‖ ▿ u (t) ‖ \sqrt{G (t)} {(2 (g ◦ ▿ u) (t))}^{1 / 2} .

By (A2) we have $G (t) \leq G_{\infty} ≔ \int_{0}^{\infty} g (s) d s < \infty$ ⁠, hence

| J (t) | \leq ‖ ▿ u (t) ‖ \sqrt{G_{\infty}} {(2 (g ◦ ▿ u) (t))}^{1 / 2} .

Finally, apply Young's inequality to the product

a ≔ ‖ ▿ u (t) ‖ \sqrt{G_{\infty}}, b ≔ {(2 (g ◦ ▿ u) (t))}^{1 / 2},

which yields, for any α > 0,

a b \leq \frac{a^{2}}{2 α} + \frac{α b^{2}}{2} = \frac{G_{\infty}}{2 α} ‖ ▿ u (t) ‖^{2} + α (g ◦ ▿ u) (t) .

This proves

| J (t) | \leq \frac{G_{\infty}}{2 α} ‖ ▿ u (t) ‖^{2} + α (g ◦ ▿ u) (t) .

(4.3)

To further estimate $J (t)$ ⁠, we use the coercivity of the bilinear form a(⋅, ⋅). By Lemma 2.1, there exist constants K₁, K₂ > 0 such that

K_{1} ‖ ▿ u (t) ‖^{2} \leq a (u (t), u (t)) \leq K_{2} ‖ ▿ u (t) ‖^{2} .

In particular this implies

‖ ▿ u (t) ‖^{2} \leq \frac{1}{K_{1}} a (u (t), u (t)) .

Substituting this bound into inequality (4.3) we obtain

| J (t) | \leq \frac{G_{\infty}}{2 α K_{1}} a (u (t), u (t)) + α (g ◦ ▿ u) (t) .

(4.4)

Step 3: Lower bound for $V$ ⁠. Starting from $V = E (t) + ε J (t)$ and using the triangle inequality,

V (t) = E (t) + ε J (t) \geq E (t) - ε | J (t) | .

Invoking (4.4), we obtain

V (t) \geq \frac{1}{2} ‖ u_{t} (t) ‖^{2} + (\frac{1}{2} - ε \frac{G_{\infty}}{2 α K_{1}}) a (u (t), u (t)) + (1 - ε α) (g ◦ ▿ u) (t) .

It is convenient to write the elastic coefficient relatively to $\frac{1}{2}$ ⁠, namely

\frac{1}{2} - ε \frac{G_{\infty}}{2 α K_{1}} = \frac{1}{2} (1 - ε \frac{G_{\infty}}{α K_{1}}),

so that the three coefficients in front of the energy components are, respectively,

\underset{for \frac{1}{2} ‖ u_{t} ‖^{2}}{\underset{︸}{1}}, \underset{for \frac{1}{2} a (u, u)}{\underset{︸}{1 - ε \frac{G_{\infty}}{α K_{1}}}}, \underset{for (g ◦ ▿ u)}{\underset{︸}{1 - ε α}} .

Choose α = 1 to simplify the expressions. Then select ɛ₀ > 0 so small that

ε_{0} \frac{G_{\infty}}{2 K_{1}} < \frac{1}{4} and ε_{0} < \frac{1}{2} .

These two conditions imply, for every $ε \in (0, ε_{0}]$ ⁠,

1 - ε \frac{G_{\infty}}{K_{1}} \geq \frac{1}{2} and 1 - ε \geq \frac{1}{2},

hence all three coefficients above are positive and uniformly bounded away from zero.

Now define

m (ε) ≔ \min \{1, 1 - ε \frac{G_{\infty}}{K_{1}}, 1 - ε\} .

With this choice we have, componentwise,

\frac{1}{2} ‖ u_{t} ‖^{2} \geq m (ε) \cdot \frac{1}{2} ‖ u_{t} ‖^{2}, \frac{1}{2} (1 - ε \frac{G_{\infty}}{K_{1}}) a (u, u) \geq m (ε) \cdot \frac{1}{2} a (u, u),

(1 - ε) (g ◦ ▿ u) (t) \geq m (ε) \cdot (g ◦ ▿ u) (t) .

Summing the three inequalities yields

V (t) \geq m (ε) (\frac{1}{2} ‖ u_{t} (t) ‖^{2} + \frac{1}{2} a (u (t), u (t)) + (g ◦ ▿ u) (t)) = m (ε) E (t) .

Since $m (ε) \geq \frac{1}{2}$ for all $ε \in (0, ε_{0}]$ ⁠, we may set $c_{1} ≔ \frac{1}{2}$ (or any $c_{1} \in (0, \frac{1}{2}])$ ⁠, independent of t, and conclude

V (t) \geq c_{1} E (t), \forall t \geq 0 .

Step 4: Upper bound for $V$ ⁠. Starting from $V = E (t) + ε J (t)$ and using (4.4) with the triangle inequality,

\begin{aligned} V (t) & = E (t) + ε J (t) \leq E (t) + ε | J (t) | \\ \leq \frac{1}{2} ‖ u_{t} (t) ‖^{2} + (\frac{1}{2} + ε \frac{G_{\infty}}{2 α K_{1}}) a (u (t), u (t)) + (1 + ε α) (g ◦ ▿ u) (t) . \end{aligned}

Choose α = 1 for simplicity. Then

V (t) \leq \frac{1}{2} ‖ u_{t} (t) ‖^{2} + (\frac{1}{2} + ε \frac{G_{\infty}}{2 K_{1}}) a (u (t), u (t)) + (1 + ε) (g ◦ ▿ u) (t) .

We now compare the coefficients with those of $E (t) = \frac{1}{2} ‖ u_{t} ‖^{2} + \frac{1}{2} a (u, u) + (g ◦ ▿ u) (t)$ ⁠. Let

c_{2} (ε) ≔ \max \{1, 1 + ε \frac{G_{\infty}}{K_{1}}, 1 + ε\} .

Then, componentwise,

\frac{1}{2} \leq c_{2} (ε) \cdot \frac{1}{2}, \frac{1}{2} + ε \frac{G_{\infty}}{2 K_{1}} \leq c_{2} (ε) \cdot \frac{1}{2}, 1 + ε \leq c_{2} (ε) \cdot 1 .

Therefore

V (t) \leq c_{2} (ε) (\frac{1}{2} ‖ u_{t} (t) ‖^{2} + \frac{1}{2} a (u (t), u (t)) + (g ◦ ▿ u) (t)) = c_{2} (ε) E (t) .

Finally, since $ε \in (0, ε_{0}]$ ⁠, we may fix

c_{2} ≔ \max \{1, 1 + ε_{0} \frac{G_{\infty}}{K_{1}}, 1 + ε_{0}\} > 1,

which is independent of t. Hence

V (t) \leq c_{2} E (t), \forall t \geq 0 .

Combining the lower and upper bounds obtained in Steps 3 and 4, we conclude (4.2) with constants c₁, c₂ > 0 independent of t. □

Proposition 4.2.

(Differential inequality for $V$ ⁠). Assume (A1)–(A4). There exist ɛ₀ ∈ (0, 1)

and c_∗, c* > 0 such that for all $ε \in (0, ε_{0}]$ and a.e. t > 0,

V^{'} (t) \leq - \int_{Ω} σ (u_{t}) \cdot u_{t} d x - c_{*} g (t) ‖ ▿ u (t) ‖^{2} - c^{*} \int_{0}^{t} (- g^{'} (t - s)) ‖ ▿ u (t) - ▿ u (s) ‖^{2} d s .

(4.5)

Proof. We argue first for smooth solutions and then pass to weak solutions by Galerkin approximation and lower semicontinuity, which preserves the differential inequality.

Recall $V (t) = E (t) + ε J (t)$ with ɛ > 0. Differentiating,

V^{'} (t) = E^{'} (t) + ε J^{'} (t) .

From the energy identity (2.5), for a.e. t > 0,

E^{'} (t) = - \int_{Ω} σ (u_{t} (t)) \cdot u_{t} (t) d x - \frac{g (t)}{2} ‖ ▿ u (t) ‖^{2} - \frac{1}{2} \int_{0}^{t} (- g^{'} (t - s)) ‖ ▿ u (t) - ▿ u (s) ‖^{2} d s .

Moreover, using Dafermos' calculus under (A2)–(A3),

J^{'} (t) = \frac{g (t)}{2} ‖ ▿ u (t) ‖^{2} - \frac{1}{2} \int_{0}^{t} (- g^{'} (t - s)) ‖ ▿ u (t) - ▿ u (s) ‖^{2} d s + \int_{0}^{t} g (t - s) (▿ u_{t} (t), ▿ u (t) - ▿ u (s)) d s .

Combining,

\begin{aligned} V^{'} (t) & \leq - \int_{Ω} σ (u_{t} (t)) \cdot u_{t} (t) d x \\ - (\frac{1}{2} - \frac{ε}{2}) g (t) ‖ ▿ u (t) ‖^{2} - (\frac{1}{2} + \frac{ε}{2}) \int_{0}^{t} (- g^{'} (t - s)) ‖ ▿ u (t) - ▿ u (s) ‖^{2} d s \\ + ε \int_{0}^{t} g (t - s) (▿ u_{t} (t), ▿ u (t) - ▿ u (s)) d s . \end{aligned}

We now estimate the mixed term. By Cauchy–Schwarz with weight g(t − s) ds and Young's inequality, for any δ > 0,

\begin{aligned} | \int_{0}^{t} g (t - s) & (▿ u_{t} (t), ▿ u (t) - ▿ u (s)) d s | \\ \leq {(\int_{0}^{t} g (t - s) d s)}^{1 / 2} ‖ ▿ u_{t} (t) ‖ {(\int_{0}^{t} g (t - s) ‖ ▿ u (t) - ▿ u (s) ‖^{2} d s)}^{1 / 2} \\ \leq \frac{δ}{2} ‖ ▿ u_{t} (t) ‖^{2} + \frac{1}{2 δ} (\int_{0}^{t} g (t - s) d s) \int_{0}^{t} g (t - s) ‖ ▿ u (t) - ▿ u (s) ‖^{2} d s . \end{aligned}

Multiplying by ɛ and keeping all terms, we have, for any δ > 0,

ε \int_{0}^{t} g (t - s) (▿ u_{t} (t), ▿ u (t) - ▿ u (s)) d s \leq \frac{ε δ}{2} ‖ ▿ u_{t} (t) ‖^{2} + \frac{ε}{2 δ} G (t) \int_{0}^{t} g (t - s) ‖ ▿ u (t) - ▿ u (s) ‖^{2} d s,

with $G (t) = \int_{0}^{t} g (t - s) d s \leq G_{\infty}$ by (A2). Using (A3), we also have

\int_{0}^{t} g (t - s) ‖ ▿ u (t) - ▿ u (s) ‖^{2} d s \leq \frac{1}{ξ (t)} \int_{0}^{t} (- g^{'} (t - s)) ‖ ▿ u (t) - ▿ u (s) ‖^{2} d s .

Hence,

ε \int_{0}^{t} g (t - s) (▿ u_{t} (t), ▿ u (t) - ▿ u (s)) d s \leq \frac{ε δ}{2} ‖ ▿ u_{t} (t) ‖^{2} + \frac{ε}{2 δ} \frac{G (t)}{ξ (t)} \int_{0}^{t} (- g^{'} (t - s)) ‖ ▿ u (t) - ▿ u (s) ‖^{2} d s .

Choosing δ = 1 and G(t) ≤ G_∞ yields

ε \int_{0}^{t} g (t - s) (▿ u_{t} (t), ▿ u (t) - ▿ u (s)) d s \leq \frac{ε}{2} ‖ ▿ u_{t} (t) ‖^{2} + ε \frac{G_{\infty}}{2 ξ (t)} \int_{0}^{t} (- g^{'} (t - s)) ‖ ▿ u (t) - ▿ u (s) ‖^{2} d s .

Inserting this bound into the differential inequality for $V^{'} (t)$ and grouping like terms yields

\begin{aligned} V^{'} (t) \leq & - \int_{Ω} σ (u_{t}) \cdot u_{t} d x - (\frac{1}{2} - \frac{ε}{2}) g (t) ‖ ▿ u (t) ‖^{2} \\ - (\frac{1}{2} + \frac{ε}{2} - ε \frac{G_{\infty}}{2 ξ (t)}) \int_{0}^{t} (- g^{'} (t - s)) ‖ ▿ u (t) - ▿ u (s) ‖^{2} d s + \frac{ε}{2} ‖ ▿ u_{t} (t) ‖^{2} . \end{aligned}

Then

\begin{aligned} V^{'} (t) - \frac{ε}{2} ‖ ▿ u_{t} (t) ‖^{2} \leq & - \int_{Ω} σ (u_{t}) \cdot u_{t} d x - (\frac{1}{2} - \frac{ε}{2}) g (t) ‖ ▿ u (t) ‖^{2} \\ - (\frac{1}{2} + \frac{ε}{2} - ε \frac{G_{\infty}}{2 ξ (t)}) \int_{0}^{t} (- g^{'} (t - s)) ‖ ▿ u (t) - ▿ u (s) ‖^{2} d s . \end{aligned}

Since $- \frac{ε}{2} ‖ ▿ u_{t} (t) ‖^{2} \leq 0$ ⁠, dropping it from the left-hand side weakens the inequality, hence

\begin{aligned} V^{'} (t) \leq & - \int_{Ω} σ (u_{t}) \cdot u_{t} d x - (\frac{1}{2} - \frac{ε}{2}) g (t) ‖ ▿ u (t) ‖^{2} \\ - (\frac{1}{2} + \frac{ε}{2} - ε \frac{G_{\infty}}{2 ξ (t)}) \int_{0}^{t} (- g^{'} (t - s)) ‖ ▿ u (t) - ▿ u (s) ‖^{2} d s . \end{aligned}

Fix T > 0 and set ξ_T ≔ inf_{0≤s ≤ T}ξ(s) = ξ(T) (since ξ is nonincreasing). Choose

ε_{0} ≔ \min \{\frac{1}{2}, \frac{ξ_{T}}{4 G_{\infty}}\}, and take ε \in (0, ε_{0}] .

Then, for all t ∈ (0, T],

\frac{1}{2} - \frac{ε}{2} \geq \frac{1}{4} and \frac{1}{2} + \frac{ε}{2} - ε \frac{G_{\infty}}{2 ξ (t)} \geq \frac{1}{2} - ε \frac{G_{\infty}}{2 ξ_{T}} \geq \frac{1}{2} - \frac{1}{8} = \frac{3}{8} .

Therefore we can take the explicit constants

c_{*} ≔ \frac{1}{4}, c^{*} ≔ \frac{3}{8},

which depend only on T through ξ_T (and on G_∞) but are independent of t ∈ (0, T]. Hence,

V^{'} (t) \leq - \int_{Ω} σ (u_{t} (t)) \cdot u_{t} (t) d x - c_{*} g (t) ‖ ▿ u (t) ‖^{2} - c^{*} \int_{0}^{t} (- g^{'} (t - s)) ‖ ▿ u (t) - ▿ u (s) ‖^{2} d s,

which is exactly (4.5).

Finally, the above computations are rigorous for Galerkin approximants. Passing to the limit uses the weak lower–semicontinuity of norms together with the monotonicity of σ (A4) and the Dafermos calculus under (A2)–(A3), preserving the differential inequality for weak solutions. □

Theorem 4.3.

(General decay estimate). Let assumptions (A1)–(A4) hold. Suppose further that the relaxation kernel g satisfies (A2)–(A3) with a nonincreasing function ξ such that

- g^{'} (t) \geq ξ (t) g (t), t \geq 0,

and ξ fulfills the integrability conditions in (A3). Then there exist constants C > 0 and κ ∈ (0, 1), depending only on the structural parameters in (A1)–(A4), such that

E (t) \leq C \exp (- κ \int_{0}^{t} ξ (s) d s), t \geq 0 .

(4.6)

In particular:

if ξ(t) ≡ ξ₀ > 0, then $E (t) \leq C e^{- κ ξ_{0} t}$ (exponential decay);
if ξ(t) ∼ c(1 + t)⁻¹, then E(t) ≤ C(1 + t)^−κc (polynomial decay).

Proof of Theorem 4.3. The proof combines Proposition 4.2 with the kernel structure (A3) and the equivalence Lemma 4.1.

Step 1: Differential inequality for $V^{'}$ ⁠. By Proposition 4.2, for all $ε \in (0, ε_{0}]$ and a.e. t > 0,

V^{'} (t) \leq - \underset{D_{1} (t)}{\underset{︸}{\int_{Ω} σ (u_{t}) \cdot u_{t} d x}} - \underset{D_{2} (t)}{\underset{︸}{c_{*} g (t) ‖ ▿ u (t) ‖^{2}}} - \underset{{\tilde{D}}_{3} (t)}{\underset{︸}{c^{*} \int_{0}^{t} (- g^{'} (t - s)) ‖ ▿ u (t) - ▿ u (s) ‖^{2} d s}} .

Step 2: From ${\tilde{D}}_{3}$ to a ξ(t)–weighted memory dissipation. Using (A3) that −g′(r) ≥ ξ(r) g(r) with ξ nonincreasing, for each fixed t > 0,

{\tilde{D}}_{3} (t) \geq c^{*} ξ (t) \int_{0}^{t} g (t - s) ‖ ▿ u (t) - ▿ u (s) ‖^{2} d s = 2 c^{*} ξ (t) (g ◦ ▿ u) (t) ≕ D_{3} (t) .

Hence

V^{'} (t) \leq - D_{1} (t) - D_{2} (t) - D_{3} (t) with D_{3} (t) = 2 c^{*} ξ (t) (g ◦ ▿ u) (t) .

(4.7)

Step 3: Absorbing the elastic energy through $D_{2} (t)$ ⁠. By Lemma 2.1 there is K₂ > 0 such that a(u, u) ≤ K₂‖▿u‖². Fix an arbitrary T > 0. Since g is nonincreasing and g(0) > 0, we have g(t) ≥ g(T) > 0 for t ∈ [0, T]. Therefore,

D_{2} (t) = c_{*} g (t) ‖ ▿ u (t) ‖^{2} \geq c_{*} g (T) ‖ ▿ u (t) ‖^{2} \geq \frac{c_{*} g (T)}{K_{2}} a (u (t), u (t)) \forall t \in [0, T] .

(4.8)

Step 4: Splitting the memory dissipation. Fix θ ∈ (0, 1). Split $D_{3}$ as

D_{3} (t) = \underset{D_{3 a} (t)}{\underset{︸}{2 θ c^{*} ξ (t) (g ◦ ▿ u) (t)}} + \underset{D_{3 b} (t)}{\underset{︸}{2 (1 - θ) c^{*} ξ (t) (g ◦ ▿ u) (t)}} .

We keep $D_{3 a}$ to manufacture the variable rate ξ(t) in the final inequality, while $D_{3 b}$ remains an additional (helpful) negative term.

With (4.7) and (4.8), for t ∈ [0, T],

V^{'} (t) \leq - D_{1} (t) - \frac{c_{*} g (T)}{K_{2}} a (u (t), u (t)) - D_{3 a} (t) - D_{3 b} (t) .

(4.9)

Step 5: A time-interval Lyapunov functional. Define the time-interval functional

Φ (t) ≔ \int_{t}^{t + 1} V (s) d s, t \geq 0 .

Then $Φ^{'} (t) = V (t + 1) - V (t)$ ⁠. Integrating (4.9) over (t, t + 1) gives, for t ∈ [0, T − 1],

Φ^{'} (t) \leq - \int_{t}^{t + 1} \int_{Ω} σ (u_{s}) \cdot u_{s} d x d s - \frac{c_{*} g (T)}{K_{2}} \int_{t}^{t + 1} a (u (s), u (s)) d s - 2 θ c^{*} \int_{t}^{t + 1} ξ (s) (g ◦ ▿ u) (s) d s .

(4.10)

Since ξ is nonincreasing, for s ∈ [t, t + 1] ⊂ [0, T] we have ξ(s) ≥ ξ(t + 1) ≥ ξ(T); hence

\int_{t}^{t + 1} ξ (s) (g ◦ ▿ u) (s) d s \geq ξ (t + 1) \int_{t}^{t + 1} (g ◦ ▿ u) (s) d s .

(4.11)

Combining (4.10)–(4.11) yields

Φ^{'} (t) \leq - \int_{t}^{t + 1} \int_{Ω} σ (u_{s}) \cdot u_{s} d x d s - \frac{c_{*} g (T)}{K_{2}} \int_{t}^{t + 1} a (u, u) d s - 2 θ c^{*} ξ (t + 1) \int_{t}^{t + 1} (g ◦ ▿ u) (s) d s .

(4.12)

Step 6: Lower bounding Φ(t) by the averaged dissipations. By Lemma 4.1 (equivalence) there exist c₁, c₂ > 0 independent of t such that

c_{1} E (s) \leq V (s) \leq c_{2} E (s) for a.e. s \geq 0,

with $E (s) = \frac{1}{2} ‖ u_{s} (s) ‖^{2} + \frac{1}{2} a (u (s), u (s)) + (g ◦ ▿ u) (s)$ ⁠. Therefore

Φ (t) = \int_{t}^{t + 1} V (s) d s \leq c_{2} \int_{t}^{t + 1} (\frac{1}{2} ‖ u_{s} (s) ‖^{2} + \frac{1}{2} a (u, u) + (g ◦ ▿ u) (s)) d s .

(4.13)

To compare the kinetic part with its dissipation, we use the pointwise inequality (valid for all y ≥ 0):

y^{2} \leq y^{2} \log (1 + y^{2}) + 1 .

Integrating over Ω and then over (t, t + 1) gives

\int_{t}^{t + 1} ‖ u_{s} (s) ‖^{2} d s \leq \int_{t}^{t + 1} \int_{Ω} σ (u_{s}) \cdot u_{s} d x d s + | Ω | .

(4.14)

Using (4.13) and (4.14), we deduce

Φ (t) \leq \frac{c_{2}}{2} \int_{t}^{t + 1} \int_{Ω} σ (u_{s}) \cdot u_{s} d x d s + \frac{c_{2}}{2} \int_{t}^{t + 1} a (u, u) d s + c_{2} \int_{t}^{t + 1} (g ◦ ▿ u) (s) d s + \frac{c_{2}}{2} | Ω | .

(4.15)

Step 7: Closing the differential inequality for Φ. Multiply (4.15) by

η_{T} ≔ \min \{\frac{2}{c_{2}}, \frac{2 c_{*} g (T)}{c_{2} K_{2}}, \frac{2 θ c^{*} ξ (T)}{c_{2}}\},

and compare with (4.12). Using ξ(s) ≥ ξ(t + 1) for s ∈ [t, t + 1], we obtain

η_{T} Φ (t) \leq \int_{t}^{t + 1} \int_{Ω} σ (u_{s}) \cdot u_{s} d x d s + \frac{c_{*} g (T)}{K_{2}} \int_{t}^{t + 1} a (u, u) d s + 2 θ c^{*} \int_{t}^{t + 1} (g ◦ ▿ u) (s) d s + C_{T},

with $C_{T} ≔ η_{T} \frac{c_{2}}{2} | Ω |$ ⁠. Hence, by (4.12),

Φ^{'} (t) \leq - η_{T} Φ (t) + C_{T} (0 \leq t \leq T - 1) .

Moreover, using ℓ > 0 in (A2) and the pointwise bound in (A4), there exist constants C₁, C₂ > 0 (independent of t) such that for a.e. s,

E (s) \leq c_{1} \int_{Ω} σ (u_{s}) \cdot u_{s} d x + C_{2} (g ◦ ▿ u) (s) .

Averaging on (t, t + 1) and invoking c₁E ≤ V ≤ c₂E (Lemma 4.1) yields

\int_{t}^{t + 1} (g ◦ ▿ u) (s) d s \geq \frac{1}{C_{2}} \frac{1}{c_{2}} Φ (t) - \frac{C_{1}}{C_{2}} \int_{t}^{t + 1} \int_{Ω} σ (u_{s}) \cdot u_{s} d x d s - C_{0},

for some C₀ > 0 depending only on structural data. Plugging this in (4.12) (and absorbing a small multiple of ∫σ into its negative coefficient) gives the **variable-rate** inequality

Φ^{'} (t) \leq - κ ξ (t + 1) Φ (t) + \bar{C} ξ (t + 1) (0 \leq t \leq T - 1),

for some $κ, \bar{C} > 0$ depending only on (A1)–(A4). Applying Grönwall with variable rate and using that ξ is nonincreasing with $\int_{0}^{\infty} ξ = + \infty$ ⁠, we get

Φ (t) \leq (Φ (0) + \frac{\bar{C}}{κ}) \exp (- κ \int_{0}^{t} ξ (s) d s) (0 \leq t \leq T - 1) .

Time-interval constant rate. Fix $k \in N$ and set T ≔ k + 1. Since ξ is nonincreasing, for every t ∈ [k, k + 1] we have ξ(s) ≥ ξ(t + 1) ≥ ξ(k + 1) for all s ∈ [t, t + 1]. Define

λ_{k} ≔ η_{k + 1}, C_{k + 1} ≔ \frac{c_{2}}{2} | Ω | .

Then, for all t ∈ [k, k + 1], the differential inequality (4.12) together with (4.15) (and the definition of η_k+1 in Step 7) yields

Φ^{'} (t) \leq - λ_{k} Φ (t) + λ_{k} C_{k + 1} .

(4.16)

Solving (4.16) on the interval [k, k + 1] gives

Φ (t) \leq e^{- λ_{k} (t - k)} Φ (k) + (1 - e^{- λ_{k} (t - k)}) C_{k + 1}, t \in [k, k + 1],

and, in particular,

Φ (k + 1) \leq e^{- λ_{k}} Φ (k) + (1 - e^{- λ_{k}}) C_{k + 1} .

(4.17)

Step 8: From Φ back to E and time-interval iteration. Iterating (4.17) over k = 0, 1, …, m − 1 yields

Φ (m) \leq e^{- \sum_{j = 0}^{m - 1} λ_{j}} Φ (0) + \sum_{j = 0}^{m - 1} (1 - e^{- λ_{j}}) C_{j + 1} e^{- \sum_{i = j + 1}^{m - 1} λ_{i}} .

Using the equivalence $c_{1} E \leq V \leq c_{2} E$ and the monotonicity of E, we get

E (m + 1) \leq \frac{1}{c_{1}} Φ (m) \leq \frac{Φ (0)}{c_{1}} e^{- \sum_{j = 0}^{m - 1} λ_{j}} + \frac{1}{c_{1}} \sum_{j = 0}^{m - 1} (1 - e^{- λ_{j}}) C_{j + 1} e^{- \sum_{i = j + 1}^{m - 1} λ_{i}} .

By construction,

η_{k + 1} = \min \{\frac{2}{c_{2}}, \frac{2 c^{*} g (k + 1)}{c_{2} K_{2}}, \frac{2 θ c^{*}}{c_{2}} ξ (k + 1)\} \geq 0,

and we set λ_k≔η_k+1. Since ξ(k + 1) ↓ 0 while g(k + 1) and 1 are bounded, there exists k₀ such that, for all k ≥ k₀,

η_{k + 1} = \frac{2 θ c^{*}}{c_{2}} ξ (k + 1) .

Hence, for k ≥ k₀,

λ_{k} \geq κ ξ (k + 1), κ ≔ \frac{2 θ c^{*}}{c_{2}} \in (0,1) .

Consequently,

\sum_{j = 0}^{m - 1} λ_{j} \geq \sum_{j = k_{0}}^{m - 1} λ_{j} \geq κ \sum_{j = k_{0}}^{m - 1} ξ (j + 1) \geq κ \int_{k_{0}}^{m} ξ (s) d s .

Absorbing the finite window [0, k₀] and the geometric tail into the constants yields, for all t ≥ 0,

E (t) \leq c \exp (- κ \int_{0}^{t} ξ (s) d s) .

and choosing $θ = \frac{1}{2}$ yields the explicit value $κ = \frac{c^{*}}{c_{2}}$ ⁠. This completes the proof. □

Corollary 4.4 (Explicit decay rates). Under (A1)–(A4) and the structural inequality −g′ ≥ ξ g with nonincreasing ξ, there exists κ ∈ (0, 1) such that

E (t) \leq c \exp (- κ \int_{0}^{t} ξ (s) d s), t \geq 0 .

In particular:

If ξ(t) ≥ ξ₀ > 0, then $E (t) \leq C e^{- κ ξ_{0} t}$ ⁠.
If $ξ (t) = \frac{p}{1 + t}$ with p > 1, then E(t) ≤ C (1 + t)^−κp.

Moreover, one can take the explicit value $κ = \frac{c^{*}}{c_{2}}$ by choosing $θ = \frac{1}{2}$ , where $c^{*} = \frac{3}{8}$ and c₂ = max{1, 1 + ɛ₀ G_∞/K₁, 1 + ɛ₀} for any $ε_{0} \in (0, \min {\frac{1}{2}, \frac{K_{1}}{2 G_{\infty}}}]$ ⁠.

Remark: While Theorem 4.3 establishes the general form of the decay, this corollary provides explicit expressions for the decay rate constant κ in terms of the problem's parameters, which is valuable for quantitative analysis.

Remark 4.5 (Logarithmic damping with power θ > 0). Consider the generalized damping

σ_{θ} (v) ≔ v {(\log (1 + | v |^{2}))}^{θ}, v \in R^{n}, θ > 0 .

Then all the results proved above for the case θ = 1 remain valid for any θ > 0, with constants possibly depending on θ. In particular, assumptions (A4) hold for σ_θ. For completeness we collect the key properties and short proofs.

Convex potential and monotonicity. Define $Φ_{θ} : [0, \infty) \to [0, \infty)$ by

Φ_{θ} (s) ≔ \int_{0}^{s} {(\log (1 + τ))}^{θ} d τ, s \geq 0,

and set the convex potential

Ψ_{θ} (v) ≔ \frac{1}{2} Φ_{θ} (| v |^{2}), v \in R^{n} .

Then

▿_{v} Ψ_{θ} (v) = Φ_{θ}^{'} (| v |^{2}) v = {(\log (1 + | v |^{2}))}^{θ} v = σ_{θ} (v) .

Since $Φ_{θ}^{''} (s) = \frac{θ}{1 + s} {(\log (1 + s))}^{θ - 1} \geq 0$ ⁠, the map Φ_θ is convex, hence Ψ_θ is convex and σ_θ = ▿Ψ_θ is (maximal) monotone:

(σ_{θ} (v) - σ_{θ} (w)) \cdot (v - w) \geq 0, \forall v, w \in R^{n} .

Equivalently, the Jacobian matrix reads

D σ_{θ} (v) = {(\log (1 + | v |^{2}))}^{θ} I + θ {(\log (1 + | v |^{2}))}^{θ - 1} \frac{2}{1 + | v |^{2}} v \otimes v,

which is positive semidefinite for all v.

Dissipativity and pointwise lower bounds. For all $v \in R^{n}$ ⁠,

σ_{θ} (v) \cdot v = | v |^{2} {(\log (1 + | v |^{2}))}^{θ} \geq 0 .

Moreover, we have the following explicit bounds:

\begin{array}{l} (near the origin) & if | v | \leq 1, \log (1 + | v |^{2}) \geq \frac{| v |^{2}}{2} \Rightarrow σ_{θ} (v) \cdot v \geq 2^{- θ} | v |^{2 + 2 θ}; \\ (at infinity) & if | v | \geq \sqrt{e - 1}, \log (1 + | v |^{2}) \geq 1 \Rightarrow σ_{θ} (v) \cdot v \geq | v |^{2} . \end{array}

Consequently, there exists a universal c_θ > 0 such that

σ_{θ} (v) \cdot v \geq c_{θ} \min \{| v |^{2 + 2 θ}, | v |^{2}\}, \forall v \in R^{n} .

Local Lipschitz continuity. For any R > 0 there exists L_θ,R > 0 with

| σ_{θ} (v) - σ_{θ} (w) | \leq L_{θ, R} | v - w | whenever | v |, | w | \leq R .

Indeed, ‖Dσ_θ(v)‖ is bounded on {|v| ≤ R} and one can take the explicit bound

L_{θ, R} ≔ {(\log (1 + R^{2}))}^{θ} + 2 θ {(\log (1 + R^{2}))}^{θ - 1} \frac{R^{2}}{1 + R^{2}} .

Consequences for the PDE. Replacing σ by σ_θ in the weak formulation (3.1) and in the Galerkin scheme, all arguments used for existence, uniqueness, energy dissipation, and decay estimates remain valid. The proof steps that rely on monotonicity, dissipativity, and local Lipschitz continuity carry over verbatim by (1)–(3) above. In particular, the Lyapunov analysis of Section 4 yields the same differential inequality for $V$ and the same decay law

E (t) \leq c \exp (- \int_{0}^{t} ξ (s) d s),

so that exponential (resp. polynomial) decay occurs according to the kernel condition (A3). The exponent θ affects only the multiplicative constants (through local Lipschitz bounds and Young-type estimates), but not the qualitative type of decay, which is governed by g via ξ.

5. Conclusion

In this work, we analyzed a Lamé-type viscoelastic wave equation subject to a logarithmic damping term. The model differs from earlier contributions on Lamé systems where damping mechanisms were mainly polynomial, exponential, or incorporated as nonlinear sources. By introducing a logarithmic dissipation directly on the velocity, we highlighted a qualitatively different stabilization mechanism.

Through the Galerkin method and compactness arguments, we established the global well-posedness of weak solutions. The construction of a Lyapunov functional equivalent to the natural energy allowed us to derive a differential inequality, from which a general decay estimate was obtained. The resulting decay rates depend explicitly on the relaxation kernel: exponential when the kernel satisfies a uniform inequality, and polynomial otherwise.

Although the logarithmic damping is weaker than polynomial dissipation, our results demonstrate that it still guarantees stability in the presence of memory effects. This provides a complementary perspective to existing studies on viscoelastic systems, showing that logarithmic mechanisms can serve as effective alternatives in modeling energy decay.

Future work may focus on extending the analysis to stronger variants of logarithmic damping (e.g. powers of the logarithm), or to coupled systems with boundary feedback, in order to further clarify the interplay between different damping structures and viscoelastic memory.

The authors would like to express their sincere gratitude to Professor Salim A. Messaoudi for suggesting the initial idea of this work, which helped shape the direction of the study.

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Well-posedness and general decay for a Lamé viscoelastic system with logarithmic damping

1. Introduction

2. Preliminaries

2.1 Function spaces

2.2 Problem setting

2.3 Lamé operator and associated bilinear form

2.4 Phase space

2.5 Relaxation kernel and memory functional

2.6 Logarithmic damping: definition and lemmas

2.7 Energy functional and dissipation

3. Existence and uniqueness of solutions

3.1 Weak formulation

4. Energy decay

4.1 Lyapunov functional

5. Conclusion

References

Further reading

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Cited By

Well-posedness and general decay for a Lamé viscoelastic system with logarithmic damping

1. Introduction

2. Preliminaries

2.1 Function spaces

2.2 Problem setting

2.3 Lamé operator and associated bilinear form

2.4 Phase space

2.5 Relaxation kernel and memory functional

2.6 Logarithmic damping: definition and lemmas

2.7 Energy functional and dissipation

3. Existence and uniqueness of solutions

3.1 Weak formulation

4. Energy decay

4.1 Lyapunov functional

5. Conclusion

References

Further reading

Email Alerts

Suggested Reading

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