Well-posedness and decay property of regularity-loss type for the Cauchy problem of the standard linear solid model with Gurtin

Abstract

In this paper, we consider the Cauchy problem related to the standard linear solid model with Gurtin–Pipkin thermal law in the whole space. Under some assumptions on the relaxation function g, we establish the well-posedness result by using semigroup theory. Besides, by using the energy method in the Fourier space, we prove the decay estimate result under the non-critical case. Our result indicates that the decay property is of the regularity-loss type, which is in line with the decay property of Cattaneo system.

Keywords

Well-posedness the standard linear solid model Gurtin–Pipkin thermal law decay estimate

1. Introduction

In this paper, we study the following Cauchy problem related to the standard linear solid model with Gurtin–Pipkin thermal law $\begin{matrix} (1.1) & \{\begin{matrix} τ u_{t t t} + u_{t t} - a^{2} Δ u - a^{2} β Δ u_{t} + δ Δ θ = 0, & (x, t) \in R^{n} \times R^{+}, \\ θ_{t} - \frac{1}{k} \int_{0}^{\infty} g (s) Δ θ (t - s) d s - τ δ Δ u_{t t} - δ Δ u_{t} = 0, & (x, t) \in R^{n} \times R^{+}, \end{matrix} \end{matrix}$ with the initial data $\begin{matrix} (1.2) & \begin{array}{l} u (x, 0) = u_{0} (x), u_{t} (x, 0) = u_{1} (x), u_{t t} (x, 0) = u_{2} (x), \\ θ (x, t) |_{t ⩽ 0} = θ_{0} (x, t), x \in R^{n}, \end{array} \end{matrix}$ where u and θ denote the vibration of flexible structures and relative temperature respectively, $a > 0$ is the constant wave velocity, $θ_{0}$ is a prescribed past history of θ for $t ⩽ 0$ , $g (s)$ is the heat conductivity relaxation kernel, whose properties will be specified later and τ, β, δ, k are positive constants.

Here we consider a model of vibrations governed by the standard linear model of viscoelasticity. The model, which describes a linear spring is connected in series with a combination of another linear spring and a dashpot in parallel, has the following differential equation $\begin{matrix} σ + τ σ_{t} = E (e + β e_{t}), \end{matrix}$ where σ is the stress and e is the strain. Then the vibrations of flexible structures are governed by the linear differential equation $\begin{matrix} (1.3) & τ u_{t t t} + u_{t t} - a^{2} (Δ u + β Δ u_{t}) = 0 . \end{matrix}$ (1.3) can be seen as (1.1) without heat coupling, that is $δ = 0$ in (1.1). We refer the reader to [3] for details on description and derivation of equation (1.3). Meanwhile, (1.3) is known as Moore–Gibson–Thompson (MGT) equation. See [15–17,23,25] for more information about MGT equation.

In recent years, the standard linear solid model has become an active area of research. Gorain [13] introduced a distributed uncertain force as input disturbance and took into account an internal material damping of the structure into (1.1). The author obtained uniform exponential stabilization and an explicit form for the energy decay rate when the input disturbances are insignificant. Alves et al. [1] considered vibrations modeled by the standard linear solid model of viscoelasticity which are coupled to a heat equation governed by Fourier law of heat conduction $\begin{matrix} (1.4) & \{\begin{matrix} τ u_{t t t} + u_{t t} - a^{2} Δ u - a^{2} β Δ u_{t} + δ Δ θ = 0, & (x, t) \in Ω \times R^{+}, \\ θ_{t} - Δ θ - τ η Δ u_{t t} - δ Δ u_{t} = 0, & (x, t) \in Ω \times R^{+}, \end{matrix} \end{matrix}$ where Ω is a bounded open connected set in $R^{n}$ having a smooth boundary $\partial Ω$ . They showed the exponential stability by using multiplier techniques. Pellicer and Said-Houari [29] investigated the standard linear solid model in $R^{n}$ , that is $\begin{matrix} (1.5) & τ u_{t t t} + u_{t t} - a^{2} Δ u - a^{2} β Δ u_{t} = 0, \end{matrix}$ where $(x, t) \in R^{n} \times R^{+}$ . For $0 < τ < β$ , the authors obtained well-posedness and the optimal decay rate by using the energy method in the Fourier space. For other related problems, see [2,4–6,9,11,17,21,22,24,30,31] and the references therein.

With respect to a system coupled with Gurtin–Pipkin thermal law, a lot of interesting results have been established. Khader and Said-Houari [18] considered the Cauchy problem for the one-dimensional Timoshenko system coupled with Gurtin–Pipkin thermal law $\begin{matrix} (1.6) & \{\begin{matrix} φ_{t t} - {(φ_{x} - ψ)}_{x} = 0, \\ ψ_{t t} - a^{2} ψ_{x x} - (φ_{x} - ψ) + δ θ_{x} = 0, \\ θ_{t} - \frac{1}{β} \int_{0}^{\infty} g (s) θ_{x x} (t - s) d s + δ ψ_{t x} = 0, \end{matrix} \end{matrix}$ where a, δ and β are positive constants. They showed that the number $α_{g}$ also plays a role in an unbounded domain like in a bounded domain (see [10]), where $\begin{matrix} α_{g} = (\frac{β}{g (0)} - 1) (1 - a^{2}) - δ^{2} \frac{β}{g (0)} \end{matrix}$ has been discovered in [10]. Recently, they [20] got an optimal decay rate of the $L^{2}$ -norm of the solution. In [19], Khader and Said-Houari studied the Cauchy problem for the one-dimensional Bresse system coupled with Gurtin–Pipkin thermal law. They obtained the decay properties of the solution by using the energy method in the Fourier space. In addition, they showed that in the absence of the frictional damping, the memory damping term is not strong enough to produce a decay rate for the solution. For other related results, we refer the reader to [8,26,27,32].

Based on the above results, we are interested in whether the decay property of problem (1.1)–(1.2) is the regularity-loss type or not. For our purpose, we first use the semigroup theory to prove the well-posedness, and then use the energy method in the Fourier space to get the decay result under the non-critical case $0 < τ < β$ . Our result indicates that the decay property problem (1.1)–(1.2) is of the regularity-loss type, which is in line with the decay property of Cattaneo system in [30].

The paper is organized as follows. In Section 2, we give some materials needed for our work and state our main results. In Section 3, we prove the well-posedness of the problem. The decay estimate result is established in Section 4.

2. Preliminaries and main results

In this section, we first introduce some notations and present our hypotheses. Then we give our main results.

Without loss of generality, we take $a = 1$ . To deal with the memory, we introduce the following new variable (see [7,12]) $\begin{matrix} (2.1) & η (x, t, s) = \int_{0}^{s} θ (x, t - σ) d σ = \int_{t - s}^{t} θ (x, σ) d σ, (x, t, s) \in R^{n} \times [0, + \infty) \times R^{+}, \end{matrix}$ which satisfies $\begin{matrix} η_{t} = - η_{s} + θ, (x, t, s) \in R^{n} \times R^{+} \times R^{+}, \end{matrix}$ and the conditions $\begin{matrix} lim_{s \to 0} η (x, t, s) = 0, (x, t) \in R^{n} \times [0, + \infty), \end{matrix}$ and $\begin{matrix} η (x, 0, s) = η_{0} (s) = \int_{0}^{s} θ_{0} (x, σ) d σ, (x, s) \in R^{n} \times R^{+} . \end{matrix}$ Assume $g (\infty) = 0$ , by integration by parts, we have $\begin{matrix} \int_{0}^{\infty} g (s) Δ θ (t - s) d s = - \int_{0}^{\infty} g^{'} (s) Δ η (s) d s . \end{matrix}$ Denoting $μ (s) = - g^{'} (s)$ , we obtain $\begin{matrix} \int_{0}^{\infty} g (s) Δ θ (t - s) d s = \int_{0}^{\infty} μ (s) Δ η (s) d s . \end{matrix}$ Let the linear operator T defined as $T η = - η_{s}$ . Then, system (1.1)–(1.2) is equivalent to the following: $\begin{matrix} (2.2) & \{\begin{matrix} τ u_{t t t} + u_{t t} - Δ u - β Δ u_{t} + δ Δ θ = 0, & (x, t) \in R^{n} \times R^{+}, \\ θ_{t} - \frac{1}{k} \int_{0}^{\infty} μ (s) Δ η (s) d s - τ δ Δ u_{t t} - δ Δ u_{t} = 0, & (x, t) \in R^{n} \times R^{+}, \\ η_{t} = T η + θ, & (x, t, s) \in R^{n} \times R^{+} \times R^{+}, \end{matrix} \end{matrix}$ with the conditions $\begin{matrix} (2.3) & \{\begin{matrix} (u, u_{t}, u_{t t}, θ) (x, 0) = (u_{0}, u_{1}, u_{2}, θ_{0}) (x), & x \in R^{n}, \\ {lim}_{s \to 0} η (x, t, s) = 0, & (x, t) \in R^{n} \times [0, + \infty), \\ η_{0} (x, s) = \int_{0}^{s} θ_{0} (x, σ) d σ, & (x, s) \in R^{n} \times R^{+} . \end{matrix} \end{matrix}$ For the memory kernel g, we use the following assumptions in [12]:

$g^{'}$ is an absolutely continuous function on $R^{+}$ so that $\begin{matrix} g^{'} (s) ⩽ 0, g^{″} (s) ⩾ 0, g^{'} (0) = lim_{s \to 0} g^{'} (s) \in (- \infty, 0) . \end{matrix}$

There exists $ν > 0$ such that the differential inequality $\begin{matrix} g^{″} (s) + ν g^{'} (s) ⩾ 0 \end{matrix}$ holds for almost every $s > 0$ .

Remark 2.1.
In particular, μ is summable on $R^{+}$ with $\begin{matrix} \int_{0}^{\infty} μ (s) d s = g (0) . \end{matrix}$
Let $v = u_{t}$ , $w = u_{t t}$ , then system (2.2)–(2.3) can be written as $\begin{matrix} (2.4) & \{\begin{matrix} \frac{d}{d t} W (t) = A W (t), t \in [0, + \infty), \\ W (0) = W_{0} = {(u_{0}, u_{1}, u_{2}, θ_{0}, η_{0})}^{T}, \end{matrix} \end{matrix}$ where $W = {(u, v, w, θ, η)}^{T}$ and $A : D (A) \subset H \to H$ is a linear operator defined by $\begin{matrix} A W = (\begin{matrix} v \\ w \\ \frac{1}{τ} (Δ u + β Δ v - δ Δ θ - w) \\ \frac{1}{k} \int_{0}^{\infty} μ (s) Δ η (s) d s + τ δ Δ w + δ Δ v \\ T η + θ \end{matrix}) . \end{matrix}$ We give the following space $\begin{matrix} M = L_{μ}^{2} (R^{+}, H^{1} (R^{n})) = {η : R^{+} \to H^{1} (R^{n}) | \int_{0}^{\infty} μ (s) {‖ η (s) ‖}_{H^{1} (R^{n})}^{2} d s < \infty} \end{matrix}$ equipped with the norm $\begin{matrix} ‖ φ ‖_{M}^{2} = \int_{0}^{\infty} μ (s) ‖ φ ‖_{H^{1} (R^{n})}^{2} d s \end{matrix}$ and inner product $\begin{matrix} {⟨ φ, ψ ⟩}_{M} = \int_{0}^{\infty} μ (s) \int_{R^{n}} φ \bar{ψ} d x d s + \int_{0}^{\infty} μ (s) \int_{R^{n}} \nabla φ \cdot \nabla \bar{ψ} d x d s, \end{matrix}$ where $\bar{ψ}$ is the complex conjugate of ψ. And we define the following energy space $\begin{matrix} H = H^{1} (R^{n}) \times H^{1} (R^{n}) \times L^{2} (R^{n}) \times L^{2} (R^{n}) \times M \end{matrix}$ equipped with the norm $\begin{array}{l} {‖ (u, v, w, θ, η) ‖}_{H}^{2} \\ = τ (β - τ) ‖ \nabla v ‖_{L^{2} (R^{n})}^{2} + {‖ \nabla (u + τ v) ‖}_{L^{2} (R^{n})}^{2} + ‖ v + τ w ‖_{L^{2} (R^{n})}^{2} \\ + ‖ u + τ v ‖_{L^{2} (R^{n})}^{2} + ‖ v ‖_{L^{2} (R^{n})}^{2} + ‖ θ ‖_{L^{2} (R^{n})}^{2} + \frac{1}{k} ‖ η ‖_{M}^{2} \end{array}$ and inner product $\begin{array}{l} {⟨ (u, v, w, θ, η), (u_{1}, v_{1}, w_{1}, θ_{1}, η_{1}) ⟩}_{H} \\ = τ (β - τ) \int_{R^{n}} \nabla v \cdot \nabla {\bar{v}}_{1} d x + \int_{R^{n}} \nabla (u + τ v) \cdot \nabla ({\bar{u}}_{1} + τ {\bar{v}}_{1}) d x + \int_{R^{n}} (v + τ w) ({\bar{v}}_{1} + τ {\bar{w}}_{1}) d x \\ + \int_{R^{n}} (u + τ v) ({\bar{u}}_{1} + τ {\bar{v}}_{1}) d x + \int_{R^{n}} v {\bar{v}}_{1} d x + \int_{R^{n}} θ {\bar{θ}}_{1} d x + \frac{1}{k} \int_{0}^{\infty} μ (s) \int_{R^{n}} \nabla η \cdot \nabla {\bar{η}}_{1} d x d s \\ + \frac{1}{k} \int_{0}^{\infty} μ (s) \int_{R^{n}} η {\bar{η}}_{1} d x d s . \end{array}$
Remark 2.2.
We notice that the definition of energy space here is different from that of the bounded domain case in [1] despite the fact that there’s a thermal term here. Therefore the definition of the norm needs some extra terms to be consistent with the definition of energy space. Since Poincaré inequality is no longer valid in our problem, a combination of the extra terms of the norm and perturbed problem considered later can dedicate to our problem. When we pay attention to the space of $M$ , which is different from the definition of space in some references about Gurtin–Pipkin thermal law (e.g. [10,33]), it is also based on this reason.

The domain of $A$ is given by $\begin{array}{l} D (A) & = {W \in H | w, θ \in H^{1} (R^{n}), η \in D (T), u + β v - δ θ \in H^{2} (R^{n}), \\ \frac{1}{k} \int_{0}^{\infty} μ (s) η (s) d s + τ δ w + δ v \in H^{2} (R^{n})}, \end{array}$ where $D (T) = {η \in M | η_{s} \in M, {lim}_{s \to 0} η (x, t, s) = 0}$ . Clearly, $D (A)$ is dense in $H$ .

In what follows, we state the well-posedness result of problem (2.4).
Theorem 2.1.
Suppose that $0 < τ < β$ . Let $W_{0} \in H$ , then problem ( 2.4 ) has a unique weak solution $W \in C ([0, \infty); H)$ . Moreover, if $W_{0} \in D (A)$ , then $\begin{matrix} W \in C ([0, \infty); D (A)) \cap C^{1} ([0, \infty); H) . \end{matrix}$

Define $\begin{matrix} V = {(u_{t} + τ u_{t t}, \nabla u_{t}, \nabla (u + τ u_{t}), θ, \nabla η)}^{T}, \end{matrix}$ where $(u (x, t), θ (x, t))$ is the solution of (2.2)–(2.3).

Our decay result reads as follows.
Theorem 2.2.
Suppose that $0 < τ < β$ . Let $V_{0} = V (x, 0) \in H^{s} (R^{n}) \cap L^{1} (R^{n})$ , where s is nonnegative integer, then for all $0 ⩽ k + l ⩽ s$ , V satisfies the following decay estimate $\begin{matrix} {‖ \nabla^{k} V (t) ‖}_{L^{2} (R^{n})}^{2} ⩽ C {(1 + t)}^{- \frac{n}{6} - \frac{k}{3}} ‖ V_{0} ‖_{L^{1} (R^{n})}^{2} + C {(1 + t)}^{- \frac{l}{3}} {‖ \nabla^{k + l} V_{0} ‖}_{L^{2} (R^{n})}^{2} . \end{matrix}$

In this paper, we denote the Fourier transform $\hat{f} (ξ)$ of the function $f (x)$ by $\begin{matrix} F [f] (ξ) = \hat{f} (ξ) = \int_{R^{n}} e^{- i x \cdot ξ} f (x) d x, \end{matrix}$ where $x \cdot ξ = \begin{matrix} \sum_{j = 1}^{n} x_{j} ξ_{j} \end{matrix}$ for $x = (x_{1}, \dots, x_{n})$ and $ξ = (ξ_{1}, \dots, ξ_{n})$ . And we use C and c to denote generic positive constants, the values of which may vary from one place to another.
3. Well-posedness

In this section, we prove the well-posedness of problem (2.4) by using the Lumer–Phillips theorem.

Proof of Theorem 2.1.
Since the domain here is $R^{n}$ , Poincaré inequality is no longer useful. Inspired by [29], we need to consider the following perturbed problem $\begin{matrix} (3.1) & \{\begin{matrix} \frac{d}{d t} W (t) = A_{p} W (t), t \in [0, + \infty), \\ W (0) = W_{0}, \end{matrix} \end{matrix}$ where $\begin{matrix} A_{p} (\begin{matrix} u \\ v \\ w \\ θ \\ η \end{matrix}) = (\begin{matrix} v \\ w \\ \frac{1}{τ} (Δ u + β Δ v - δ Δ θ - w) - \frac{1}{τ} u - v - \frac{1}{τ^{2}} v \\ \frac{1}{k} \int_{0}^{\infty} μ (s) Δ η (s) d s + τ δ Δ w + δ Δ v - \frac{1}{k} \int_{0}^{\infty} μ (s) η (s) d s \\ T η + θ \end{matrix}) . \end{matrix}$

First, we prove that the operator $A_{p}$ is dissipative. $\begin{array}{l} {⟨ A_{p} W, W ⟩}_{H} \\ = τ (β - τ) \int_{R^{n}} \nabla w \cdot \nabla \bar{v} d x + \int_{R^{n}} \nabla (v + τ w) \cdot \nabla (\bar{u} + τ \bar{v}) d x \\ + \int_{R^{n}} (w + Δ u + β Δ v - δ Δ θ - w - u - τ v - \frac{1}{τ} v) (\bar{v} + τ \bar{w}) d x \\ + \int_{R^{n}} (v + τ w) (\bar{u} + τ \bar{v}) d x + \int_{R^{n}} w \bar{v} d x \\ + \int_{R^{n}} (\frac{1}{k} \int_{0}^{\infty} μ (s) Δ η (s) d s + τ δ Δ w + δ Δ v - \frac{1}{k} \int_{0}^{\infty} μ (s) η (s) d s) \bar{θ} d x \\ + \frac{1}{k} \int_{0}^{\infty} μ (s) \int_{R^{n}} (- \nabla η_{s} + \nabla θ) \cdot \nabla \bar{η} d x d s + \frac{1}{k} \int_{0}^{\infty} μ (s) \int_{R^{n}} (- η_{s} + θ) \bar{η} d x d s \\ = \int_{R^{n}} (β τ \nabla w \cdot \nabla \bar{v} + \nabla v \cdot \nabla \bar{u} + τ \nabla v \cdot \nabla \bar{v} + τ \nabla w \cdot \nabla \bar{u}) d x \\ + \int_{R^{n}} (- \nabla u \cdot \nabla \bar{v} - τ \nabla u \cdot \nabla \bar{w} - β \nabla v \cdot \nabla \bar{v} - β τ \nabla v \cdot \nabla \bar{w} + δ \nabla θ \cdot \nabla \bar{v} + δ τ \nabla θ \cdot \nabla \bar{w} \\ - u \bar{v} - τ u \bar{w} - τ^{2} v \bar{w} - \frac{1}{τ} v \bar{v} - v \bar{w}) d x - \frac{1}{k} \int_{0}^{\infty} μ (s) \int_{R^{n}} η \bar{θ} d x d s \\ - \frac{1}{k} \int_{0}^{\infty} μ (s) \int_{R^{n}} \nabla η \cdot \nabla \bar{θ} d x d s + \frac{1}{k} \int_{0}^{\infty} μ (s) \int_{R^{n}} \nabla θ \cdot \nabla \bar{η} d x d s \\ + \int_{R^{n}} (v \bar{u} + τ w \bar{u} + τ^{2} w \bar{v}) d x + \int_{R^{n}} (w \bar{v} - τ δ \nabla w \cdot \nabla \bar{θ} - δ \nabla v \cdot \nabla \bar{θ}) d x \\ - \frac{1}{k} \int_{0}^{\infty} μ (s) \int_{R^{n}} \nabla η_{s} \cdot \nabla \bar{η} d x d s + \frac{1}{k} \int_{0}^{\infty} μ (s) \int_{R^{n}} θ \bar{η} d x d s \\ (3.2) & - \frac{1}{k} \int_{0}^{\infty} μ (s) \int_{R^{n}} η_{s} \bar{η} d x d s . \end{array}$ From [14], it follows from integration by parts and a limit argument that $\begin{matrix} Re (- \frac{1}{k} \int_{0}^{\infty} μ (s) \int_{R^{n}} \nabla η_{s} (s) \cdot \nabla \bar{η} (s) d x d s) = \frac{1}{2 k} \int_{0}^{\infty} μ^{'} (s) \int_{R^{n}} {| \nabla η (s) |}^{2} d x d s \end{matrix}$ and $\begin{matrix} Re (- \frac{1}{k} \int_{0}^{\infty} μ (s) \int_{R^{n}} η_{s} (s) \bar{η} (s) d x d s) = \frac{1}{2 k} \int_{0}^{\infty} μ^{'} (s) \int_{R^{n}} {| η (s) |}^{2} d x d s . \end{matrix}$ So taking the real part of (3.2), we have $\begin{array}{l} Re ({⟨ A_{p} W, W ⟩}_{H}) \\ = - (β - τ) \int_{R^{n}} | \nabla v |^{2} d x - \frac{1}{τ} \int_{R^{n}} | v |^{2} d x + \frac{1}{2 k} \int_{0}^{\infty} μ^{'} (s) \int_{R^{n}} | \nabla η |^{2} d x d s \\ + \frac{1}{2 k} \int_{0}^{\infty} μ^{'} (s) \int_{R^{n}} | η |^{2} d x d s . \end{array}$ Since $0 < τ < β$ and (H1), $Re ({⟨ A_{p} W, W ⟩}_{H}) ⩽ 0$ . Hence, the operator $A_{p}$ is dissipative.

Next, we want to prove $I - A_{p}$ is surjective. Given $F = {(f_{1}, f_{2}, f_{3}, f_{4}, f_{5})}^{T} \in H$ , we prove that there exists a unique $W = {(u, v, w, θ, η)}^{T} \in D (A)$ such that $\begin{matrix} (3.3) & (I - A_{p}) W = F, \end{matrix}$ that is, $\begin{matrix} (3.4) & \{\begin{matrix} u - v = f_{1} \in H^{1} (R^{n}), \\ v - w = f_{2} \in H^{1} (R^{n}), \\ w - \frac{1}{τ} (Δ u + β Δ v - δ Δ θ - w) + \frac{1}{τ} u + v + \frac{1}{τ^{2}} v = f_{3} \in L^{2} (R^{n}), \\ θ - \frac{1}{k} \int_{0}^{\infty} μ (s) Δ η (s) d s - τ δ Δ w - δ Δ v + \frac{1}{k} \int_{0}^{\infty} μ (s) η (s) d s = f_{4} \in L^{2} (R^{n}), \\ η + η_{s} - θ = f_{5} \in M . \end{matrix} \end{matrix}$ From ${(3.4)}_{1}$ , ${(3.4)}_{2}$ , ${(3.4)}_{5}$ and ${lim}_{s \to 0} η (x, t, s) = 0$ , we obtain $\begin{matrix} (3.5) & \{\begin{matrix} v = u - f_{1}, \\ w = v - f_{2} = u - f_{1} - f_{2}, \\ η = (1 - e^{- s}) θ + \int_{0}^{s} e^{τ - s} f_{5} (τ) d τ . \end{matrix} \end{matrix}$ Inserting (3.5) into ${(3.4)}_{3}$ and ${(3.4)}_{4}$ , we get $\begin{matrix} (3.6) & \{\begin{matrix} (τ + 2 + \frac{τ^{2} + 1}{τ}) u - (1 + β) Δ u + δ Δ θ = (τ + 1 + \frac{τ^{2} + 1}{τ}) f_{1} + (τ + 1) f_{2} + τ f_{3} - β Δ f_{1}, \\ θ - \frac{1}{k} \int_{0}^{\infty} μ (s) (1 - e^{- s}) Δ θ d s - (τ + 1) δ Δ u + \frac{1}{k} \int_{0}^{\infty} μ (s) (1 - e^{- s}) θ d s \\ = - (τ + 1) δ Δ f_{1} - τ δ Δ f_{2} + f_{4} + \frac{1}{k} \int_{0}^{\infty} μ (s) \int_{0}^{s} e^{τ - s} Δ f_{5} (τ) d τ d s \\ + \frac{1}{k} \int_{0}^{\infty} μ (s) \int_{0}^{s} e^{τ - s} f_{5} (τ) d τ d s . \end{matrix} \end{matrix}$ In order to solve (3.6), we consider the following variational formulation $\begin{matrix} (3.7) & B ((u, θ), (p_{1}, p_{2})) = G (p_{1}, p_{2}), \end{matrix}$ where $B : (H^{1} (R^{n}) \times H^{1} (R^{n})) \times (H^{1} (R^{n}) \times H^{1} (R^{n})) \to C$ is the bilinear form defined by $\begin{array}{l} B ((u, θ), (p_{1}, p_{2})) \\ = (τ + 1) (τ + 2 + \frac{τ^{2} + 1}{τ}) \int_{R^{n}} u {\bar{p}}_{1} d x + (τ + 1) (1 + β) \int_{R^{n}} \nabla u \cdot \nabla {\bar{p}}_{1} d x \\ - (τ + 1) δ \int_{R^{n}} \nabla θ \cdot \nabla {\bar{p}}_{1} d x + \int_{R^{n}} θ {\bar{p}}_{2} d x + \frac{1}{k} \int_{R^{n}} \int_{0}^{\infty} μ (s) (1 - e^{- s}) d s \nabla θ \cdot \nabla {\bar{p}}_{2} d x \\ + (τ + 1) δ \int_{R^{n}} \nabla u \cdot \nabla {\bar{p}}_{2} d x + \frac{1}{k} \int_{R^{n}} \int_{0}^{\infty} μ (s) (1 - e^{- s}) d s θ {\bar{p}}_{2} d x \end{array}$ and $G : H^{1} (R^{n}) \times H^{1} (R^{n}) \to C$ is the linear form given by $\begin{array}{l} G (p_{1}, p_{2}) \\ = (τ + 1) (τ + 1 + \frac{τ^{2} + 1}{τ}) \int_{R^{n}} f_{1} {\bar{p}}_{1} d x + {(τ + 1)}^{2} \int_{R^{n}} f_{2} {\bar{p}}_{1} d x + τ (τ + 1) \int_{R^{n}} f_{3} {\bar{p}}_{1} d x \\ + β (τ + 1) \int_{R^{n}} \nabla f_{1} \cdot \nabla {\bar{p}}_{1} d x + (τ + 1) δ \int_{R^{n}} \nabla f_{1} \cdot \nabla {\bar{p}}_{2} d x + τ δ \int_{R^{n}} \nabla f_{2} \cdot \nabla {\bar{p}}_{2} d x \\ + \int_{R^{n}} f_{4} {\bar{p}}_{2} d x + \frac{1}{k} \int_{R^{n}} {\bar{p}}_{2} \int_{0}^{\infty} μ (s) \int_{0}^{s} e^{τ - s} Δ f_{5} (τ) d τ d s d x \\ + \frac{1}{k} \int_{R^{n}} {\bar{p}}_{2} \int_{0}^{\infty} μ (s) \int_{0}^{s} e^{τ - s} f_{5} (τ) d τ d s d x . \end{array}$ Using the definition of B, we have $\begin{array}{l} Re (B ((u, θ), (u, θ))) \\ = (τ + 1) (τ + 2 + \frac{τ^{2} + 1}{τ}) \int_{R^{n}} u^{2} d x + (τ + 1) (1 + β) \int_{R^{n}} | \nabla u |^{2} d x + \int_{R^{n}} θ^{2} d x \\ + \frac{1}{k} (g (0) - \int_{0}^{\infty} e^{- s} μ (s) d s) \int_{R^{n}} | \nabla θ |^{2} d x + \frac{1}{k} (g (0) - \int_{0}^{\infty} e^{- s} μ (s) d s) \int_{R^{n}} | θ |^{2} d x \\ ⩾ C (‖ u ‖_{H^{1} (R^{n})}^{2} + ‖ θ ‖_{H^{1} (R^{n})}^{2}), for some C > 0 . \end{array}$ Hence, B is coercive. Furthermore, it’s easy to get that B and G are bounded by using Hölder inequality.

As a consequence, by applying Lax–Milgram Lemma, we obtain that (3.6) has a unique solution $(u, θ) \in H^{1} (R^{n}) \times H^{1} (R^{n})$ . Substituting u and θ into ${(3.5)}_{1}$ , ${(3.5)}_{2}$ , we get $\begin{matrix} (v, w) \in H^{1} (R^{n}) \times H^{1} (R^{n}) . \end{matrix}$ Using ${(3.5)}_{3}$ and the method in [33, Proposition 2.2], we get $\begin{array}{l} \int_{0}^{\infty} μ (s) {‖ η (s) ‖}_{H^{1} (R^{n})}^{2} d s \\ ⩽ 2 \int_{0}^{\infty} μ (s) {‖ (1 - e^{- s}) θ ‖}_{H^{1} (R^{n})}^{2} d s + 2 \int_{0}^{\infty} μ (s) {‖ \int_{0}^{s} e^{τ - s} f_{5} (τ) d τ ‖}_{H^{1} (R^{n})}^{2} d s \\ = 2 \int_{0}^{\infty} (1 - e^{- s}) μ (s) d s ‖ θ ‖_{H^{1} (R^{n})}^{2} + 2 {‖ \int_{0}^{s} e^{τ - s} f_{5} (τ) d τ ‖}_{M}^{2} \\ ⩽ 2 g (0) ‖ θ ‖_{H^{1} (R^{n})}^{2} + 2 ‖ f_{5} ‖_{M}^{2}, \end{array}$ which implies that $η \in M$ . Then by recalling ${(3.4)}_{5}$ , we arrive at $\begin{matrix} η_{s} = θ - η + f_{5} \in M . \end{matrix}$ Therefore, $η \in D (T)$ . If $p_{2} = 0 \in H^{1} (R^{n})$ , then (3.7) yields that $\begin{array}{l} \int_{R^{n}} \nabla (u + β v - δ θ) \cdot \nabla p_{1} d x = \int_{R^{n}} [τ f_{3} - (1 + τ) w - u - τ v - \frac{1}{τ} v] p_{1} d x \end{array}$ for all $p_{1} \in H^{1} (R^{n})$ , which gives us $\begin{matrix} - Δ (u + β v - δ θ) = τ f_{3} - (1 + τ) w - u - τ v - \frac{1}{τ} v \in L^{2} (R^{n}) . \end{matrix}$ Therefore, we obtain $\begin{matrix} u + β v - δ θ \in H^{2} (R^{n}) . \end{matrix}$ Similarly, we get $\begin{matrix} \frac{1}{k} \int_{0}^{\infty} μ (s) η (s) d s + τ δ w + δ v \in H^{2} (R^{n}) . \end{matrix}$ Hence, there exists a unique $W \in D (A)$ such that (3.3) is satisfied. So we conclude that the operator $I - A_{p}$ is surjective.

Consequently, $A_{p}$ is a maximal monotone operator. Therefore, $A_{p}$ is the generator of a $C_{0}$ -semigroup of contractions on $H$ , which follows from the Lumer–Phillips theorem (see [28]). Since $A_{p}$ is a bounded perturbation of $A$ , $A$ is the generator of a $C_{0}$ -semigroup on $H$ . □

4. Decay estimate

In this section, we consider the decay result of the norm related to (2.2) for the non-critical case $0 < τ < β$ .

Taking the Fourier transform of system (2.2)–(2.3), we obtain $\begin{matrix} (4.1) & \{\begin{matrix} τ {\hat{u}}_{t t t} + u_{t t} + ξ^{2} \hat{u} + β ξ^{2} {\hat{u}}_{t} - δ ξ^{2} \hat{θ} = 0, \\ {\hat{θ}}_{t} + \frac{ξ^{2}}{k} \int_{0}^{\infty} μ (s) \hat{η} (s) d s + τ δ ξ^{2} {\hat{u}}_{t t} + δ ξ^{2} {\hat{u}}_{t} = 0, \\ {\hat{η}}_{t} + {\hat{η}}_{s} = \hat{θ}, \end{matrix} \end{matrix}$ with the initial data $\begin{matrix} (4.2) & \{\begin{matrix} (\hat{u}, {\hat{u}}_{t}, {\hat{u}}_{t t}, \hat{θ}) (ξ, 0) = ({\hat{u}}_{0}, {\hat{u}}_{1}, {\hat{u}}_{2}, {\hat{θ}}_{0}) (ξ), \\ {\hat{η}}_{0} (ξ, s) = \int_{0}^{s} {\hat{θ}}_{0} (ξ, σ) d σ, \end{matrix} \end{matrix}$ where $ξ \in R^{n}$ . As in [29], we introduce the following new variables $\begin{matrix} (4.3) & \hat{v} = {\hat{u}}_{t}, \hat{w} = {\hat{u}}_{t t} . \end{matrix}$ Thus, (4.1) can be written as $\begin{matrix} (4.4) & \{\begin{matrix} {\hat{u}}_{t} - \hat{v} = 0, \\ {\hat{v}}_{t} - \hat{w} = 0, \\ τ {\hat{w}}_{t} + \hat{w} + ξ^{2} \hat{u} + β ξ^{2} \hat{v} - δ ξ^{2} \hat{θ} = 0, \\ {\hat{θ}}_{t} + \frac{ξ^{2}}{k} \int_{0}^{\infty} μ (s) \hat{η} (s) d s + τ δ ξ^{2} \hat{w} + δ ξ^{2} \hat{v} = 0, \\ {\hat{η}}_{t} + {\hat{η}}_{s} = \hat{θ} . \end{matrix} \end{matrix}$

Before proving Theorem 2.2, we need the following pointwise estimate result:

Proposition 4.1 (Pointwise estimate).

Suppose that $0 < τ < β$ . Then $\begin{matrix} (4.5) & {| \hat{V} (ξ, t) |}^{2} ⩽ C e^{- c ρ_{1} (ξ) t} {| {\hat{V}}_{0} (ξ) |}^{2}, \end{matrix}$ for any $t ⩾ 0$ , where $ρ_{1} (ξ) : = \frac{ξ^{6}}{{(1 + ξ^{2})}^{6}}$ .

In order to prove Proposition 4.1, we first introduce the following energy functional of system (4.4) $\begin{matrix} (4.6) & \hat{E} (ξ, t) : = | \hat{v} + τ \hat{w} |^{2} + τ (β - τ) ξ^{2} | \hat{v} |^{2} + ξ^{2} | \hat{u} + τ \hat{v} |^{2} + | \hat{θ} |^{2} + \frac{ξ^{2}}{k} \int_{0}^{\infty} μ (s) {| \hat{η} (ξ, t, s) |}^{2} d s . \end{matrix}$

Lemma 4.1.

Let $(\hat{u}, \hat{v}, \hat{w}, \hat{θ}, \hat{η})$ be the solution of ( 4.4 ), then $\hat{E} (ξ, t)$ satisfies $\begin{matrix} (4.7) & \frac{d}{d t} \hat{E} (ξ, t) = - (β - τ) ξ^{2} | \hat{v} |^{2} + \frac{ξ^{2}}{2 k} \int_{0}^{\infty} μ^{'} (s) {| \hat{η} (ξ, t, s) |}^{2} d s . \end{matrix}$

Proof.

Adding ${(4.4)}_{2}$ and ${(4.4)}_{3}$ together, we have $\begin{matrix} (4.8) & {(\hat{v} + τ \hat{w})}_{t} = - ξ^{2} \hat{u} - β ξ^{2} \hat{v} + δ ξ^{2} \hat{θ} . \end{matrix}$ Multiplying (4.8) by $({\hat{v}}^{*} + τ {\hat{w}}^{*})$ and taking the real part, we get $\begin{array}{l} \frac{1}{2} \frac{d}{d t} | \hat{v} + τ \hat{w} |^{2} & = - ξ^{2} Re (\hat{u} {\hat{v}}^{*}) - τ ξ^{2} Re (\hat{u} {\hat{w}}^{*}) - β ξ^{2} | \hat{v} |^{2} \\ (4.9) & - β τ ξ^{2} Re (\hat{v} {\hat{w}}^{*}) + δ ξ^{2} Re (\hat{θ} {\hat{v}}^{*}) + τ δ ξ^{2} Re (\hat{θ} {\hat{w}}^{*}) . \end{array}$ Multiplying ${(4.4)}_{2}$ by $τ (β - τ) {\hat{v}}^{*}$ and taking the real part, we obtain $\begin{matrix} (4.10) & \frac{1}{2} τ (β - τ) \frac{d}{d t} | \hat{v} |^{2} = τ (β - τ) Re (\hat{w} {\hat{v}}^{*}) . \end{matrix}$ Multiplying ${(4.4)}_{2}$ by τ, and summing up the resulting equality and ${(4.4)}_{1}$ , we obtain $\begin{matrix} (4.11) & {(\hat{u} + τ \hat{v})}_{t} = \hat{v} + τ \hat{w} . \end{matrix}$ Multiplying (4.11) by $({\hat{u}}^{*} + τ {\hat{v}}^{*})$ and taking the real part, we get $\begin{matrix} (4.12) & \frac{1}{2} \frac{d}{d t} | \hat{u} + τ \hat{v} |^{2} = Re (\hat{v} {\hat{u}}^{*}) + τ | \hat{v} |^{2} + τ Re (\hat{w} {\hat{u}}^{*}) + τ^{2} Re (\hat{w} {\hat{v}}^{*}) . \end{matrix}$ Multiplying ${(4.4)}_{4}$ by ${\hat{θ}}^{*}$ and taking the real part, we have $\begin{matrix} (4.13) & \frac{1}{2} \frac{d}{d t} | \hat{θ} |^{2} = - δ ξ^{2} Re (\hat{v} {\hat{θ}}^{*}) - τ δ ξ^{2} Re (\hat{w} {\hat{θ}}^{*}) - Re (\frac{ξ^{2}}{k} {\hat{θ}}^{*} \int_{0}^{\infty} μ (s) \hat{η} (s) d s) . \end{matrix}$ Multiplying ${(4.4)}_{5}$ by $μ (s) {\hat{η}}^{*} (ξ, t, s)$ , integrating over $(0, \infty)$ respect to s and taking the real part, we arrive at $\begin{array}{l} Re (\int_{0}^{\infty} μ (s) {\hat{η}}^{*} (ξ, t, s) {\hat{η}}_{t} (ξ, t, s) d s) \\ (4.14) & = - Re (\int_{0}^{\infty} μ (s) {\hat{η}}^{*} (ξ, t, s) {\hat{η}}_{s} (ξ, t, s) d s) + Re (\int_{0}^{\infty} μ (s) {\hat{η}}^{*} (ξ, t, s) d s \hat{θ} (ξ, t)) . \end{array}$ By computing $(4.13) + (4.14) \times \frac{ξ^{2}}{k}$ , we get $\begin{array}{l} \frac{1}{2} \frac{d}{d t} | \hat{θ} |^{2} & = - δ ξ^{2} Re (\hat{v} {\hat{θ}}^{*}) - τ δ ξ^{2} Re (\hat{w} {\hat{θ}}^{*}) - Re (\frac{ξ^{2}}{k} \int_{0}^{\infty} μ (s) {\hat{η}}^{*} (ξ, t, s) {\hat{η}}_{t} (ξ, t, s) d s) \\ - Re (\frac{ξ^{2}}{k} \int_{0}^{\infty} μ (s) {\hat{η}}^{*} (ξ, t, s) {\hat{η}}_{s} (ξ, t, s) d s) \\ = - δ ξ^{2} Re (\hat{v} {\hat{θ}}^{*}) - τ δ ξ^{2} Re (\hat{w} {\hat{θ}}^{*}) - \frac{1}{2} \frac{d}{d t} {\frac{ξ^{2}}{k} \int_{0}^{\infty} μ (s) {| \hat{η} (ξ, t, s) |}^{2} d s} \\ - Re (\frac{ξ^{2}}{k} \int_{0}^{\infty} μ (s) {\hat{η}}^{*} (ξ, t, s) {\hat{η}}_{s} (ξ, t, s) d s) \\ = - δ ξ^{2} Re (\hat{v} {\hat{θ}}^{*}) - τ δ ξ^{2} Re (\hat{w} {\hat{θ}}^{*}) - \frac{1}{2} \frac{d}{d t} {\frac{ξ^{2}}{k} \int_{0}^{\infty} μ (s) {| \hat{η} (ξ, t, s) |}^{2} d s} \\ (4.15) & + \frac{ξ^{2}}{2 k} \int_{0}^{\infty} μ^{'} (s) {| \hat{η} (ξ, t, s) |}^{2} d s, \end{array}$ where we used integration by parts. Computing $(4.9) + ξ^{2} \times (4.10) + ξ^{2} \times (4.12) + (4.15)$ , we obtain our desired estimate (4.7). The proof is complete. □

Lemma 4.2.

Define the functional $\begin{matrix} F_{1} (ξ, t) : = Re ((\hat{v} + τ \hat{w}) ({\hat{u}}^{*} + τ {\hat{v}}^{*})), \end{matrix}$ then for any $ε_{1} > 0$ , there exists $C (ε_{1}) > 0$ such that $\begin{matrix} (4.16) & \frac{d}{d t} F_{1} (ξ, t) + (1 - 2 ε_{1}) ξ^{2} | \hat{u} + τ \hat{v} |^{2} ⩽ | \hat{v} + τ \hat{w} |^{2} + C (ε_{1}) ξ^{2} (| \hat{v} |^{2} + | \hat{θ} |^{2}) . \end{matrix}$

Proof.

Multiplying (4.8) by $({\hat{u}}^{*} + τ {\hat{v}}^{*})$ , we obtain $\begin{matrix} {(\hat{v} + τ \hat{w})}_{t} ({\hat{u}}^{*} + τ {\hat{v}}^{*}) = (- ξ^{2} \hat{u} - β ξ^{2} \hat{v} - τ ξ^{2} \hat{v} + τ ξ^{2} \hat{v} + δ ξ^{2} \hat{θ}) ({\hat{u}}^{*} + τ {\hat{v}}^{*}) . \end{matrix}$ Multiplying (4.11) by $({\hat{v}}^{*} + τ {\hat{w}}^{*})$ , we have $\begin{matrix} {(\hat{u} + τ \hat{v})}_{t} ({\hat{v}}^{*} + τ {\hat{w}}^{*}) = (\hat{v} + τ \hat{w}) ({\hat{v}}^{*} + τ {\hat{w}}^{*}) . \end{matrix}$ Adding the above two equalities and taking the real part, we get $\begin{matrix} (4.17) & \frac{d}{d t} F_{1} (ξ, t) + ξ^{2} | \hat{u} + τ \hat{v} |^{2} - | \hat{v} + τ \hat{w} |^{2} = ξ^{2} (τ - β) Re (\hat{v} ({\hat{u}}^{*} + τ {\hat{v}}^{*})) + δ ξ^{2} Re (\hat{θ} ({\hat{u}}^{*} + τ {\hat{v}}^{*})) . \end{matrix}$ By virtue of Young’s inequality, we have for any $ε_{1} > 0$ , $\begin{array}{l} (4.18) & ξ^{2} (τ - β) Re (\hat{v} ({\hat{u}}^{*} + τ {\hat{v}}^{*})) ⩽ ε_{1} ξ^{2} | \hat{u} + τ \hat{v} |^{2} + C (ε_{1}) ξ^{2} | \hat{v} |^{2}, \\ (4.19) & δ ξ^{2} Re (\hat{θ} ({\hat{u}}^{*} + τ {\hat{v}}^{*})) ⩽ ε_{1} ξ^{2} | \hat{u} + τ \hat{v} |^{2} + C (ε_{1}) ξ^{2} | \hat{θ} |^{2} . \end{array}$ Inserting (4.18) and (4.19) into (4.17), we deduce (4.16). □

Lemma 4.3.

The functional $\begin{matrix} F_{2} (ξ, t) : = Re (- τ (\hat{v} + τ \hat{w}) {\hat{v}}^{*}), \end{matrix}$ satisfies $\begin{matrix} (4.20) & \frac{d}{d t} F_{2} (ξ, t) + (1 - ε_{2}) | \hat{v} + τ \hat{w} |^{2} ⩽ C (ε_{2}, ε_{2}^{'}) (1 + ξ^{2}) | \hat{v} |^{2} + ε_{2}^{'} \frac{ξ^{4}}{1 + ξ^{2}} | \hat{u} + τ \hat{v} |^{2} + ε_{2}^{'} \frac{ξ^{4}}{1 + ξ^{2}} | \hat{θ} |^{2}, \end{matrix}$ for any $ε_{2}, ε_{2}^{'} > 0$ and $C (ε_{2}, ε_{2}^{'}) > 0$ .

Proof.

Multiplying ${(4.4)}_{2}$ by $- τ ({\hat{v}}^{*} + τ {\hat{w}}^{*})$ , we get $\begin{matrix} - τ {\hat{v}}_{t} ({\hat{v}}^{*} + τ {\hat{w}}^{*}) = - τ \hat{w} ({\hat{v}}^{*} + τ {\hat{w}}^{*}) . \end{matrix}$ Multiplying (4.8) by $- τ {\hat{v}}^{*}$ , we obtain $\begin{matrix} - τ {(\hat{v} + τ \hat{w})}_{t} {\hat{v}}^{*} = [τ ξ^{2} \hat{u} + β τ ξ^{2} \hat{v} + τ^{2} ξ^{2} \hat{v} - τ^{2} ξ^{2} \hat{v} - δ τ ξ^{2} \hat{θ} + (\hat{v} + τ \hat{w}) - (\hat{v} + τ \hat{w})] {\hat{v}}^{*} . \end{matrix}$ Adding the above two equalities and taking the real part, we arrive at $\begin{matrix} (4.21) & \begin{matrix} \frac{d}{d t} F_{2} (ξ, t) + | \hat{v} + τ \hat{w} |^{2} - τ (β - τ) ξ^{2} | \hat{v} |^{2} \\ = τ ξ^{2} Re ((\hat{u} + τ \hat{v}) {\hat{v}}^{*}) + Re ((\hat{v} + τ \hat{w}) {\hat{v}}^{*}) - δ τ ξ^{2} Re (\hat{θ} {\hat{v}}^{*}) . \end{matrix} \end{matrix}$ Young’s inequality yields, for any $ε_{2}, ε_{2}^{'} > 0$ , $\begin{array}{l} (4.22) & τ ξ^{2} Re ((\hat{u} + τ \hat{v}) {\hat{v}}^{*}) ⩽ ε_{2}^{'} \frac{ξ^{4}}{1 + ξ^{2}} | \hat{u} + τ \hat{v} |^{2} + C (ε_{2}^{'}) (1 + ξ^{2}) | \hat{v} |^{2}, \\ (4.23) & Re ((\hat{v} + τ \hat{w}) {\hat{v}}^{*}) ⩽ ε_{2} | \hat{v} + τ \hat{w} |^{2} + C (ε_{2}) | \hat{v} |^{2}, \\ (4.24) & - δ τ ξ^{2} Re (\hat{θ} {\hat{v}}^{*}) ⩽ ε_{2}^{'} \frac{ξ^{4}}{1 + ξ^{2}} | \hat{θ} |^{2} + C (ε_{2}^{'}) (1 + ξ^{2}) | \hat{v} |^{2} . \end{array}$ Plugging (4.22)–(4.24) into (4.21), we obtain (4.20). □

Lemma 4.4.

For any $ε_{3}, ε_{3}^{'} > 0$ , there exist $C (ε_{3}), C (ε_{3}^{'}) > 0$ such that the following inequality holds true $\begin{array}{l} \frac{d}{d t} F_{3} (ξ, t) + (g (0) - ε_{3}) | \hat{θ} |^{2} \\ ⩽ (\frac{1}{k} + C (ε_{3}^{'})) g (0) {(1 + ξ^{2})}^{3} \int_{0}^{\infty} μ (s) {| \hat{η} (s) |}^{2} d s + ε_{3}^{'} \frac{ξ^{4}}{{(1 + ξ^{2})}^{3}} | \hat{v} + τ \hat{w} |^{2} \\ (4.25) & + C (ε_{3}) g^{'} (0) \int_{0}^{\infty} μ^{'} (s) {| \hat{η} (s) |}^{2} d s, \end{array}$ where $\begin{matrix} F_{3} (ξ, t) : = Re (- \int_{0}^{\infty} μ (s) {\hat{η}}^{*} (s) d s \hat{θ}) . \end{matrix}$

Proof.

Multiplying ${(4.4)}_{4}$ and ${(4.4)}_{5}$ by $- μ (s) {\hat{η}}^{*} (s)$ and $- μ (s) {\hat{θ}}^{*}$ , respectively, integrating the results respect to s, adding the results and taking the real part, we get $\begin{array}{l} \frac{d}{d t} F_{3} (ξ, t) + \int_{0}^{\infty} μ (s) d s | \hat{θ} |^{2} \\ = \frac{ξ^{2}}{k} Re ({(\int_{0}^{\infty} μ (s) \hat{η} (s) d s)}^{2}) + Re (\int_{0}^{\infty} μ (s) {\hat{η}}_{s} d s {\hat{θ}}^{*}) \\ (4.26) & + τ δ ξ^{2} Re (\int_{0}^{\infty} μ (s) {\hat{η}}^{*} (s) d s \hat{w}) + δ ξ^{2} Re (\int_{0}^{\infty} μ (s) {\hat{η}}^{*} (s) d s \hat{v}) . \end{array}$ Integration by parts yields that $\begin{array}{l} \frac{d}{d t} F_{3} (ξ, t) + g (0) | \hat{θ} |^{2} \\ = \frac{ξ^{2}}{k} Re ({(\int_{0}^{\infty} μ (s) \hat{η} (s) d s)}^{2}) - Re (\int_{0}^{\infty} μ^{'} (s) \hat{η} d s {\hat{θ}}^{*}) \\ (4.27) & + δ ξ^{2} Re (\int_{0}^{\infty} μ (s) {\hat{η}}^{*} (s) d s (\hat{v} + τ \hat{w})) . \end{array}$ Applying Young’s inequality with $ε_{3}, ε_{3}^{'} > 0$ , we have $\begin{array}{l} (4.28) & - Re (\int_{0}^{\infty} μ^{'} (s) \hat{η} d s {\hat{θ}}^{*}) ⩽ ε_{3} | \hat{θ} |^{2} + C (ε_{3}) g^{'} (0) \int_{0}^{\infty} μ^{'} (s) {| \hat{η} (s) |}^{2} d s, \\ (4.29) & \begin{matrix} δ ξ^{2} Re (\int_{0}^{\infty} μ (s) {\hat{η}}^{*} (s) d s (\hat{v} + τ \hat{w})) \\ ⩽ ε_{3}^{'} \frac{ξ^{4}}{{(1 + ξ^{2})}^{3}} | \hat{v} + τ \hat{w} |^{2} + C (ε_{3}^{'}) g (0) {(1 + ξ^{2})}^{3} \int_{0}^{\infty} μ (s) {| \hat{η} (s) |}^{2} d s . \end{matrix} \end{array}$ Combining (4.28)–(4.29) with (4.27), we arrive at (4.25). □

Now, we begin to prove Proposition 4.1.

Proof of Proposition 4.1.

We define the Lyapunov functional $\begin{matrix} L_{1} (ξ, t) : = N_{1} \frac{ξ^{6}}{{(1 + ξ^{2})}^{3}} F_{1} (ξ, t) + N_{2} \frac{ξ^{4}}{{(1 + ξ^{2})}^{2}} F_{2} (ξ, t) + N_{3} ξ^{2} F_{3} (ξ, t), \end{matrix}$ where $N_{1}$ , $N_{2}$ and $N_{3}$ are positive constants that will be fixed later. Taking advantage of the above lemmas, we have $\begin{array}{l} \frac{d}{d t} L_{1} (ξ, t) + [N_{1} (1 - 2 ε_{1}) - N_{2} ε_{2}^{'}] \frac{ξ^{6}}{{(1 + ξ^{2})}^{3}} ξ^{2} | \hat{u} + τ \hat{v} |^{2} \\ + [N_{2} (1 - ε_{2}) - N_{1} - N_{3} ε_{3}^{'}] \frac{ξ^{6}}{{(1 + ξ^{2})}^{3}} | \hat{v} + τ \hat{w} |^{2} \\ + [- N_{2} C (ε_{2}, ε_{2}^{'}) - N_{1} C (ε_{1})] ξ^{2} | \hat{v} |^{2} \\ + [N_{3} (g (0) - ε_{3}) - N_{2} ε_{2}^{'} - N_{1} C (ε_{1})] ξ^{2} | \hat{θ} |^{2} \\ - N_{3} (\frac{1}{k} + C (ε_{3}^{'})) g (0) {(1 + ξ^{2})}^{3} \int_{0}^{\infty} ξ^{2} μ (s) {| \hat{η} (s) |}^{2} d s \\ + N_{3} C (ε_{3}) ξ^{2} g^{'} (0) \int_{0}^{\infty} (- μ^{'} (s)) {| \hat{η} (s) |}^{2} d s \\ (4.30) & ⩽ 0, \end{array}$ where we used the fact that $\frac{ξ^{2}}{1 + ξ^{2}} ⩽ 1$ . Moreover, according to (H2), we deduce $\begin{matrix} (4.31) & \int_{0}^{\infty} ξ^{2} μ (s) {| \hat{η} (ξ, t, s) |}^{2} d s ⩽ \frac{1}{ν} \int_{0}^{\infty} (- μ^{'} (s)) ξ^{2} {| \hat{η} (ξ, t, s) |}^{2} d s . \end{matrix}$ Hence, (4.30) becomes $\begin{array}{l} \frac{d}{d t} L_{1} (ξ, t) + [N_{1} (1 - 2 ε_{1}) - N_{2} ε_{2}^{'}] \frac{ξ^{6}}{{(1 + ξ^{2})}^{3}} ξ^{2} | \hat{u} + τ \hat{v} |^{2} \\ + [N_{2} (1 - ε_{2}) - N_{1} - N_{3} ε_{3}^{'}] \frac{ξ^{6}}{{(1 + ξ^{2})}^{3}} | \hat{v} + τ \hat{w} |^{2} \\ + [N_{3} (g (0) - ε_{3}) - N_{2} ε_{2}^{'} - N_{1} C (ε_{1})] ξ^{2} | \hat{θ} |^{2} \\ - C_{1} (N_{1}, N_{2}, ε_{1}, ε_{2}, ε_{2}^{'}) ξ^{2} | \hat{v} |^{2} - C_{2} (N_{3}, ε_{3}, ε_{3}^{'}, ν) {(1 + ξ^{2})}^{3} \int_{0}^{\infty} ξ^{2} (- μ^{'} (s)) {| \hat{η} (ξ, t, s) |}^{2} d s \\ (4.32) & ⩽ 0 . \end{array}$ At this point, we want to choose the constants in (4.32). First, we choose $\begin{matrix} ε_{1} < \frac{1}{2}, ε_{2} < 1, ε_{3} < g (0) . \end{matrix}$ Next, we fix $N_{1} = 1$ , $N_{2}$ and $N_{3}$ large enough such that $\begin{matrix} N_{2} > \frac{N_{1}}{1 - ε_{2}} and N_{3} > \frac{N_{1} C (ε_{1})}{g (0) - ε_{3}} . \end{matrix}$ Then, we pick $ε_{2}^{'}$ and $ε_{3}^{'}$ satisfying $\begin{matrix} ε_{2}^{'} < min {\frac{N_{1} (1 - 2 ε_{1})}{N_{2}}, \frac{N_{3} (g (0) - ε_{3}) - N_{1} C (ε_{1})}{N_{2}}} and ε_{3}^{'} < \frac{N_{2} (1 - ε_{2}) - N_{1}}{N_{3}} . \end{matrix}$ From the above, we deduce that there exists a positive constant $α_{1}$ such that (4.32) becomes $\begin{array}{l} \frac{d}{d t} L_{1} (ξ, t) + α_{1} M_{1} (ξ, t) \\ ⩽ C_{1} (N_{1}, N_{2}, ε_{1}, ε_{2}, ε_{2}^{'}) {(1 + ξ^{2})}^{3} ξ^{2} | \hat{v} |^{2} \\ (4.33) & + C_{2} (N_{3}, ε_{3}, ε_{3}^{'}, ν) {(1 + ξ^{2})}^{3} \int_{0}^{\infty} ξ^{2} (- μ^{'} (s)) {| \hat{η} (ξ, t, s) |}^{2} d s, \end{array}$ where $\begin{array}{l} M_{1} (ξ, t) & = \frac{ξ^{6}}{{(1 + ξ^{2})}^{3}} ξ^{2} | \hat{u} + τ \hat{v} |^{2} + \frac{ξ^{6}}{{(1 + ξ^{2})}^{3}} | \hat{v} + τ \hat{w} |^{2} + ξ^{2} | \hat{θ} |^{2} \\ ⩾ \frac{ξ^{6}}{{(1 + ξ^{2})}^{3}} (ξ^{2} | \hat{u} + τ \hat{v} |^{2} + | \hat{v} + τ \hat{w} |^{2} + | \hat{θ} |^{2}) . \end{array}$

We define the Lyapunov functional $L_{1} (ξ, t)$ satisfies $\begin{matrix} L_{1} (ξ, t) : = N {(1 + ξ^{2})}^{3} \hat{E} (ξ, t) + L_{1} (ξ, t), \end{matrix}$ where N is a positive constant that will be fixed later. By virtue of Lemma 4.1 and (4.33), we have $\begin{array}{l} \frac{d}{d t} L_{1} (ξ, t) + α_{1} M_{1} (ξ, t) + [N (β - τ) - C_{1} (N_{1}, N_{2}, ε_{1}, ε_{2}, ε_{2}^{'})] {(1 + ξ^{2})}^{3} ξ^{2} | \hat{v} |^{2} \\ + [\frac{N}{2 k} - C_{2} (N_{3}, ε_{3}, ε_{3}^{'}, ν)] {(1 + ξ^{2})}^{3} ξ^{2} \int_{0}^{\infty} (- μ^{'} (s)) {| \hat{η} (ξ, t, s) |}^{2} d s \\ (4.34) & ⩽ 0 . \end{array}$ We choose N large enough such that $\begin{matrix} N > max {\frac{C_{1} (N_{1}, N_{2}, ε_{1}, ε_{2}, ε_{2}^{'})}{β - τ}, 2 k C_{2} (N_{3}, ε_{3}, ε_{3}^{'}, ν)} . \end{matrix}$ Finally, (4.34) becomes $\begin{matrix} (4.35) & \frac{d}{d t} L_{1} (ξ, t) + C \frac{ξ^{6}}{{(1 + ξ^{2})}^{3}} \hat{E} (ξ, t) ⩽ 0, \forall t ⩾ 0 . \end{matrix}$

On the other hand, from the definitions of $L_{1} (ξ, t)$ , $L_{1} (ξ, t)$ and $\hat{E} (ξ, t)$ , we find that $\begin{array}{l} | L_{1} (ξ, t) - N {(1 + ξ^{2})}^{3} \hat{E} (ξ, t) | \\ = | L_{1} (ξ, t) | \\ = | N_{1} \frac{ξ^{6}}{{(1 + ξ^{2})}^{3}} F_{1} (ξ, t) + N_{2} \frac{ξ^{4}}{{(1 + ξ^{2})}^{2}} F_{2} (ξ, t) + N_{3} ξ^{2} F_{3} (ξ, t) | \\ ⩽ C_{3} \hat{E} (ξ, t) . \end{array}$ Then we arrive at $\begin{matrix} (N - C_{4}) {(1 + ξ^{2})}^{3} \hat{E} (ξ, t) ⩽ L_{1} (ξ, t) ⩽ (N + C_{4}) {(1 + ξ^{2})}^{3} \hat{E} (ξ, t) . \end{matrix}$ For N large enough, we conclude that there exist two positive constants $C_{5}$ and $C_{6}$ such that $\begin{matrix} C_{5} {(1 + ξ^{2})}^{3} \hat{E} (ξ, t) ⩽ L_{1} (ξ, t) ⩽ C_{6} {(1 + ξ^{2})}^{3} \hat{E} (ξ, t), \forall t ⩾ 0 . \end{matrix}$ Therefore, (4.35) becomes $\begin{matrix} (4.36) & \frac{d}{d t} \hat{E} (ξ, t) + C ρ_{1} (ξ) \hat{E} (ξ, t) ⩽ 0, \forall t ⩾ 0 . \end{matrix}$ Consequently, making use of the equivalence of $\hat{E} (ξ, t)$ and $| \hat{V} |^{2}$ and Gronwall’s inequality, we obtain the desired result (4.5). □

Lemma 4.5.

For all $t ⩾ 0$ and all $j ⩾ 0$ , the following estimate holds $\begin{matrix} (4.37) & \int_{| ξ | ⩽ 1} ξ^{j} e^{- c | ξ |^{6} t} d ξ ⩽ C {(1 + t)}^{- \frac{n}{6} - \frac{j}{6}}, n ⩾ 1 . \end{matrix}$

Proof.

From Lemma 2.1 in [18], we have $\begin{matrix} (4.38) & \int_{0}^{1} r^{k} e^{- c r^{6} t} d r ⩽ C {(1 + t)}^{- (k + 1) / 6} . \end{matrix}$ Employing the change of variables $r = | ξ |$ and $d ξ = C | ξ |^{n - 1} d r$ , we get our desired estimate (4.37). □

Proof of Theorem 2.2.

Applying the Plancherel theorem and the pointwise estimate (4.5), we have $\begin{array}{l} {‖ \nabla^{k} V (t) ‖}_{L^{2} (R^{n})}^{2} \\ ⩽ C \int_{R^{n}} ξ^{2 k} {| \hat{V} (ξ, t) |}^{2} d ξ \\ ⩽ C \int_{R^{n}} ξ^{2 k} e^{- c ρ_{1} (ξ) t} {| {\hat{V}}_{0} (ξ) |}^{2} d ξ \\ = C \int_{| ξ | ⩽ 1} ξ^{2 k} e^{- c ρ_{1} (ξ) t} {| {\hat{V}}_{0} (ξ) |}^{2} d ξ + C \int_{| ξ | ⩾ 1} ξ^{2 k} e^{- c ρ_{1} (ξ) t} {| {\hat{V}}_{0} (ξ) |}^{2} d ξ \\ (4.39) & = : J_{1} + J_{2} . \end{array}$ And by $ρ_{1} (ξ) ⩾ C | ξ |^{6}$ for $| ξ | ⩽ 1$ and Lemma 4.5, we arrive at $\begin{array}{l} J_{1} & ⩽ C \int_{| ξ | ⩽ 1} ξ^{2 k} e^{- c | ξ |^{6} t} {| {\hat{V}}_{0} (ξ) |}^{2} d ξ \\ ⩽ C ‖ {\hat{V}}_{0} ‖_{L^{\infty} (R^{n})}^{2} \int_{| ξ | ⩽ 1} ξ^{2 k} e^{- c | ξ |^{6} t} d ξ \\ (4.40) & ⩽ C {(1 + t)}^{- \frac{n}{6} - \frac{k}{3}} ‖ V_{0} ‖_{L^{1} (R^{n})}^{2} . \end{array}$

On the other hand, using $ρ_{1} (ξ) ⩾ C | ξ |^{- 6}$ for $| ξ | ⩾ 1$ , we get $\begin{array}{l} J_{2} & ⩽ C \int_{| ξ | ⩾ 1} ξ^{2 k} e^{- c | ξ |^{- 6} t} {| {\hat{V}}_{0} (ξ) |}^{2} d ξ \\ ⩽ C sup_{| ξ | ⩾ 1} {ξ^{- 2 l} e^{- c | ξ |^{- 6} t}} \int_{| ξ | ⩾ 1} ξ^{2 (k + l)} {| {\hat{V}}_{0} (ξ) |}^{2} d ξ \\ (4.41) & ⩽ C {(1 + t)}^{- \frac{l}{3}} {‖ \nabla^{k + l} V_{0} ‖}_{L^{2} (R^{n})}^{2} . \end{array}$ Substituting (4.40)–(4.41) into (4.39) gives the desired estimate in Theorem 2.2. □

Footnotes

Acknowledgements

This work was supported by the National Natural Science Foundation of China [grant number 11771216], the Key Research and Development Program of Jiangsu Province (Social Development) [grant number BE2019725], the Six Talent Peaks Project in Jiangsu Province [grant number 2015-XCL-020] and the Postgraduate Research and Practice Innovation Program of Jiangsu Province [grant number KYCX20_0906].

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Well-posedness and decay property of regularity-loss type for the Cauchy problem of the standard linear solid model with Gurtin–Pipkin thermal law

Abstract

Keywords

1. Introduction

2. Preliminaries and main results

Proposition 4.1 (Pointwise estimate).

Footnotes

Acknowledgements

References