On the Cauchy problem of the standard linear solid model with Cattaneo heat conduction

Abstract

In the present paper we consider the Standard Linear Solid model in $R^{N}$ coupled with the Cattaneo law of heat conduction. We show the well-posedness and asymptotic stability of the problem, giving decay rates for a norm related to the solution. These results are compared with those given for the Fourier problem in (Pellicer and Said-Houari (2020)) and the ones of the problem without heat conduction (see previous work (Appl Math. Optim 80 (2019) 447–478)). The main difference is that the Cattaneo system exhibits the well-known regularity-loss phenomenon. The methods used to prove these results are the energy method in the Fourier space and the eigenvalues expansion method.

Keywords

Standard Linear Solid model Moore–Gibson–Thompson equation Cattaneo heat conduction decay rate asymptotic stability regularity-loss

1. Introduction

The Standard Linear Solid model $\begin{matrix} (1.1) & τ u_{t t t} + u_{t t} - a^{2} Δ u - a^{2} β Δ u_{t} = 0 \end{matrix}$ is a mechanical model derived by Gorain & Bose [14] (see also [4]) as a more realistic approach to model flexible structural systems possessing internal damping. To obtain this model, the following relation between the stress σ and the strain e is used $\begin{matrix} σ + τ σ^{'} = E (e + β e^{'}) \end{matrix}$ where E is the Young modulus of the elastic structure and the constants τ and β are small and satisfying $0 < τ < β$ . The case $τ = β = 0$ corresponds to $σ = E e$ and, hence, to vibrations governed by the free wave equation $\begin{matrix} u_{t t} - a^{2} Δ u = 0 . \end{matrix}$ (see [13] or [27] for more details of the modelling). It is interesting to observe that the same equation (1.1) was first introduced in [40] as a model for the acoustic velocity potential in thermally relaxing fluids (see also [16] and the references therein, among others), then under the name of Moore–Gibson–Thompson equation (MGT equation). It has recently been seen that (1.1) can also be obtained as a model for the heat conduction after introducing a relaxation parameter in the type III Green–Naghdi heat conduction model (see [10,30]).

Equation (1.1) together with its abstract form has attracted a lot of work in the last years, specially in bounded domains. The well-posedness and asymptotic behaviour of (1.1) is very interesting, as it strongly depends on the choice of the parameters. For instance, it is needed $β > 0$ to have a well-posed problem if $τ > 0$ (which is a difference with the same equation with $τ = 0$ , that is, the strongly damped wave equation). Or, also, it is known that the solutions decay exponentially fast or grow expontentially fast depending whether $τ < β$ or $τ > β$ , respectively. The kind of the generated semigroup also depends on the choice of the parameters. We refer the reader to the following works and the references therein, among others, for a deeper insight on this equation: [7,11,12,16,19,21,27,29].

On the other hand, the Fourier law is a well-known and commonly used model for heat conduction. If θ and q stand for the temperature difference and heat flux vector, respectively, we have: $\begin{matrix} θ_{t} + γ \nabla \cdot q = 0 \end{matrix}$ and $\begin{matrix} q + κ \nabla θ = 0 \end{matrix}$ for $γ, β > 0$ . Combining the previous two equations we obtain the parabolic heat equation: $\begin{matrix} θ_{t} - γ κ Δ θ = 0 \end{matrix}$ that allows an infinite speed for thermal signals. To overcome this drawback in the Fourier law, we can consider the Cattaneo law (also known as Maxwell–Cattaneo law) of heat conduction that differs from the Fourier one by the presence of the relaxation term $q_{t}$ as $\begin{matrix} (1.2) & τ_{0} q_{t} + q + κ \nabla θ = 0 \end{matrix}$ with $τ_{0} > 0$ . This now models the heat propagation as a damped wave equation. This phenomenon is known as second sound effect and it is experimentally observed in materials at a very low temperature, where heat seems to propagate as a thermal wave, which is the reason for this name (see [1,3] or [23], and the review in [32]). The parameter $τ_{0} > 0$ is the relaxation time of the heat flux (usually small compared to other parameters) and arises due to the delay in the response of the heat flux to the temperature gradient.

There are many references of thermoelastic and thermoviscoelastic problems in the literature. These models are, for instance, the strongly damped wave equation (or wave equation with Kelvin–Voigt damping) with heat conduction, using either the Fourier or the Cattaneo law of heat conduction (see [22] as a reference for the former, or [2] or [31] as references for the latter). Or also the models of classical thermoelasticity (that is, wave equation coupled with heat conduction), where both Fourier and Cattaneo laws have been used for the heat coupling (see [6,20] or [31], among many others, as references for the asymptotic stability of these models). Finally, we can find works on the asymptotic stability of thermoelastic plates, with also either Fourier or Cattaneo laws being used (see [36] and [35], respectively, or [37] for a comparison between both approaches). However, there are not so many references of the Standard Linear Solid model coupled with heat conduction, although there have started to appear works on thermoelastic and thermoviscoelastic models where the displacement part is modelled by the Standard Linear Solid model, but none of them in $R^{N}$ (see references given in [28] of such recent works where the standard linear solid structure is used either in the mechanical part, in thermal part or both, such as [8–10,26]). To our knowledge, the first model to consider this viscoelastic model with heat conduction is the recent work [1], where the authors consider the equation (1.1) coupled with the Fourier law of heat conduction in a bounded domain: $\begin{array}{rcl} \{\begin{matrix} τ u_{t t t} + u_{t t} - a^{2} Δ u - a^{2} β Δ u_{t} + η Δ θ = 0, \\ θ_{t} - Δ θ - τ η Δ u_{t t} - η Δ u_{t} = 0, \end{matrix} \end{array}$ where $x \in Ω$ (a bounded domain of $R^{N}$ ), $t \in (0, \infty)$ and $η ⩾ 0$ being the coupling coefficient. Observe that when $η = 0$ we recover the standard linear solid model of viscoelasticity (1.1). The authors prove the well-posedness of the previous problem and the exponential decay of their solutions, under certain assumptions on the parameters (more concretely, the dissipativeness or stability condition $0 < τ < β$ ).

In the even more recent work [28] we consider the previous system in $R^{N}$ , that is $\begin{array}{rcl} (1.3) & \{\begin{matrix} τ u_{t t t} + u_{t t} - Δ u - β Δ u_{t} + η Δ θ = 0, \\ θ_{t} - Δ θ - τ η Δ u_{t t} - η Δ u_{t} = 0, \end{matrix} \end{array}$ with $x \in R^{N}$ , with $0 < τ ⩽ β$ and $η ⩾ 0$ . We provide the appropriate functional framework where this system is well-posed and show its asymptotic stability by giving the optimal decay rate of a norm related to the solution. More concretely, we show that when $0 < τ < β$ the decay rate turns out to be the same as the one for the Cauchy problem without heat conduction (see [27]), that is: for $V_{F}^{0} \in L^{2} (R^{N}) \cap L^{1} (R^{N})$ , we have $\begin{matrix} (1.4) & {‖ V_{F} (t) ‖}_{L^{2} (R^{N})} ⩽ C {(1 + t)}^{- N / 4} {‖ V_{F}^{0} ‖}_{L^{1} (R^{N})} + C e^{- c t} {‖ V_{F}^{0} ‖}_{L^{2} (R^{N})}, \end{matrix}$ for any $t ⩾ 0$ , where $V_{F} (x, t) = {(τ u_{t t} + u_{t}, \nabla (τ u_{t} + u), \nabla u_{t}, θ)}^{T} (x, t)$ and $V_{F}^{0} = V_{F} (x, 0)$ (see Theorem 3.1 in [28]). In particular, this means that adding heat conduction under the Fourier law does not increase the corresponding decay rate. The authors also show that when $τ = β > 0$ , however, the decay turns out to be slower than before, which makes sense since when $τ = β$ the heat conduction is the only source for dissipation. More concretely, we show that for $W_{F}^{0} \in L^{2} (R^{N}) \cap L^{1} (R^{N})$ we have $\begin{matrix} (1.5) & {‖ W_{F} (t) ‖}_{L^{2} (R^{N})} ⩽ C {(1 + t)}^{- N / 8} {‖ W_{F}^{0} ‖}_{L^{1} (R^{N})} + C e^{- c t} {‖ W_{F}^{0} ‖}_{L^{2} (R^{N})}, \end{matrix}$ for any $t ⩾ 0$ , where $W_{F} (x, t) = {(τ u_{t t} + u_{t}, \nabla (τ u_{t} + u), θ)}^{T} (x, t)$ and $W_{F}^{0} = W_{F} (x, 0)$ (see Theorem 5.1 in [28]). In particular, this also means that adding heat conduction in the case $τ = β$ makes the problem become asymptotically stable. Observe that no regularity-loss is observed in none of the two cases, that is, we do not need higher regularity in the initial data to obtained the desired decay.

In the present paper we consider the Standard Linear Solid model (1.1) in $R^{N}$ coupled with the Cattaneo law of the heat conduction (1.2), instead of the Fourier law: $\begin{array}{rcl} (1.6) & \{\begin{matrix} τ u_{t t t} + u_{t t} - a^{2} Δ u - a^{2} β Δ u_{t} + η Δ θ = 0, \\ θ_{t} + γ \nabla \cdot q - τ η Δ u_{t t} - η Δ u_{t} = 0, \\ τ_{0} q_{t} + q + κ \nabla θ = 0, \end{matrix} \end{array}$ where $x \in R^{N}$ , $t ⩾ 0$ and $τ, γ, κ, β, τ_{0}, η, a^{2} > 0$ . After providing the appropriate functional setting where the problem becomes well-posed, we give the decay rate for a norm related to the solution, both when $0 < τ < β$ (subcritical case) and when $0 < τ = β$ (critical case), and compare the results with those of the Fourier heat law problem in [28] to see how the heat conduction term and its type affect the asymptotic behaviour of the solutions. More concretely, our decay results for system (1.6) are the following. When $0 < τ < β$ we see that for $V_{C}^{0} \in H^{ℓ} (R^{N}) \cap L^{1} (R^{N})$ , it holds that $\begin{matrix} (1.7) & {‖ V_{C} (t) ‖}_{L^{2} (R^{N})} ⩽ C {(1 + t)}^{- N / 4} {‖ V_{C}^{0} ‖}_{L^{1} (R^{N})} + C {(1 + t)}^{- ℓ / 2} {‖ \nabla^{ℓ} V_{C}^{0} ‖}_{L^{2} (R^{N})}, \end{matrix}$ for any $t ⩾ 0$ , where $V_{C} (x, t) = {(τ u_{t t} + u_{t}, \nabla (τ u_{t} + u), \nabla u_{t}, θ, q)}^{T} (x, t)$ , $V_{C}^{0} = V_{C} (x, 0)$ (see Theorem 3.1). And when $0 < τ = β$ we see that for $W_{C}^{0} \in H^{ℓ} (R^{N}) \cap L^{1} (R^{N})$ , it holds that $\begin{matrix} (1.8) & {‖ W_{C} (t) ‖}_{L^{2} (R^{N})} ⩽ C {(1 + t)}^{- N / 8} {‖ W_{C}^{0} ‖}_{L^{1} (R^{N})} + C {(1 + t)}^{- ℓ / 3} {‖ \nabla^{ℓ} W_{C}^{0} ‖}_{L^{2} (R^{N})}, \end{matrix}$ where $W_{C} (x, t) = {(τ u_{t t} + u_{t}, \nabla (τ u_{t} + u), θ, q)}^{T} (x, t)$ , $W_{C}^{0} = W_{C} (x, 0)$ and ℓ is a nonnegative integer. (See Theorem 4.1).

These results mean that adding heat conduction under Cattaneo law instead of the Fourier one to the Standard Linear Solid model gives the same decay rate exponent (although it has to be noticed that the spaces and the norms are slightly different in both problems). More concretely, we mean that the decay rate of the Fourier model can be recovered in the Cattaneo model if we pay extra regularity for the initial data (the difference is in the regularity required for the decay, not in the decay rate). For instance, we recall that for initial data $V_{F}^{0} \in L^{1} (R^{N}) \cap L^{2} (R^{N})$ the $L^{2}$ -norm of the solution of the Fourier model decays like ${(1 + t)}^{- N / 4}$ (see (1.4)). Then, from (1.7), we can obtain the same decay rate for the Cattaneo model provided that $V_{C}^{0} \in L^{1} (R^{N}) \cap H^{ℓ} (R^{N})$ with $ℓ ⩾ [N / 2] + 1$ (where $[N / 2]$ stands for the least integer part of $N / 2$ ).

In particular, this means that it does not improve the decay rate of the Cauchy problem without heat conduction either. This may not be surprising, as it is known that for many mathematical models that the dissipation induced by the heat conduction is usually weaker than the viscoelastic (or frictional) dissipation. As an example, see similar decay rates recently obtained for the Timoshenko system coupled with heat conduction in [24] and [18].

However, we can see from the decay rates (1.7) and (1.8) that adding heat conduction under Cattaneo law to the Standard Linear Solid model yields the regularity-loss phenomenon in both cases ( $0 < τ < β$ and $0 < τ = β$ ). This does not happen for the Fourier coupling (see estimates (1.4) and (1.5)), but it’s something that can be expected in Cattaneo-type problems (see for instance [39]). As we said, this is a significant difference with the Fourier coupling problem. Such a difference cannot be seen when $τ = 0$ (strongly damped wave equation for the mechanical part), which makes the limit problem $τ \to 0$ very interesting.

Finally, we would like to add that the previous results (1.4)–(1.5) (Fourier case) and (1.7)–(1.8) (Cattaneo case) are given in [28] and in the present paper (Theorems 3.1 and 4.1), respectively, for more regular functions (that is to say, for initial conditions in $H^{s} (R^{N}) \cap L^{1} (R^{N})$ for any s non-negative integer) and their derivatives. The general conclusions stated above remain in this more general case.

As in [28], using the eigenvalues expansion method we are able to prove that the decay rate when $0 < τ < β$ given in (1.7) is optimal. But using this same method we have not able to prove optimality of the decay (1.8) when $0 < τ = β$ . This represents an important difference with the Fourier results for this case (see [28]).

Finally, we want to remark here that the use of the Routh–Hurwitz theorem in [28] allowed us to prove that $0 < τ < β$ is not only a sufficient (which was also known) but also necessary condition for the stability of the Cauchy problem without heat conduction. The same theorem also allowed us to prove in [28] that $0 < τ ⩽ β$ is a sufficient but not necessary condition for the asymptotic stability of the Standard Linear Solid model with the Fourier heat coupling. However, we have to mention that the same techniques do not give any similar information for the Cattaneo problem (1.6). But we will see in Sections 3.2 and 4.2 that $0 < τ ⩽ β$ is a sufficient condition for its stability.

This paper is organized as follows. In Section 2 we show the well-posedness for the Cattaneo model, and Sections 3 and 4 are devoted to the stability results for the Cattaneo model when $0 < τ < β$ and $0 < τ = β$ , respectively.

2. Functional setting and well-posedness of the Cattaneo-law model

In this section, we consider the Standard Linear Solid model for viscoelasticity in $R^{N}$ coupled with heat conduction of Cattaneo type (see Section 1 for more details). That is, we consider system (1.6) with initial data $\begin{matrix} (2.1) & \begin{array}{l} u (x, 0) = u_{0} (x), u_{t} (x, 0) = u_{1} (x), u_{t t} (x, 0) = u_{2} (x), \\ θ (x, 0) = θ_{0} (x), θ_{t} (x, 0) = θ_{1} (x), q (x, 0) = q_{0} (x), q_{t} (x, 0) = q_{1} (x) . \end{array} \end{matrix}$ Without loss of generality, we now take $a = 1$ in (1.6).

The functional setting and the proof for the well-posedness in the Cattaneo case will follow the same ideas as the Fourier one (see Section 2 of [28]). Following [1] and [27], we introduce the energy space $\begin{matrix} (2.2) & H = H^{1} (R^{N}) \times H^{1} (R^{N}) \times L^{2} (R^{N}) \times L^{2} (R^{N}) \times {(L^{2} (R^{N}))}^{N} \end{matrix}$ with the following inner product $\begin{array}{l} {⟨ (u, v, w, θ, q), (u_{1}, v_{1}, w_{1}, θ_{1}, q_{1}) ⟩}_{H} \\ = τ (β - τ) \int_{R^{N}} \nabla v \cdot \nabla {\overline{v}}_{1} d x + \int_{R^{N}} \nabla (u + τ v) \cdot \nabla ({\overline{u}}_{1} + τ {\overline{v}}_{1}) d x \\ + \int_{R^{N}} (v + τ w) ({\overline{v}}_{1} + τ {\overline{w}}_{1}) d x + \int_{R^{N}} (u + τ v) ({\overline{u}}_{1} + τ {\overline{v}}_{1}) d x \\ + \int_{R^{N}} v {\overline{v}}_{1} d x + \int_{R^{N}} θ {\overline{θ}}_{1} d x + τ_{0} \frac{γ}{κ} \int_{R^{N}} q \cdot {\overline{q}}_{1} d x . \end{array}$

That is, we have corrected the inner product for the Fourier case with the last term (the one with $τ_{0}$ ), that appears as we have the Cattaneo heat conduction law (see [28]).

The corresponding norm is $\begin{matrix} (2.3) & \begin{matrix} {‖ (u, v, w, θ, q) ‖}_{H}^{2} & = τ (β - τ) ‖ \nabla v ‖_{L^{2} (R^{N})}^{2} + {‖ \nabla (u + τ v) ‖}_{L^{2} (R^{N})}^{2} + ‖ v + τ w ‖_{L^{2} (R^{N})}^{2} \\ + ‖ u + τ v ‖_{L^{2} (R^{N})}^{2} + ‖ v ‖_{L^{2} (R^{N})}^{2} + ‖ θ ‖_{L^{2} (R^{N})}^{2} + τ_{0} \frac{γ}{κ} ‖ q ‖_{{(L^{2} (R^{N}))}^{N}}^{2}, \end{matrix} \end{matrix}$ where $\begin{matrix} ‖ q ‖_{{(L^{2} (R^{N}))}^{N}}^{2} = \sum_{i = 1}^{N} ‖ q_{i} ‖_{L^{2} (R^{N})}^{2} . \end{matrix}$

Remark 2.1.
When $0 < τ < β$ , the norm defined in (2.3) is equivalent to the usual one in $H$ . When $τ = β > 0$ , however, the absence of the first term in (2.3) makes it impossible to be equivalent to the usual norm in $H$ . This fact makes the critical case $τ = β$ to be treated separatedly in the well-posedness of the problem (see Section 2.1).

Let’s first consider the subcritical case $0 < τ < β$ . As in the Fourier problem, the first thing we need is to write problem (1.6) with initial conditions (2.1) as a first-order evolution equation. By taking $v = u_{t}$ and $w = u_{t t}$ , the previous system can be written as $\begin{matrix} (2.4) & \{\begin{matrix} \frac{d}{d t} U (t) = A U (t), t \in [0, + \infty) \\ U (0) = U_{0} \end{matrix} \end{matrix}$ where $U (t) = {(u, v, w, θ, q)}^{T}$ , $U_{0} = {(u_{0}, u_{1}, u_{2}, θ_{0}, q_{0})}^{T}$ and $A : D (A) \subset H ⟶ H$ is the following linear operator $\begin{matrix} A (\begin{matrix} u \\ v \\ w \\ θ \\ q \end{matrix}) = (\begin{matrix} v \\ w \\ \frac{1}{τ} Δ (u + β v - η θ) - \frac{1}{τ} w \\ η Δ (v + τ w) - γ \nabla \cdot q \\ - \frac{1}{τ_{0}} (q + κ \nabla θ) \end{matrix}) \end{matrix}$ (the dot stands for the usual inner product in $R^{N}$ ).
Remark 2.2.
It can be seen that this is an equivalent norm to the natural one in $H$ . Observe also that the norms and spaces are slightly different from the Fourier case studied in [28] (see also Remark 2.1 of [28]).

We consider (2.4) in the Hilbert space $H$ with domain $\begin{array}{rcl} (2.5) & D (A) = {(u, v, w, θ, q) \in H; w, θ, q \in H^{1} (R^{N}), u + β v - η θ, v + τ w \in H^{2} (R^{N})} . \end{array}$ Theorem 2.3.
Under the condition $0 < τ < β$ , the operator $A$ is the generator of a $C_{0}$ -semigroup on $H$ . In particular, for any $U_{0} \in D (A)$ , there exists a unique solution $U \in C^{1} ([0, + \infty); H) \cap C ([0, + \infty); D (A))$ satisfying ( 2.4 ).
Proof.
The proof follows the same steps as the proof of Theorem 2.2 of [28]. As in that theorem, we are going to work with a perturbed problem, which now is $\begin{matrix} \{\begin{matrix} \frac{d}{d t} U (t) = A^{B} U (t), t \in [0, + \infty) \\ U (0) = U_{0} \end{matrix} \end{matrix}$ where $\begin{matrix} A^{B} (\begin{matrix} u \\ v \\ w \\ θ \\ q \end{matrix}) = (A + B) (\begin{matrix} u \\ v \\ w \\ θ \\ q \end{matrix}) = (\begin{matrix} v \\ w \\ \frac{1}{τ} Δ (u + β v - η θ) - \frac{1}{τ} w - \frac{1}{τ} u - v - \frac{1}{τ^{2}} v \\ η Δ (v + τ w) - γ \nabla \cdot q \\ - \frac{1}{τ_{0}} (q + κ \nabla θ) \end{matrix}) . \end{matrix}$

Using the same ideas as in Theorem 2.2 of [28], we can see that, for any $U \in D (A^{B})$ , $\begin{matrix} Re {⟨ A^{B} U, U ⟩}_{H} = - (β - τ) \int_{R^{N}} | \nabla v |^{2} d x - \frac{1}{τ} \int_{R^{N}} | v |^{2} d x - \frac{γ}{κ} \int_{R^{N}} | q |^{2} d x ⩽ 0 \end{matrix}$ since $0 < τ < β$ . Hence, the operator $A^{B}$ is dissipative when $0 < τ < β$ .

We are now going to prove that $(λ Id - A^{B})$ is surjective in $H$ for $\begin{matrix} (2.6) & λ_{} = \frac{- 1 + \sqrt{1 + 4 τ}}{2 τ} > 0, \end{matrix}$ which means that $A^{B}$ is maximal in $H$ . We consider a given $F = (f, g, h, j, p) \in H$ . We want to see that there exists a unique $U = (u, v, w, θ, q) \in D (A)$ such that $(λ Id - A^{B}) U = F$ , that is $\begin{array}{l} λ u - v = f \in H^{1} (R^{N}) \\ λ v - w = g \in H^{1} (R^{N}) \\ (2.7) & λ w - \frac{1}{τ} Δ (u + β v - η θ) + \frac{1}{τ} w + \frac{1}{τ} u + v + \frac{1}{τ^{2}} v = h \in L^{2} (R^{N}) \\ (2.8) & λ θ - η Δ (v + τ w) + γ \nabla \cdot q = j \in L^{2} (R^{N}) \\ (2.9) & (λ τ_{0} + 1) q + κ \nabla θ = τ_{0} p \in {(L^{2} (R^{N}))}^{N} . \end{array}$ By taking $v = λ u - f$ , $w = λ v - g = λ^{2} u - λ f - g$ and plugging it into (2.7), (2.8) and (2.9) we arrive at the following system: $\begin{array}{l} (2.10) & (λ^{3} τ + λ^{2} + λ \frac{τ^{2} + 1}{τ} + 1) u - (1 + λ β) Δ u + η Δ θ \\ = (λ^{2} τ + λ + \frac{τ^{2} + 1}{τ}) f - β Δ f + (λ τ + 1) g + τ h \\ - λ η (1 + τ λ) Δ u + λ θ + γ \nabla \cdot q \\ (2.11) & = j - η (1 + λ τ) Δ f - τ η Δ g \\ (2.12) & \frac{γ}{κ} (λ τ_{0} + 1) q + γ \nabla θ = \frac{γ}{κ} τ_{0} p . \end{array}$ In order to solve this problem, we consider its weak formulation given by the bilinear form $\begin{matrix} M : (H^{1} (R^{N}) \times H^{1} (R^{N}) \times {(H^{1} (R^{N}))}^{N}) \times (H^{1} (R^{N}) \times H^{1} (R^{N}) \times {(H^{1} (R^{N}))}^{N}) ⟶ R \end{matrix}$ given by $\begin{array}{rcl} M ((u, θ, q), (φ, χ, ϕ)) & = & (λ^{3} τ + λ^{2} + λ \frac{τ^{2} + 1}{τ} + 1) \int_{R^{N}} u \overline{φ} d x \\ + (1 + β λ) \int_{R^{N}} \nabla u \cdot \nabla \overline{φ} d x - η \int_{R^{N}} \nabla θ \cdot \nabla \overline{φ} d x \\ + λ η (1 + τ λ) \int_{R^{N}} \nabla u \cdot \nabla \overline{χ} d x + λ \int_{R^{N}} θ \overline{χ} d x + γ \int_{R^{N}} (\nabla \cdot q) \overline{χ} d x \\ + \frac{γ}{κ} (τ_{0} λ + 1) \int_{R^{N}} q \overline{ϕ} d x + γ \int_{R^{N}} \nabla θ \cdot \overline{ϕ} d x \end{array}$ and the linear one $\begin{matrix} K : (H^{1} (R^{N}) \times H^{1} (R^{N}) \times {(H^{1} (R^{N}))}^{N}) ⟶ R \end{matrix}$ given by $\begin{array}{rcl} K (φ, χ, ϕ) & = & (λ^{2} τ + λ + \frac{τ^{2} + 1}{τ}) \int_{R^{N}} f \overline{φ} d x + β \int_{R^{N}} \nabla f \cdot \nabla \overline{φ} d x \\ + (λ τ + 1) \int_{R^{N}} g \overline{φ} d x + τ \int_{R^{N}} h \overline{φ} d x + \int_{R^{N}} j \overline{χ} d x \\ + η (1 + τ λ) \int_{R^{N}} \nabla f \cdot \nabla \overline{χ} d x + τ η \int_{R^{N}} \nabla g \cdot \nabla \overline{χ} d x + \frac{γ τ_{0}}{κ} \int_{R^{N}} p \cdot \overline{ϕ} d x . \end{array}$ We can see that $M$ is coercive. Integrating by parts using that $u, θ \in H^{1} (R^{N})$ and $q \in {(H^{1} (R^{N}))}^{N}$ , we obtain: $\begin{array}{l} Re (M ((u, θ, q), (u, θ, q))) \\ = (λ^{3} τ + λ^{2} + λ \frac{τ^{2} + 1}{τ} + 1) \int_{R^{N}} | u |^{2} d x \\ + (1 + β λ) \int_{R^{N}} | \nabla u |^{2} d x \\ + η (τ λ^{2} + λ - 1) \int_{R^{N}} \nabla u \cdot \nabla \overline{θ} + λ \int_{R^{N}} | θ |^{2} d x + \frac{γ}{κ} (λ τ_{0} + 1) \int_{R^{N}} | q |^{2} d x . \end{array}$ Taking $λ_{} = \frac{- 1 + \sqrt{1 + 4 τ}}{2 τ} > 0$ (see (2.6)) the third term vanishes and hence, $\begin{matrix} Re (M ((u, θ, q), (u, θ, q))) ⩾ C (‖ u ‖_{H^{1} (R^{N})}^{2} + ‖ θ ‖_{H^{1} (R^{N})}^{2} + ‖ q ‖_{{(H^{1} (R^{N}))}^{N}}) \end{matrix}$ for $C = min {λ_{}^{3} τ + λ_{}^{2} + λ_{} \frac{τ^{2} + 1}{τ} + 1, 1 + β λ_{}, λ_{}, λ_{} τ_{0} + 1}$ .

We can also see that $M$ is bounded using the Hölder inequality: $\begin{array}{l} | M ((u, θ, q), (φ, χ, ϕ)) | \\ ⩽ | λ^{3} τ + λ^{2} + λ \frac{τ^{2} + 1}{τ} + 1 | ‖ u ‖_{L^{2} (R^{N})} \cdot ‖ φ ‖_{L^{2} (R^{N})} \\ + | 1 + β λ | ‖ \nabla u ‖_{L^{2} (R^{N})} ‖ \nabla φ ‖_{L^{2} (R^{N})} + η ‖ \nabla θ ‖_{L^{2} (R^{N})} ‖ \nabla φ ‖_{L^{2} (R^{N})} \\ + λ η | 1 + τ λ | ‖ \nabla u ‖_{L^{2} (R^{N})} ‖ \nabla χ ‖_{L^{2} (R^{N})} ‖ + λ ‖ θ ‖_{L^{2} (R^{N})} ‖ χ ‖_{L^{2} (R^{N})} \\ + κ ‖ q ‖_{{(L^{2} (R^{N}))}^{N}} ‖ \nabla χ ‖_{L^{2} (R^{N})} \\ + | \frac{γ}{κ} (λ τ_{0} + 1) | ‖ q ‖_{{(L^{2} (R^{N}))}^{N}} ‖ ϕ ‖_{{(L^{2} (R^{N}))}^{N}} + γ ‖ \nabla θ ‖_{L^{2} (R^{N})} ‖ ϕ ‖_{{(L^{2} (R^{N}))}^{N}} . \end{array}$ Hence, $\begin{array}{rcl} | M ((u, θ, q), (φ, χ, ϕ)) | & ⩽ & C {(‖ u ‖_{H^{1} (R^{N})}^{2} + ‖ θ ‖_{H^{1} (R^{N})}^{2} + ‖ q ‖_{{(H^{1} (R^{N}))}^{N}})}^{1 / 2} \\ \times {(‖ φ ‖_{H^{1} (R^{N})}^{2} + ‖ χ ‖_{H^{1} (R^{N})}^{2} + ‖ ϕ ‖_{{(H^{1} (R^{N}))}^{N}})}^{1 / 2} \end{array}$ for some $C > 0$ (for any $λ > 0$ and, in particular, for $λ = λ_{} > 0$ ).

Finally, as $f, g \in H^{1} (R^{N})$ , $h \in L^{2} (R^{N})$ , $p \in L^{2} {(R^{N})}^{N}$ , we can see that $K$ is bounded: $\begin{matrix} | K (φ, χ, ϕ) | & ⩽ | λ^{2} τ + λ + τ + \frac{1}{τ} | ‖ f ‖_{L^{2} (R^{N})} ‖ φ ‖_{L^{2} (R^{N})} + | β | ‖ \nabla f ‖_{L^{2} (R^{N})} ‖ \nabla φ ‖_{L^{2} (R^{N})} \\ + (| τ | ‖ h ‖_{L^{2} (R^{N})} + | τ λ + 1 | ‖ g ‖_{L^{2} (R^{N})} + ‖ j ‖_{L^{2} (R^{N})}) ‖ φ ‖_{L^{2} (R^{N})} \\ + (| η (1 + λ τ) | ‖ \nabla f ‖_{L^{2} (R^{N})} + | τ η | ‖ \nabla g ‖_{L^{2} (R^{N})}) ‖ \nabla χ ‖_{L^{2} (R^{N})} \\ + | \frac{γ}{κ} τ_{0} | ‖ p ‖_{{(L^{2} (R^{N}))}^{N}} ‖ ϕ ‖_{{(L^{2} (R^{N}))}^{N}} \\ ⩽ C {(‖ φ ‖_{H^{1} (R^{N})}^{2} + ‖ χ ‖_{H^{1} (R^{N})}^{2} + ‖ ϕ ‖_{{(H^{1} (R^{N}))}^{N}}^{2})}^{1 / 2} \end{matrix}$ for some $C > 0$ (for any $λ > 0$ and, in particular, for $λ = λ_{} > 0$ ). So, by the Lax–Milgram theorem, there exists a unique $(u, θ, q) \in H^{1} (R^{N}) \times H^{1} (R^{N}) \times {(H^{1} (R^{N}))}^{N}$ such that $\begin{matrix} M ((u, θ, q), (φ, χ, ϕ)) = K (φ, χ, ϕ) \forall (φ, χ, ϕ) \in H^{1} (R^{N}) \times H^{1} (R^{N}) \times {(H^{1} (R^{N}))}^{N} . \end{matrix}$ By the same arguments as in [1], this mild solution is, indeed, a strong solution of (2.10)–(2.12). This means the operator $λ_{*} Id - A^{B}$ is maximal in $H$ , as we claimed. Hence, by the Lummer–Phillips Theorem we can conclude that $A^{B}$ is the generator of a $C_{0}$ -semigroup of contractions on $H$ . And, as $A$ is a bounded perturbation of $A^{B}$ in $H$ , we can conclude that $A$ is the generator of a $C_{0}$ -semigroup on $H$ (see [25]). The regularity of the solution is a consequence of this fact (see [5] or [25]). □

2.1. Well-posedness in the critical case $τ = β$

We now consider the critical case $τ = β$ . As we have seen in Remark 2.1, (2.3) is not an equivalent norm to the usual one in $H$ if $τ = β$ . But, as we will see, we can adapt the same kind of proof as in Theorem 2.3 to this situation and prove the well-posedness of the problem in the critical case, which will be stated in Theorem 2.6 below. To do so, and following the idea of works as [17] or [16], among others, we use the change of variables $z = u + τ u_{t}$ , which is very natural to consider in the system (1.6) when $β = τ$ . Also, as in the subcritical case, we can take $a = 1$ without loss of generality. Then, we can recast system (1.6) as $\begin{array}{rcl} (2.13) & \{\begin{matrix} z_{t t} - Δ z + η Δ θ = 0, \\ θ_{t} + γ \nabla \cdot q - η Δ z_{t} = 0, \\ τ_{0} q_{t} + q + κ \nabla θ = 0 . \end{matrix} \end{array}$

If $z = u + τ u_{t}$ , observe that $u_{t} = \frac{1}{τ} (z - u)$ . Hence, considering $y = z_{t}$ , we can write (2.13) (and the corresponding initial conditions (2.1) after applying this change of variables to them) as a first order evolution problem: $\begin{matrix} (2.14) & \{\begin{matrix} \frac{d}{d t} Z (t) = A Z (t), t \in [0, + \infty) \\ Z (0) = Z_{0} \end{matrix} \end{matrix}$ where $\begin{matrix} A Z = A (\begin{matrix} u \\ z \\ y \\ θ \\ q \end{matrix}) = (\begin{matrix} - \frac{1}{τ} u + \frac{1}{τ} z \\ y \\ Δ (z - η θ) \\ η Δ y - γ \nabla \cdot q \\ - \frac{1}{τ_{0}} (q + κ \nabla θ) \end{matrix}) \end{matrix}$ The operator $A$ is defined in $\begin{matrix} (2.15) & H = H^{1} (R^{N}) \times H^{1} (R^{N}) \times L^{2} (R^{N}) \times L^{2} (R^{N}) \times {(L^{2} (R^{N}))}^{N} \end{matrix}$ and has domain $\begin{matrix} (2.16) & D (A) = {Z \in H; q, θ \in H^{1} (R^{N}), z - η θ, y \in H^{2} (R^{N})} \end{matrix}$

Remark 2.4.
Observe that $Z = (u, z, y, θ, q) \in H$ defined in (2.15) is equivalent to $U = (u, v, w, θ, q) \in H$ defined in (2.2). Also, $Z \in D (A)$ defined in (2.16) is equivalent to $U \in D (A)$ defined in (2.5) for $β = τ$ . We notice that, although the spaces defined in (2.15) and (2.2) are the same, we have used different names for them as the variables we are using in each case (Z and U) are different.

We consider the following inner product in $H$ : $\begin{matrix} (2.17) & \begin{matrix} {⟨ (u, z, y, θ, q), (u_{1}, z_{1}, y_{1}, θ_{1}, q_{1}) ⟩}_{H} \\ = τ \int_{R^{N}} u {\overline{u}}_{1} d x + τ \int_{R^{N}} \nabla u \cdot \nabla {\overline{u}}_{1} d x + \int_{R^{N}} z {\overline{z}}_{1} d x \\ + \int_{R^{N}} \nabla z \cdot \nabla {\overline{z}}_{1} d x + \int_{R^{N}} y {\overline{y}}_{1} d x + \int_{R^{N}} θ {\overline{θ}}_{1} d x + \frac{γ τ_{0}}{κ} \int_{R^{N}} q \cdot {\overline{q}}_{1} d x \end{matrix} \end{matrix}$ We remark that this inner product is equivalent to the usual one in $H$ .

We now consider the following perturbation of $A$ : $\begin{matrix} A^{B} Z = (A + B) (\begin{matrix} u \\ z \\ y \\ θ \\ q \end{matrix}) = (\begin{matrix} - \frac{1}{τ} u + \frac{1}{τ} z \\ y - u \\ Δ (z - η θ) - z \\ η Δ y - γ \nabla \cdot q \\ - \frac{1}{τ_{0}} (q + κ \nabla θ) \end{matrix}) . \end{matrix}$ We observe that $B$ is a bounded perturbation in $H$ and also that $A^{B}$ is closed and densely defined. We are going to prove that it is the generator of a $C_{0}$ -semigroup of contractions. To do so, we are going to use the following result, which is a corollary of the Lumer–Phillips Theorem.
Theorem 2.5 (Corollary of Lumer–Phillips Theorem, [20]).

Let A be a linear operator with dense domain $D (A)$ in a Hilbert space H. If A is dissipative and $0 \in ρ (A)$ (the resolvent set of A), then A is the infinitesimal generator of a $C_{0}$ -semigroup of contractions on H.

With the previous result, we are now able to prove the well-posedness of the problem in the critical case, as the following theorem states:

Theorem 2.6.
Assume that $τ = β > 0$ (critical case). Then, the operator $A$ is the generator of a $C_{0}$ -semigroup on $H$ . In particular, for any $Z_{0} \in D (A)$ , there exists a unique solution $Z \in C^{1} ([0, + \infty); H) \cap C ([0, + \infty); D (A))$ satisfying ( 2.14 ).
Proof.
We start by proving that $A^{B}$ fulfills the requirements of Theorem 2.5 and, hence, it generates a $C_{0}$ -semigroup of contractions on $H$ .

First, it is easy to check that with the previous inner product (2.17) the operator $A^{B}$ is dissipative, as we have $\begin{matrix} Re {⟨ A^{B} Z, Z ⟩}_{H} = - \int_{R^{N}} | u |^{2} - \int_{R^{N}} | \nabla u |^{2} - \frac{γ}{κ} \int_{R^{N}} | q |^{2} ⩽ 0 . \end{matrix}$ Second, we can see that for any $F = (f, g, h, j, p) \in H$ we have unique $Z \in D (A)$ such that $A^{B} Z = F$ . This can be done using the Lax–Milgram Theorem and seeing that the unique mild solution we obtain is, indeed, a strong solution of our problem (as in the proof of Theorem 2.3). Hence, $0 \in ρ (A^{B})$ . Theorem 2.5 allows us to conclude that $A_{B}$ is the generator of a $C_{0}$ -semigroup of contractions on $H$ .

As $B$ is a bounded perturbation on $H$ , this means that $A$ generates a $C_{0}$ -semigroup on $H$ (see [25]). The regularity of the solution for $β = τ$ is a consequence of this fact (see [5] or [25]). □

3. Stability results: The case $0 < τ < β$

We show some stability results of a norm related with the solution of (1.6). Without loss of generality, we take again $a = 1$ and, now, $γ = κ$ . We discuss the two cases separatedly: $0 < τ < β$ in the present section and $0 < τ = β$ in Section 4. In this section, we show that the $L^{2}$ -norm of the norm related to the solution decays with the rate ${(1 + t)}^{- N / 4}$ . But, unlike the Fourier model studied in [28], the Cattaneo model yields a regularity loss. As we said in Section 1, this phenomenon can be expected for this type of models.

Using the change of variables $v = u_{t}$ and $w = u_{t t}$ , we may rewrite system (1.6) as $\begin{array}{l} (3.1a) & u_{t} - v = 0, \\ (3.1b) & v_{t} - w = 0, \\ (3.1c) & τ w_{t} + w - Δ u - β Δ v + η Δ θ = 0, \\ (3.1d) & θ_{t} + κ \nabla \cdot q - η τ Δ w - η Δ v = 0, \\ (3.1e) & τ_{0} q_{t} + q + κ \nabla θ = 0, \end{array}$ with the initial data $\begin{matrix} u (x, 0) = u_{0} (x), v (x, 0) = v_{0} (x), w (x, 0) = w_{0} (x), \\ θ (x, 0) = θ_{0} (x), q (x, 0) = q_{0} . \end{matrix}$ Taking the Fourier transform of the previous system (3.1), we obtain $\begin{array}{l} (3.2a) & {\hat{u}}_{t} - \hat{v} = 0, \\ (3.2b) & {\hat{v}}_{t} - \hat{w} = 0, \\ (3.2c) & τ {\hat{w}}_{t} + \hat{w} + | ξ |^{2} \hat{u} + β | ξ |^{2} \hat{v} - η | ξ |^{2} \hat{θ} = 0, \\ (3.2d) & {\hat{θ}}_{t} + κ i ξ \cdot \hat{q} + η τ | ξ |^{2} \hat{w} + η | ξ |^{2} \hat{v} = 0, \\ (3.2e) & τ_{0} {\hat{q}}_{t} + \hat{q} + i ξ κ \hat{θ} = 0, \end{array}$ with the initial data $\begin{array}{l} \hat{u} (ξ, 0) = {\hat{u}}_{0} (ξ), \hat{v} (ξ, 0) = v_{0} (ξ), \hat{w} (ξ, 0) = {\hat{w}}_{0} (ξ), \\ \hat{θ} (ξ, 0) = \hat{θ} (ξ), \hat{q} (ξ, 0) = \hat{q} (ξ) . \end{array}$

3.1. The Lyapunov functional

Our main result in this section follows the same ideas as in the Fourier problem in [28], and reads as:

Theorem 3.1.
Assume that $0 < τ < β$ . Let $V_{C} (x, t) = {(τ u_{t t} + u_{t}, \nabla (τ u_{t} + u), \nabla u_{t}, θ, q)}^{T} (x, t)$ , where $(u, θ, q)$ is the solution of ( 1.6 ). Let s be a nonnegative integer and let $V_{C}^{0} = V_{C} (x, 0) \in H^{s} (R^{N}) \cap L^{1} (R^{N})$ . Then, the following decay estimate $\begin{matrix} (3.3) & {‖ \nabla^{k} V_{C} (t) ‖}_{L^{2} (R^{N})} ⩽ C {(1 + t)}^{- N / 4 - k / 2} {‖ V_{C}^{0} ‖}_{L^{1} (R^{N})} + C {(1 + t)}^{- ℓ / 2} {‖ \nabla^{k + ℓ} V_{C}^{0} ‖}_{L^{2} (R^{N})} \end{matrix}$ holds, for any $t ⩾ 0$ and $0 ⩽ k + ℓ ⩽ s$ , where C and c are two positive constants independent of t and $V_{C}^{0}$ .
Remark 3.2.
Observe that we have the same decay rate as in the Fourier case when $0 < τ < β$ but, in that case, there was not any regularity loss phenomenon, although the norm and the spaces of the functional framework are not the same in both cases (see Theorem 3.1 of [28]). Hence, the fact that the heat coupling is of Cattaneo type does not imply a slower decay rate, but a loss of regularity in the solution. Such regularity loss is one of the major difficulties for proving global existence in nonlinear problems. In many cases higher regularity assumption is needed to close the estimates. See for instance [15] and [33].
Remark 3.3.
Note that after the first version of our paper, the authors in [41] have proved a decay rate of a regularity-loss type for the Cauchy problem of the standard linear solid model with Gurtin–Pipkin thermal law in the subcritical case $0 < τ < β$ . As it is known, the Gurtin–Pipkin law with a negative exponential is nothing but the Maxwell–Cattaneo law. However, the decay result obtained in [41] seems not optimal compared to (3.3). We believe that by following the estimates derived in this paper, it might be possible to obtain the same decay rate (3.3) for the Gurtin–Pipkin law and, hence, improve the result in [41].

To justify rigorously the decay rate (3.3) without computing the solution we use the energy method in the Fourier space to build an appropriate Lyapunov functional that will give us the decay rate of the Fourier image of $V_{C} (x, t)$ and, hence, the above decay rate. In Section 3.2, we use eigenvalues expansion to show that the decay rate obtained in Theorem 3.1 is optimal.

We will follow the same ideas as in Section 3.1 of [28]. The decay rate of the Fourier image of $V_{C}$ is the one given in the following proposition.
Proposition 3.4.
Assume that $0 < τ < β$ . Let $V_{C} (x, t) = {(τ u_{t t} + u_{t}, \nabla (τ u_{t} + u), \nabla u_{t}, θ, q)}^{T} (x, t)$ , where $(u (x, t), θ (x, t), q (x, t))$ is the solution of ( 1.6 ) for $η > 0$ . Then we have $\begin{matrix} (3.4) & {| \hat{V_{C}} (ξ, t) |}^{2} ⩽ C e^{- c ρ (ξ) t} {| \hat{V_{C}} (ξ, 0) |}^{2}, \end{matrix}$ where $\begin{matrix} ρ (ξ) = \frac{| ξ |^{2}}{1 + | ξ |^{2} + | ξ |^{4}}, \end{matrix}$ and C and c are two positive constants.

The proof of this the above proposition will be done using the following lemmas, which are similar to those in Section 3.1 of [28].
Lemma 3.5.
Assume that $0 < τ < β$ . Let $(\hat{u}, \hat{v}, \hat{w}, \hat{θ}, \hat{q}) (ξ, t)$ be the solution of ( 3.2 ). The energy functional associated to this system is defined as: $\begin{matrix} (3.5) & \hat{E} (ξ, t) : = \frac{1}{2} {| \hat{v} + τ \hat{w} |^{2} + | ξ |^{2} | \hat{u} + τ \hat{v} |^{2} + τ (β - τ) | ξ |^{2} | \hat{v} |^{2} + | \hat{θ} |^{2} + τ_{0} | \hat{q} |^{2}} . \end{matrix}$ Then for all $t ⩾ 0$ , $\begin{matrix} \hat{E} (ξ, t) ⩾ 0 \end{matrix}$ and there exist two positive constants $d_{1}$ and $d_{2}$ such that $\begin{matrix} (3.6) & d_{1} {| \hat{V_{C}} (ξ, t) |}^{2} ⩽ \hat{E} (ξ, t) ⩽ d_{2} {| \hat{V_{C}} (ξ, t) |}^{2} \end{matrix}$ and $\begin{matrix} (3.7) & \frac{d}{d t} \hat{E} (ξ, t) = - (β - τ) | ξ |^{2} | \hat{v} |^{2} - | \hat{q} |^{2}, \end{matrix}$ where $\begin{matrix} {| \hat{V_{C}} (ξ, t) |}^{2} = | \hat{v} + τ \hat{w} |^{2} + | ξ |^{2} | \hat{u} + τ \hat{v} |^{2} + | ξ |^{2} | \hat{v} |^{2} + | \hat{θ} |^{2} + | \hat{q} |^{2} . \end{matrix}$

The proof of this Lemma 3.5 is similar to the one of Lemma 3.4 of [28]. So, we omit the details. Proof of Proposition 3.4.
As in the proof of Proposition 3.3 of [28], we consider the functionals $\begin{matrix} F_{1} (ξ, t) = Re ((\bar{\hat{u}} + τ \bar{\hat{v}}) (\hat{v} + τ \hat{w})) \end{matrix}$ and $\begin{matrix} F_{2} (ξ, t) = - τ Re (\bar{\hat{v}} (\hat{v} + τ \hat{w})) . \end{matrix}$ We observe that the estimates (3.16) and (3.19) in [28] for $F_{1} (ξ, t)$ and $F_{2} (ξ, t)$ , respectively, remain valid for the new system (3.2). We include them here for the completeness of this work: $\begin{matrix} (3.8) & \frac{d}{d t} F_{1} (ξ, t) + (1 - ϵ_{0}) | ξ |^{2} | \hat{u} + τ \hat{v} |^{2} ⩽ | \hat{v} + τ \hat{w} |^{2} + C (ϵ_{0}) | ξ |^{2} (| \hat{v} |^{2} + | \hat{θ} |^{2}) . \end{matrix}$ for any $ϵ_{0} > 0$ , and $\begin{matrix} (3.9) & \frac{d}{d t} F_{2} (ξ, t) + (1 - ϵ_{1}) | \hat{v} + τ \hat{w} |^{2} ⩽ C (ϵ_{1}, ϵ_{2}, ϵ_{3}) (1 + | ξ |^{2}) | \hat{v} |^{2} + ϵ_{2} | ξ |^{2} | \hat{u} + τ \hat{v} |^{2} + ϵ_{3} | ξ |^{2} | \hat{θ} |^{2} \end{matrix}$ for any $ϵ_{1}, ϵ_{2}, ϵ_{3} > 0$ . The estimate (3.17) of [28] also holds, that is $\begin{matrix} (3.10) & \frac{d}{d t} F_{1} (ξ, t) + | ξ |^{2} | \hat{u} + τ \hat{v} |^{2} - | \hat{v} + τ \hat{w} |^{2} = | ξ |^{2} (τ - β) Re (\hat{v} (\bar{\hat{u}} + τ \bar{\hat{v}})) + η | ξ |^{2} Re (\hat{θ} (\bar{\hat{u}} + τ \bar{\hat{v}})) . \end{matrix}$

We can see that (3.9) can be estimated as $\begin{array}{l} \frac{d}{d t} F_{2} (ξ, t) + (1 - ε_{1}) | \hat{v} + τ \hat{w} |^{2} \\ (3.11) & ⩽ C (ε_{0}, ε_{1}, ε_{2}) (1 + | ξ |^{2} + | ξ |^{4}) | \hat{v} |^{2} + ε_{0} | \hat{θ} |^{2} + ε_{2} \frac{| ξ |^{2}}{1 + | ξ |^{2}} | \hat{u} + τ \hat{v} |^{2} \end{array}$ for any $ε_{0}, ε_{1}, ε_{2} > 0$ . The previous estimate comes from the fact that: $\begin{array}{l} \frac{d}{d t} F_{2} (ξ, t) + | \hat{v} + τ \hat{w} |^{2} - τ (β - τ) | ξ |^{2} | \hat{v} |^{2} \\ = τ | ξ |^{2} Re {(\hat{u} + τ \hat{v}) \bar{\hat{v}}} + Re {(\hat{v} + τ \hat{w}) \bar{\hat{v}}} - η τ | ξ |^{2} Re (\hat{θ} \bar{\hat{v}}) \end{array}$ (see proof of Lemma 3.7 in [28]). Using Young’s inequality, we have $\begin{array}{rcl} | τ | ξ |^{2} Re {(\hat{u} + τ \hat{v}) \bar{\hat{v}}} | ⩽ ε_{2} \frac{| ξ |^{2}}{1 + | ξ |^{2}} | \hat{u} + τ \hat{v} |^{2} + C (ε_{2}) | ξ |^{2} (1 + | ξ |^{2}) | \hat{v} |^{2} \end{array}$ and $\begin{matrix} | - η τ | ξ |^{2} Re (\hat{θ} \bar{\hat{v}}) | ⩽ ε_{0} | \hat{θ} |^{2} + C (ε_{0}) | ξ |^{4} | \hat{v} |^{2} \end{matrix}$ and $\begin{matrix} | Re {(\hat{v} + τ \hat{w}) \bar{\hat{v}}} | ⩽ ε_{1} {| (\hat{v} + τ \hat{w}) |}^{2} + C (ε_{1}) | \hat{v} |^{2} . \end{matrix}$ Collecting the above estimates, we obtain (3.11).

Our goal now is to build dissipative terms for $| \hat{θ} |^{2}$ . Indeed, multiplying equation (3.2e) by $- i ξ \hat{θ}$ we get $\begin{matrix} (3.12) & - τ_{0} ⟨ {\hat{q}}_{t}, i ξ \hat{θ} ⟩ - ⟨ \hat{q}, i ξ \hat{θ} ⟩ + κ | ξ |^{2} | \hat{θ} |^{2} = 0, \end{matrix}$ where $⟨ \cdot, \cdot ⟩$ is the dot product in $C^{n}$ . Multiplying (3.2d) by $τ_{0} i ξ \cdot \bar{\hat{q}} = τ_{0} ⟨ i ξ, \hat{q} ⟩$ , we get $\begin{matrix} (3.13) & τ_{0} ⟨ i ξ {\hat{θ}}_{t}, \hat{q} ⟩ - κ τ_{0} | ξ \cdot \hat{q} |^{2} + η τ τ_{0} | ξ |^{2} ⟨ i ξ \hat{w}, \hat{q} ⟩ + τ_{0} η | ξ |^{2} ⟨ i ξ \hat{v}, \hat{q} ⟩ . \end{matrix}$

Summing up (3.12) and (3.13), and taking the real part, we get $\begin{matrix} (3.14) & \frac{d}{d t} τ_{0} Re ⟨ i ξ \hat{θ}, \hat{q} ⟩ + κ | ξ |^{2} | \hat{θ} |^{2} = τ_{0} κ | ξ \cdot \hat{q} |^{2} + Re ⟨ \hat{q}, i ξ + \hat{θ} ⟩ - η τ_{0} | ξ |^{2} Re ⟨ i ξ (\hat{v} + τ \hat{w}), \hat{q} ⟩ . \end{matrix}$

Now, by using first (3.2a) and (3.2b) and, then, (3.2e), we have $\begin{array}{rcl} ⟨ i ξ (\hat{v} + τ \hat{w}), \hat{q} ⟩ & = & ⟨ i ξ ({\hat{u}}_{t} + τ {\hat{v}}_{t}), \hat{q} ⟩ \\ = & \frac{d}{d t} ⟨ i ξ (\hat{u} + τ \hat{v}), \hat{q} ⟩ - ⟨ i ξ (\hat{u} + τ \hat{v}), {\hat{q}}_{t} ⟩ \\ = & \frac{d}{d t} ⟨ i ξ (\hat{u} + τ \hat{v}), \hat{q} ⟩ + \frac{1}{τ_{0}} ⟨ i ξ (\hat{u} + τ \hat{v}), \hat{q} ⟩ + \frac{κ}{τ_{0}} ⟨ i ξ (\hat{u} + τ \hat{v}), i ξ \hat{θ} ⟩ . \end{array}$ (recall that $τ_{0} > 0$ ). Plugging these into (3.14), we get $\begin{array}{rcl} \frac{d}{d t} F_{3} (ξ, t) + κ | ξ |^{2} | \hat{θ} |^{2} & = & τ_{0} κ | ξ \cdot \hat{q} |^{2} + Re ⟨ \hat{q}, i ξ \hat{θ} ⟩ \\ (3.15) & - η | ξ |^{2} Re ⟨ i ξ (\hat{u} + τ \hat{v}), \hat{q} ⟩ - κ η | ξ |^{2} Re ⟨ i ξ (\hat{u} + τ \hat{v}), i ξ \hat{θ} ⟩ \end{array}$ where $\begin{matrix} F_{3} (t) = τ_{0} Re ⟨ i ξ \hat{θ}, \hat{q} ⟩ + η τ_{0} | ξ |^{2} Re ⟨ i ξ (\hat{u} + τ \hat{v}), \hat{q} ⟩ . \end{matrix}$ Now, we see that $\begin{matrix} ⟨ i ξ (\hat{u} + τ \hat{v}), i ξ \hat{θ} ⟩ = (\hat{u} + τ \hat{v}) \bar{\hat{θ}} ⟨ i ξ, i ξ ⟩ = (\hat{u} + τ \hat{v}) \bar{\hat{θ}} | ξ |^{2} . \end{matrix}$ Consequently, (3.15) takes the form $\begin{array}{rcl} \frac{d}{d t} F_{3} (ξ, t) + κ | ξ |^{2} | \hat{θ} |^{2} & = & τ_{0} κ | ξ \cdot \hat{q} |^{2} + Re ⟨ \hat{q}, i ξ \hat{θ} ⟩ \\ (3.16) & - η | ξ |^{2} Re ⟨ i ξ (\hat{u} + τ \hat{v}), \hat{q} ⟩ - κ η | ξ |^{4} Re ((\hat{u} + τ \hat{v}) \bar{\hat{θ}}) . \end{array}$ Now, we define the functional $\begin{matrix} F_{4} (ξ, t) = κ | ξ |^{2} F_{1} (ξ, t) + F_{3} (ξ, t) . \end{matrix}$ Then, we have from (3.10) and (3.16) $\begin{array}{l} \frac{d}{d t} F_{4} (ξ, t) + κ | ξ |^{2} | \hat{θ} |^{2} + κ | ξ |^{4} | \hat{u} + τ \hat{v} |^{2} - κ | ξ |^{2} | \hat{v} + τ \hat{w} |^{2} \\ = τ_{0} κ | ξ \cdot \hat{q} |^{2} + Re ⟨ \hat{q}, i ξ \hat{θ} ⟩ - η | ξ |^{2} Re ⟨ i ξ (\hat{u} + τ \hat{v}), \hat{q} ⟩ \\ (3.17) & + κ | ξ |^{4} (τ - β) Re (\hat{v} (\bar{\hat{u}} + τ \bar{\hat{v}})) . \end{array}$ Using Cauchy–Schwarz inequality together with Young’s inequality, we have $\begin{array}{l} | ξ \cdot \hat{q} |^{2} ⩽ | ξ |^{2} | \hat{q} |^{2} \\ | ⟨ \hat{q}, i ξ \hat{θ} ⟩ | ⩽ | ξ | | \hat{θ} | | \hat{q} | ⩽ ε_{3} | ξ |^{2} | \hat{θ} |^{2} + C (ε_{3}) | \hat{q} |^{2} \end{array}$ and $\begin{array}{l} | - η | ξ |^{2} Re ⟨ i ξ (\hat{u} + τ \hat{v}), \hat{q} ⟩ | ⩽ \frac{ε_{4}}{2} | ξ |^{4} | \hat{u} + τ \hat{v} |^{2} + C (ε_{4}) | ξ |^{2} | \hat{q} |^{2}, \\ | κ | ξ |^{4} (τ - β) Re (\hat{v} (\bar{\hat{u}} + τ \bar{\hat{v}})) | ⩽ \frac{ε_{4}}{2} | ξ |^{4} | \hat{u} + τ \hat{v} |^{2} + C (ε_{4}) | ξ |^{4} | \hat{v} |^{2} \end{array}$ where $ε_{3}$ and $ε_{4}$ are arbitrary positive constants. Plugging the above estimates into (3.17), we get $\begin{array}{l} \frac{d}{d t} F_{4} (t) + (κ - ε_{3}) | ξ |^{2} | \hat{θ} |^{2} + (κ - ε_{4}) | ξ |^{4} | \hat{u} + τ \hat{v} |^{2} \\ (3.18) & ⩽ κ | ξ |^{2} | \hat{v} + τ \hat{w} |^{2} + C (ε_{3}, ε_{4}) (1 + | ξ |^{2}) | \hat{q} |^{2} + C (ε_{4}) | ξ |^{4} | \hat{v} |^{2} . \end{array}$ Now, define the functional $L (ξ, t)$ as $\begin{matrix} L (ξ, t) : = γ_{0} (1 + | ξ |^{2} + | ξ |^{4}) \hat{E} (ξ, t) + | ξ |^{2} F_{2} (ξ, t) + γ_{1} F_{4} (ξ, t), \end{matrix}$ where $γ_{0}$ and $γ_{1}$ are positive constants that will be fixed later. Taking the derivative of $L (ξ, t)$ with respect to t and using (3.7), (3.11) and (3.18), we obtain $\begin{array}{l} \frac{d}{d t} L (ξ, t) + ((1 - ε_{1}) - γ_{1} κ) | ξ |^{2} | \hat{v} + τ \hat{w} |^{2} \\ + (γ_{1} (κ - ε_{4}) - ε_{2}) | ξ |^{4} (| \hat{u} + τ \hat{v} |^{2}) + (γ_{1} (κ - ε_{3}) - ε_{0}) | ξ |^{2} | \hat{θ} |^{2} \\ + (γ_{0} (β - τ) - C (ε_{0}, ε_{1}, ε_{2}, ε_{4}) γ_{1}) | ξ |^{2} (1 + | ξ |^{2} + | ξ |^{4}) | \hat{v} |^{2} \\ + (γ_{0} - C (ε_{3}, ε_{4}) γ_{1}) (1 + | ξ |^{2} + | ξ |^{4}) | \hat{q} |^{2} ⩽ 0, \end{array}$ where Λ is a generic positive constant that depends on $ε_{i}$ for $i = 1, \dots, 4$ and $γ_{j}$ for $j = 0, 1$ , but independent on t and ξ. We have used the fact that $| ξ |^{2} / (1 + | ξ |^{2}) ⩽ | ξ |^{2}$ . Let us now fix the constants in the above estimate in such a way that all the coefficients in the previous inequality are strictly positive. First, we pick $ε_{1}$ and $ε_{3}$ and $ε_{4}$ small enough such that $\begin{matrix} ε_{1} < 1, ε_{3} < κ, ε_{4} < κ \end{matrix}$ After that, we choose $γ_{1}$ small enough such that $\begin{matrix} γ_{1} < \frac{1 - ε_{1}}{κ} . \end{matrix}$ Next, we choose $ε_{0}$ and $ε_{2}$ small enough such that $\begin{matrix} ε_{0} < γ_{1} (κ - ε_{3}), ε_{2} < γ_{1} (κ - ε_{4}) . \end{matrix}$ Finally and once all the above constants are fixed, we choose $γ_{0}$ large enough such that $\begin{matrix} γ_{0} > max {\frac{C (ε_{0}, ε_{1}, ε_{2}, ε_{4}) γ_{1}}{β - τ}, C (ε_{3}, ε_{4}) γ_{1}} . \end{matrix}$ Consequently, we deduce that there exists a positive constant $γ_{2}$ such that for all $t ⩾ 0$ , we have $\begin{array}{l} \frac{d}{d t} L (ξ, t) + γ_{2} | ξ |^{2} {| \hat{v} + τ \hat{w} |^{2} + | ξ |^{2} | \hat{u} + τ \hat{v} |^{2} + | \hat{θ} |^{2} + | ξ |^{2} | \hat{v} |^{2} + | \hat{q} |^{2}} ⩽ 0, \end{array}$ and, hence, $\begin{array}{l} (3.19) & \frac{d}{d t} L (ξ, t) + γ_{2} | ξ |^{2} \hat{E} (t) ⩽ 0 . \end{array}$ On the other hand using the Cauchy–Schwarz inequality together with the Young inequality, we deduce that, for $γ_{0}$ large enough, there exist two positive constants $γ_{3}$ and $γ_{4}$ such that $\begin{matrix} (3.20) & γ_{3} (1 + | ξ |^{2} + | ξ |^{4}) \hat{E} (ξ, t) ⩽ L (ξ, t) ⩽ γ_{4} (1 + | ξ |^{2} + | ξ |^{4}) \hat{E} (ξ, t), \forall t ⩾ 0 . \end{matrix}$ Consequently, combining (3.6), (3.19) and (3.20) and using Gronwall’s lemma, we deduce (3.4) because $L (ξ, t)$ and $| V_{C} (ξ, t) |^{2}$ are equivalent. □
Proof of Theorem 3.1.
The proof of Theorem 3.1 can be done with the same method as in the proof of Theorem 3.1 of [28].

Indeed, by Plancherel theorem and the estimate (3.4) it holds that $\begin{matrix} (3.21) & {‖ \nabla^{k} V_{C} (t) ‖}_{L^{2} (R^{N})}^{2} = \int_{R^{N}} | ξ |^{2 k} {| \hat{V_{C}} (ξ, t) |}^{2} d ξ ⩽ C \int_{R^{N}} | ξ |^{2 k} e^{- c ρ (ξ) t} {| \hat{V_{C}} (ξ, 0) |}^{2} d ξ, \end{matrix}$

We can see that the behavior of the function $ρ (ξ)$ is as follows: $\begin{matrix} ρ (ξ) ⩾ \{\begin{matrix} \frac{1}{3} | ξ |^{2}, & for | ξ | ⩽ 1, \\ \frac{1}{3} | ξ |^{- 2}, & for | ξ | ⩾ 1 . \end{matrix} \end{matrix}$

Then it is natural to write the integral on the right-hand side of (3.21) as $\begin{array}{l} \int_{R} | ξ |^{2 k} e^{- c ρ (ξ) t} {| \hat{V_{C}} (ξ, 0) |}^{2} d ξ \\ = \int_{| ξ | ⩽ 1} | ξ |^{2 k} e^{- c ρ (ξ) t} {| \hat{V_{C}} (ξ, 0) |}^{2} d ξ + \int_{| ξ | ⩾ 1} C | ξ |^{2 k} e^{- c ρ (ξ) t} {| \hat{V_{F}} (ξ, 0) |}^{2} d ξ \\ : = L_{1} + L_{2} . \end{array}$ Concerning the integral $L_{1}$ , we have $\begin{matrix} L_{1} & ⩽ C {‖ \hat{V_{F}^{0}} (t) ‖}_{L^{\infty}}^{2} \int_{| ξ | ⩽ 1} | ξ |^{2 k} e^{- \frac{c}{3} | ξ |^{2} t} d ξ ⩽ C {(1 + t)}^{- N / 2 - k} {‖ V_{F}^{0} ‖}_{L^{1} (R^{N})}^{2}, \end{matrix}$ where we have used the inequality (see Lemma 3.5 in [27]) $\begin{matrix} \int_{| ξ | ⩽ 1} | ξ |^{j} e^{- c | ξ |^{2} t} d ξ ⩽ C {(1 + t)}^{- (j + N) / 2}, for N ⩾ 1 . \end{matrix}$ On the other hand, using the estimate $\begin{matrix} sup_{| ξ | ⩾ 1} {| ξ |^{- 2 ℓ} e^{- c | ξ |^{- 2} t}} ⩽ C {(1 + t)}^{- ℓ}, \end{matrix}$ then the high frequency part $L_{2}$ corresponding to $| ξ | ⩾ 1$ is estimated as $\begin{array}{rcl} L_{2} & ⩽ & C sup_{| ξ | ⩾ 1} {| ξ |^{- 2 ℓ} e^{- c | ξ |^{- 2} t}} \int_{| ξ | ⩾ 1} | ξ |^{2 (k + ℓ)} {| \hat{V_{C}} (ξ, 0) |}^{2} d ξ \\ ⩽ & C {(1 + t)}^{- ℓ} {‖ \partial_{x}^{k + ℓ} V_{C}^{0} ‖}_{L^{2}}^{2} . \end{array}$ Collecting the estimates in the low and high frequency parts, we deduce (3.3). □
Remark 3.6.
It is known that the behavior of $ρ (ξ)$ for small frequencies ( $| ξ | ⩽ 1$ ) yields the decay rate of the solutions, as shown in Theorem 3.1. If we denote by $ρ_{F} (ξ)$ the corresponding exponent in the Fourier case (see [28]) and $ρ_{C} (ξ)$ the one for the Cattaneo one (see Proposition 3.4), we observe that both of them behave like $| ξ |^{2}$ for small frequencies, similar to the Fourier transform of the heat kernel. This leads to the decay rate ${(1 + t)}^{- N / 4}$ of the $L^{2}$ norm for initial data in $L^{1} (R^{N})$ . The only difference is for high frequencies ( $| ξ | ⩾ 1$ ). In this case, $ρ_{F} (ξ) \approx c$ , which yields the exponential decay rate of the high-frequency part of the $L^{2}$ norm, provided that the initial data are also in $L^{2} (R^{N})$ . However, we have $ρ_{C} (ξ) \approx | ξ |^{- 2}$ , which yields the decay rate ${(1 + t)}^{- ℓ / 2}$ provided that the initial data are in $H^{ℓ} (R^{N})$ . So, we can read this as a loss of regularity, since to recover the decay ${(1 + t)}^{- N / 4}$ as in the low frequency part, it is enough to take the initial data in the high-frequency part to be in $H^{ℓ} (R^{N})$ with $ℓ ⩾ [N / 2] + 1$ (recall that $[N / 2]$ represents the least integer part of $N / 2$ ).
Remark 3.7.
The estimates in Theorems 3.1 of [28] and (3.3) can be improved by allowing our initial data to be in some weighted spaces of $L^{1} (R^{N})$ where its total mass vanishes. See [34,38] and [33, Remark 3.2].
Remark 3.8.
Using the Gagliardo–Nirenberg inequality for $m > N / 2$ : $\begin{matrix} ‖ u ‖_{L^{\infty} (R^{N})} ⩽ C (N, m) ‖ u ‖_{L^{2} (R^{N})}^{1 - \frac{N}{2 m}} {‖ \nabla^{m} u ‖}_{L^{2} (R^{N})}^{\frac{N}{2 m}} \end{matrix}$ then we can easily obtain the decay estimates of the $L^{\infty}$ -norm. Hence, by interpolation inequalities, we obtain the decay of the $L^{p}$ -norms for $2 ⩽ p ⩽ \infty$ . Such estimates for $1 ⩽ p < 2$ are not clear because we do not know any bound for the $L^{1}$ -norm of the solution.

3.2. Asymptotic behaviour of the eigenvalues

In this section, we compute the asymptotic expansion of the eigenvalues by using the eigenvalues expansion method. This asymptotic behaviour will be in agreement with the decay rate seen in the previous section, as the following remark explains.

Remark 3.9.
The decay rate in Theorem 3.1 comes from the exponent $\begin{matrix} ρ (ξ) = \frac{| ξ |^{2}}{1 + | ξ |^{2} + | ξ |^{4}} \end{matrix}$ of Proposition 3.4. Observe that $ρ (ξ) \sim | ξ |^{2}$ when $| ξ | \to 0$ , and $ρ (ξ) \sim | ξ |^{- 2}$ when $| ξ | \to \infty$ . This is the asymptotic behaviour we expect for the real parts of the slowest eigenvalues in these cases (see Lemmas 3.10 and 3.11 below).

The first thing we want is to compute the characteristic equation associated with (3.2). Because of the presence of inner products, this equation is not as easy to obtain as in the Fourier case in [28]. But we proceed with some manipulations on this system in order to write it as a fifth order ODE, for which we will compute the corresponding characteristic equation straightforward.

First, by taking the dot product of equation (3.2e) with $i ξ$ , we get $\begin{matrix} (3.22) & τ_{0} i ξ \cdot {\hat{q}}_{t} + i ξ \cdot \hat{q} - κ | ξ |^{2} \hat{θ} = 0 . \end{matrix}$ Next, we take the time derivative of equation (3.2d), obtaining $\begin{matrix} (3.23) & {\hat{θ}}_{t t} + κ i ξ \cdot {\hat{q}}_{t} + η τ | ξ |^{2} {\hat{w}}_{t} + η | ξ |^{2} {\hat{v}}_{t} = 0 . \end{matrix}$ Plugging (3.22) into (3.23) gives $\begin{matrix} (3.24) & {\hat{θ}}_{t t} - \frac{κ}{τ_{0}} (i ξ \cdot \hat{q}) + \frac{κ^{2}}{τ_{0}} | ξ |^{2} \hat{θ} + η τ | ξ |^{2} {\hat{w}}_{t} + η | ξ |^{2} {\hat{v}}_{t} = 0 . \end{matrix}$ Using (3.2d), we obtain $\begin{matrix} (3.25) & - \frac{κ}{τ_{0}} i ξ \cdot \hat{q} = \frac{1}{τ_{0}} {\hat{θ}}_{t} + \frac{1}{τ_{0}} η τ | ξ |^{2} \hat{w} + \frac{1}{τ_{0}} η | ξ |^{2} \hat{v} . \end{matrix}$ Inserting (3.25) into (3.24), we get $\begin{matrix} (3.26) & {\hat{θ}}_{t t} + \frac{1}{τ_{0}} {\hat{θ}}_{t} + \frac{1}{τ_{0}} η τ | ξ |^{2} \hat{w} + \frac{1}{τ_{0}} η | ξ |^{2} \hat{v} + \frac{κ^{2}}{τ_{0}} | ξ |^{2} \hat{θ} + η τ | ξ |^{2} {\hat{w}}_{t} + η | ξ |^{2} {\hat{v}}_{t} = 0 . \end{matrix}$ Now, taking the second derivative of (3.2c), with respect to t, we get $\begin{matrix} (3.27) & τ {\hat{w}}_{t t t} + {\hat{w}}_{t t} + | ξ |^{2} {\hat{u}}_{t t} + β | ξ |^{2} {\hat{v}}_{t t} - η | ξ |^{2} {\hat{θ}}_{t t} = 0 . \end{matrix}$ Plugging (3.26) into (3.27), we obtain $\begin{array}{l} τ {\hat{w}}_{t t t} + {\hat{w}}_{t t} + | ξ |^{2} {\hat{u}}_{t t} + β | ξ |^{2} {\hat{v}}_{t t} + \frac{η}{τ_{0}} | ξ |^{2} {\hat{θ}}_{t} + \frac{η^{2} τ}{τ_{0}} | ξ |^{4} \hat{w} + \frac{η^{2}}{τ_{0}} | ξ |^{4} \hat{v} \\ (3.28) & + \frac{η κ^{2}}{τ_{0}} | ξ |^{4} \hat{θ} + η^{2} τ | ξ |^{4} {\hat{w}}_{t} + η^{2} | ξ |^{4} {\hat{v}}_{t} = 0 . \end{array}$ Now, taking the derivative of (3.2c) with respect to t, we get $\begin{matrix} (3.29) & η | ξ |^{2} {\hat{θ}}_{t} = τ {\hat{w}}_{t t} + {\hat{w}}_{t} + | ξ |^{2} {\hat{u}}_{t} + β | ξ |^{2} {\hat{v}}_{t} \end{matrix}$ Plugging (3.29) and (3.2c) into (3.28), we obtain $\begin{array}{l} τ {\hat{w}}_{t t t} + {\hat{w}}_{t t} + | ξ |^{2} {\hat{u}}_{t t} + β | ξ |^{2} {\hat{v}}_{t t} + \frac{η^{2} τ}{τ_{0}} | ξ |^{4} \hat{w} + \frac{η^{2}}{τ_{0}} | ξ |^{4} \hat{v} \\ + η^{2} τ | ξ |^{4} {\hat{w}}_{t} + η^{2} | ξ |^{4} {\hat{v}}_{t} \\ + \frac{1}{τ_{0}} (τ {\hat{w}}_{t t} + {\hat{w}}_{t} + | ξ |^{2} {\hat{u}}_{t} + β | ξ |^{2} {\hat{v}}_{t}) \\ (3.30) & + \frac{κ^{2}}{τ_{0}} | ξ |^{2} (τ {\hat{w}}_{t} + \hat{w} + | ξ |^{2} \hat{u} + β | ξ |^{2} \hat{v}) = 0 . \end{array}$ Now, we take the fourth derivative of (3.2a) and the third derivative of (3.2b), we obtain, respectively $\begin{matrix} (3.31) & \frac{d^{5}}{d t^{5}} \hat{u} = \frac{d^{4}}{d t^{4}} \hat{v} = w_{t t t} . \end{matrix}$ Now, plugging (3.31) into (3.30), we obtain $\begin{array}{l} τ \frac{d^{5}}{d t^{5}} \hat{u} + (1 + \frac{τ}{τ_{0}}) \frac{d^{4}}{d t^{4}} \hat{u} + (β | ξ |^{2} + \frac{1}{τ_{0}} + η^{2} τ | ξ |^{4} + \frac{κ^{2} τ}{τ_{0}} | ξ |^{2}) {\hat{u}}_{t t t} \\ + (| ξ |^{2} + \frac{η^{2} τ}{τ_{0}} | ξ |^{4} + \frac{κ^{2}}{τ_{0}} | ξ |^{2} + η^{2} | ξ |^{4} + \frac{β}{τ_{0}} | ξ |^{2}) {\hat{u}}_{t t} \\ + (\frac{η^{2}}{τ_{0}} | ξ |^{4} + \frac{1}{τ_{0}} | ξ |^{2} + \frac{κ^{2} β}{τ_{0}} | ξ |^{4}) {\hat{u}}_{t} + \frac{κ^{2}}{τ_{0}} | ξ |^{4} \hat{u} = 0 . \end{array}$ So, we have written system (3.2) as a fifth order ODE. Now, if we name $ζ = i | ξ |$ , the characteristic equation associate to the above ODE (and hence to system (3.2)) is $\begin{array}{l} λ^{5} + (\frac{1}{τ_{0}} + \frac{1}{τ}) λ^{4} + \frac{τ η^{2} τ_{0} ζ^{4} - (τ κ^{2} + β τ_{0}) ζ^{2} + 1}{τ τ_{0}} λ^{3} \\ (3.32) & + \frac{(τ + τ_{0}) η^{2} ζ^{2} - (β + τ_{0} + κ^{2})}{τ τ_{0}} ζ^{2} λ^{2} + \frac{(η^{2} + β κ^{2}) ζ^{2} - 1}{τ τ_{0}} ζ^{2} λ + \frac{κ^{2} ζ^{4}}{τ τ_{0}} = 0 . \end{array}$
Lemma 3.10.
Assume that $0 < τ < β$ . Then the real parts of the eigenvalues of ( 3.2 ) (i.e., the solutions of ( 3.32 )) satisfy for $| ξ | \to 0$ the following asymptotic expansion: $\begin{array}{rcl} (3.33) & \{\begin{matrix} Re (λ_{1}) (ξ) = Re (λ_{2}) (ξ) = \frac{τ - β}{2} | ξ |^{2} + O (| ξ |^{3}) \\ Re (λ_{3}) (ξ) = - κ^{2} | ξ |^{2} + O (| ξ |^{3}) \\ Re (λ_{4}) (ξ) = - \frac{1}{τ} + O (| ξ |^{2}) \\ Re (λ_{5}) (ξ) = - \frac{1}{τ_{0}} + O (| ξ |^{2}) . \end{matrix} \end{array}$
Proof.
We denote the eigenvalues of the previous system as $λ_{j} (| ξ |)$ , $j = 1, \dots, 5$ . We want to compute their asymptotic expansion as $| ξ | \to 0$ , that is: $\begin{matrix} λ_{j} (| ξ |) = λ_{0, j} + λ_{1, j} i | ξ | - λ_{2, j} | ξ |^{2} - λ_{3, j} i | ξ |^{3} + \dots, j = 1, \dots, 5 . \end{matrix}$ To do so, it will be more convenient to use the previous change of variables $ζ = i ξ$ and compute the asymptotic expansion of $λ_{j} (ζ)$ as $ζ \to 0$ : $\begin{matrix} (3.34) & λ_{j} (ζ) = λ_{0, j} + λ_{1, j} ζ + λ_{2, j} ζ^{2} + λ_{3, j} ζ^{3} + \dots, j = 1, \dots, 5 . \end{matrix}$ Plugging this expression into (3.32) and equating the coefficients of $ζ^{r}$ , $r = 0, 1, 2, \dots$ , we obtain an equation for each of the terms of the asymptotic expansion. That is, equating the coefficients of $ζ^{0}$ we obtain the following equation for $λ_{0, j}$ : $\begin{matrix} (3.35) & λ_{0, j}^{5} + (\frac{1}{τ_{0}} + \frac{1}{τ}) λ_{0, j}^{4} + \frac{1}{τ τ_{0}} λ_{0, j}^{3} = 0, j = 1, 2, 3, 4, 5 . \end{matrix}$

As it is a five order equation, its five solutions correspond to each of the (five) first terms of (3.34), $λ_{0, j}$ for $j = 1, \dots, 5$ . Hence, solving (3.35) we get $\begin{matrix} λ_{0, 1} = λ_{0, 2} = λ_{0, 3} = 0, λ_{0, 4} = \frac{1}{τ}, λ_{0, 5} = - \frac{1}{τ_{0}} . \end{matrix}$ As we want the leading term of the real part of the asymptotic expansions of the eigenvalues, we do not need to compute any other terms of the asymptotic expansions of the eigenvalues $λ_{4}$ and $λ_{5}$ . But we still have to proceed with additional terms of the asymptotic expansion of $λ_{j}$ for $j = 1, 2, 3$ . Now, equating the coefficients of $ζ^{1}$ and using the above values of $λ_{0, j} = 0$ , $j = 1, 2, 3$ , and after tedious computations, we obtain the following equation for their corresponding $λ_{1, j}$ : $\begin{matrix} (3.36) & \frac{λ_{1, j}^{3}}{τ τ_{0}} - \frac{λ_{1, j}}{τ τ_{0}} = 0, j = 1, 2, 3 . \end{matrix}$ Solving (3.36), we get $\begin{matrix} λ_{1, 1} = 1, λ_{1, 2} = - 1, λ_{1, 3} = 0 . \end{matrix}$ But we still have to compute at least another term of their asymptotic expansion to obtain the leading order real part of it. Applying the same procedure and using the above values of $λ_{i, j}$ , $j = 1, 2, 3$ , $i = 0, 1$ , we obtain $\begin{matrix} (3.37) & \frac{- β + τ + 2 λ_{2, j}}{τ τ_{0}} = 0, j = 1, 2 \end{matrix}$ and $\begin{matrix} λ_{2, 3} = κ^{2} . \end{matrix}$ Solving equation (3.37) yields $λ_{2, 1} = λ_{2, 2} = \frac{τ - β}{2}$ . Collecting all the previous results, we have the following asymptotic expansions of $λ_{j} (ζ)$ for $ζ \to 0$ : $\begin{array}{l} λ_{1, 2} (ζ) = \pm ζ - \frac{τ - β}{2} ζ^{2} + O (ζ^{3}), \\ λ_{3} (ζ) = κ^{2} ζ^{2} + O (ζ^{3}), \\ λ_{4} (ζ) = - \frac{1}{τ} + O (ζ^{2}), \\ λ_{5} (ζ) = - \frac{1}{τ_{0}} + O (ζ^{2}) . \end{array}$

If we now come back to ξ, we have the asymptotic behavior (3.33) for the real parts of the previous eigenvalues when $| ξ | \to 0$ . Observe that when $0 < τ < β$ (dissipative case for the standard linear model) all the previous real parts are negative. □

We now proceed with the asymptotic behaviour of the eigenvalues when $| ξ | \to \infty$ . We have the following lemma. Lemma 3.11.
Assume that $0 < τ < β$ . The real parts of the eigenvalues of ( 3.2 ) (i.e., the solutions of ( 3.32 )) satisfy for $| ξ | \to \infty$ , the following asymptotic expansion: $\begin{matrix} \{\begin{matrix} Re (λ_{1, 2}) (ξ) = - \frac{(β - τ) τ_{0}^{2} + κ^{2} τ^{2}}{2 τ^{2} η^{2} τ_{0}^{2}} | ξ |^{- 2} + O (| ξ |^{- 3}), \\ Re (λ_{3, 4, 5}) (ξ) = σ_{3, 4, 5} + O (| ξ |^{- 1}), \end{matrix} \end{matrix}$ with $σ_{3, 4, 5} < 0$ .
Proof.
Again, taking $ν = ζ^{- 1} = {(i | ξ |)}^{- 1}$ in (3.32), the characteristic equation is now $\begin{array}{l} μ^{5} + (\frac{1}{τ_{0}} + \frac{1}{τ}) ν^{2} μ^{4} + \frac{ν^{4} - (τ κ^{2} + β τ_{0}) ν^{2} + τ η^{2} τ_{0}}{τ τ_{0}} μ^{3} \\ (3.38) & + \frac{(τ + τ_{0}) η^{2} - (β + τ_{0} + κ^{2}) ν^{2}}{τ τ_{0}} ν^{2} μ^{2} + \frac{(η^{2} + β κ^{2}) - ν^{2}}{τ τ_{0}} ν^{4} μ + \frac{κ^{2} ν^{6}}{τ τ_{0}} = 0 . \end{array}$ Again, observe that if $λ (ξ)$ is a solution of (3.32), then $\begin{matrix} μ (ν) = ν^{2} λ = ζ^{- 2} λ = - | ξ |^{- 2} λ \end{matrix}$ is a solution of (3.38). Proceeding in the same way as in Lemma 3.10, we can now compute the asymptotic expansion $\begin{matrix} μ_{j} (ν) = μ_{j}^{0} + μ_{j}^{1} ν + μ_{j}^{2} ν^{2} + μ_{j}^{3} ν^{3} + μ_{j}^{4} ν^{4} + \dots, j = 1, \dots, 5 \end{matrix}$ of the roots of the characteristic polynomial (3.38) when $ν \to 0$ (that is, when $| ξ | \to \infty$ ): $\begin{array}{l} μ_{1, 2} (ν) = \pm η i + \frac{τ κ^{2} + β τ_{0}}{2 τ η τ_{0}} i ν^{2} + (\frac{(β - τ) τ_{0}^{2} + κ^{2} τ^{2}}{2 τ^{2} η^{2} τ_{0}^{2}} \mp \frac{β^{2} τ_{0}^{2} + 6 β κ^{2} τ τ_{0} + κ^{4} τ^{2}}{8 τ^{2} η^{3} τ_{0}^{2}} i) ν^{4} + O (ν^{5}) \\ μ_{3, 4, 5} (ν) = σ_{3, 4, 5} ν^{2} + O (ν^{3}) . \end{array}$ (if $η, τ, τ_{0} \neq 0$ ) where $σ_{3, 4, 5}$ represent the three roots of the following third order equation in σ $\begin{matrix} (3.39) & τ η^{2} τ_{0} σ^{3} + η^{2} (τ_{0} + τ) σ^{2} + (η^{2} + β κ^{2}) σ + κ^{2} = 0 . \end{matrix}$ Considering again the relation between μ and λ and that $ν = ζ^{- 1} = - i | ξ |^{- 1}$ , we recall that $\begin{matrix} λ_{j} (ξ) = - μ_{j}^{0} | ξ |^{2} + μ_{j}^{1} i | ξ | + μ_{j}^{2} - μ_{j}^{3} i | ξ |^{- 1} - μ_{j}^{4} | ξ |^{- 2} + O (| ξ |^{- 3}), j = 1, \dots, 4 . \end{matrix}$ So, we have that, when $| ξ | \to \infty$ : $\begin{array}{rcl} Re (λ_{1, 2}) (ξ) & = & - \frac{(β - τ) τ_{0}^{2} + κ^{2} τ^{2}}{2 τ^{2} η^{2} τ_{0}^{2}} | ξ |^{- 2} + O (| ξ |^{- 3}) \end{array}$ which is negative if $0 < τ < β$ , but: $\begin{array}{rcl} Re (λ_{3, 4, 5}) (ξ) & = & σ_{3, 4, 5} + O (| ξ |^{- 1}) . \end{array}$ For the roots of (3.39), we consider $\begin{matrix} (3.40) & f (X) = τ η^{2} τ_{0} X^{3} + η^{2} (τ_{0} + τ) X^{2} + (η^{2} + β) X + 1 . \end{matrix}$ It is well known that an algebraic equation of an odd degree with real coefficients has at least one real root $σ_{3}$ . Now, in order to know the location of $σ_{3}$ , we consider the equation (3.40) with $X \in R$ . Thus, we have $\begin{matrix} f (- \frac{1}{τ} - \frac{1}{τ_{0}}) = - \frac{(β - τ) κ^{2} τ_{0} + η^{2} (τ + τ_{0}) + β κ^{2} τ}{τ τ_{0}} < 0, \end{matrix}$ since $0 < τ < β$ . Consequently, we obtain $\begin{matrix} f (- \frac{1}{τ} - \frac{1}{τ_{0}}) f (0) < 0 . \end{matrix}$ Therefore, equation (3.40) has at least one real root $X = σ_{3}$ in the interval $(- \frac{1}{τ} - \frac{1}{τ_{0}}, 0)$ . In this case, we may write (3.40) in the form $\begin{matrix} f (X) = (X - σ_{3}) (τ η^{2} τ_{0} X^{2} + d_{1} X + d_{0}) \end{matrix}$ with $\begin{matrix} d_{1} = τ η^{2} τ_{0} σ_{3} + η^{2} (τ + τ_{0}) and d_{0} = β κ^{2} + η^{2} + d_{1} σ_{3} = β κ^{2} + η^{2} + τ η^{2} τ_{0} σ_{3}^{2} + η^{2} (τ + τ_{0}) σ_{3} \end{matrix}$ or, also, $d_{0} = - \frac{1}{σ_{3}}$ . Now, let us denote by $σ_{4}$ and $σ_{5}$ , the other two roots. Then, we have $\begin{matrix} σ_{4} + σ_{5} = - \frac{d_{1}}{τ η^{2} τ_{0}} and σ_{4} σ_{5} = \frac{d_{0}}{τ η^{2} τ_{0}} . \end{matrix}$ Since $σ_{3} \in (- \frac{1}{τ} - \frac{1}{τ_{0}}, 0)$ it is clear that $d_{1} > 0$ and $d_{0} > 0$ .

Then if $σ_{3}$ and $σ_{4}$ are real, they are both negative (since their sum is negative and their product is positive).

If $σ_{3}$ and $σ_{4}$ are complex, then they are conjugate and therefore $\begin{matrix} Re (σ_{4}) = Re (σ_{5}) . \end{matrix}$ This implies that $\begin{array}{rcl} Re (σ_{4}) & = & Re (σ_{5}) = - \frac{1}{2} (σ_{3} + \frac{1}{τ_{0}} + \frac{1}{τ}) < 0, \end{array}$ since $σ_{3} \in (- \frac{1}{τ} - \frac{1}{τ_{0}}, 0)$ . □

4. Stability results: The case $0 < τ = β$

4.1. The Lyapunov functional

As we did for the Fourier model in [28], in this section we show that the presence of the heat conduction allows us to push the above result to the case $β = τ$ . However, as in the Fourier model in [28], a slower decay rate is obtained. The main result here is given in the following theorem.

Theorem 4.1.
Assume that $0 < τ = β$ . Let $W_{C} (x, t) = {(τ u_{t t} + u_{t}, \nabla (τ u_{t} + u), θ, q)}^{T} (x, t)$ , where $(u, θ, q)$ is the solution of ( 1.6 ). Let s be a nonnegative integer and let $W_{C}^{0} = W_{C} (x, 0) \in H^{s} (R^{N}) \cap L^{1} (R^{N})$ . Then, the following decay estimate $\begin{matrix} (4.1) & {‖ \nabla^{k} W_{C} (t) ‖}_{L^{2} (R^{N})} ⩽ C {(1 + t)}^{- N / 8 - k / 4} {‖ W_{C}^{0} ‖}_{L^{1} (R^{N})} + C {(1 + t)}^{- ℓ / 3} {‖ \nabla^{k + ℓ} W_{C}^{0} ‖}_{L^{2} (R^{N})} \end{matrix}$ holds, for any $t ⩾ 0$ and $0 ⩽ k + ℓ ⩽ s$ , where C and c are two positive constants independent of t and $W_{C}^{0}$ .

The proof of this theorem follows from the Proposition 4.3 below and the same techniques as in the proofs of Theorems 5.1 in [28] and Theorem 3.1 above. Hence, we omit the details here.
Remark 4.2.
Notice that the previous result shows the same exponent for the decay rate as the Fourier case when $τ = β$ (see Theorem 5.1 in [28]) and also exhibits a regularity-loss phenomenon as in the Cattaneo case with $τ < β$ , although the loss of regularity is not the same (see Theorem 3.1). As we will see, we recall that this regularity loss part comes from the asymptotic behaviour of the eigenvalues when $| ξ | \to \infty$ . And, as we will see in Remark 4.4 in next section, this asymptotic behaviour is not the same as the one obtained for $ϱ (ξ)$ in Proposition 4.3 below (which, actually, is the exponent used to prove Theorem 4.1). Hence, unlike the previous sections, the exponent obtained in this proposition (and, hence, the regularity loss phenomenon in Theorem 4.1) is true, but may not be optimal. The exponent that gives the decay rate part in Theorem 4.1, though, is optimal.

As we said, the main result to prove the previous decay result is the following proposition. Proposition 4.3.
Assume that $0 < τ = β$ . Let $W_{C} (x, t) = {(τ u_{t t} + u_{t}, \nabla (τ u_{t} + u), θ, q)}^{T} (x, t)$ , where $(u (x, t), θ (x, t), q (x, t))$ is the solution of ( 1.6 ). Then we have $\begin{matrix} {| \hat{W_{C}} (ξ, t) |}^{2} ⩽ C e^{- c ϱ (ξ) t} {| \hat{W_{C}} (ξ, 0) |}^{2}, \end{matrix}$ where $\begin{matrix} ϱ (ξ) = \frac{| ξ |^{4}}{{(1 + | ξ |^{2})}^{2} (1 + | ξ |^{2} + | ξ |^{4} + | ξ |^{6})}, \end{matrix}$ and C and c are two positive constants.
Proof.
Now, for $τ = β$ , the energy functional (3.5) takes the form $\begin{matrix} \hat{E} (ξ, t) : = \frac{1}{2} {| \hat{v} + τ \hat{w} |^{2} + | ξ |^{2} | \hat{u} + τ \hat{v} |^{2} + | \hat{θ} |^{2} + τ_{0} | \hat{q} |^{2}} \end{matrix}$ and satisfies for all $t ⩾ 0$ , $\begin{matrix} (4.2) & \frac{d}{d t} \hat{E} (ξ, t) = - | \hat{q} |^{2} . \end{matrix}$ The estimate (5.3) in [28] also holds for the Cattaneo model, that is $\begin{array}{rcl} (4.3) & \frac{d}{d t} F_{1} (ξ, t) + (1 - ϵ_{0}) | ξ |^{2} | \hat{u} + τ \hat{v} |^{2} ⩽ | \hat{v} + τ \hat{w} |^{2} + C (ϵ_{0}) | ξ |^{2} | \hat{θ} |^{2} . \end{array}$ for any $ϵ_{0} > 0$ (where $F_{1}$ is given in Section 3.1.

Now, to get a dissipative term for $| \hat{v} + τ \hat{w} |^{2}$ , we rewrite system (3.2) when $τ = β$ as $\begin{array}{rcl} (4.4) & \{\begin{matrix} {(\hat{v} + τ \hat{w})}_{t} + | ξ |^{2} (\hat{u} + τ \hat{v}) - η | ξ |^{2} \hat{θ} = 0, \\ {\hat{θ}}_{t} + κ i ξ \cdot \hat{q} + η | ξ |^{2} (\hat{v} + τ \hat{w}) = 0, \\ τ_{0} {\hat{q}}_{t} + \hat{q} + i ξ κ \hat{θ} = 0 . \end{matrix} \end{array}$ Hence, we define the functional $F_{3} (ξ, t)$ as in (5.5) in [28], that is $\begin{array}{rcl} (4.5) & F_{3} (ξ, t) = Re (\bar{\hat{θ}} (\hat{v} + τ \hat{w})) . \end{array}$ By taking its time derivative we obtain $\begin{array}{l} \frac{d}{d t} F_{3} (ξ, t) + η | ξ |^{2} | \hat{v} + τ \hat{w} |^{2} \\ (4.6) & = κ Re ⟨ i ξ \cdot \hat{q}, \hat{v} + τ \hat{w} ⟩ + η | ξ |^{2} | \hat{θ} |^{2} - | ξ |^{2} Re (\bar{\hat{θ}} (\hat{u} + τ \hat{v})) . \end{array}$ Next, taking the dot product of the third equation in (4.4) with $i ξ η (\overline{\hat{u} + τ \hat{v}})$ , we get $\begin{matrix} τ_{0} η ⟨ i ξ \cdot {\hat{q}}_{t}, \hat{u} + τ \hat{v} ⟩ + η ⟨ i ξ \cdot \hat{q}, \hat{u} + τ \hat{v} ⟩ - | ξ |^{2} κ η \hat{θ} (\bar{\hat{u}} + τ \bar{\hat{v}}) = 0 \end{matrix}$ adding and subtracting $η τ_{0} ⟨ {(\hat{u} + τ \hat{v})}_{t}, i ξ \cdot \hat{q} ⟩ = η τ_{0} ⟨ \hat{v} + τ \hat{w}, i ξ \cdot \hat{q} ⟩$ to the above equation and taking the real part, we obtain $\begin{array}{l} τ_{0} η \frac{d}{d t} Re ⟨ i ξ (\hat{u} + τ \hat{v})), \hat{q} ⟩ \\ (4.7) & = η τ_{0} Re ⟨ i ξ (\hat{v} + τ \hat{w})), \hat{q} ⟩ + η Re (⟨ i ξ \cdot \hat{q}, \hat{u} + τ \hat{v} ⟩) - | ξ |^{2} κ η Re (\hat{θ} (\bar{\hat{u}} + τ \bar{\hat{v}})) . \end{array}$ We define the functional $\begin{array}{rcl} {\tilde{F}}_{3} (ξ, t) = | ξ |^{2} (F_{3} (ξ, t) - \frac{τ_{0}}{κ} Re ⟨ i ξ (\hat{u} + τ \hat{v}), \hat{q} ⟩) . \end{array}$ Hence, from (4.6) and (4.7), we deduce that $\begin{array}{l} \frac{d}{d t} {\tilde{F}}_{3} (ξ, t) + η | ξ |^{4} | \hat{v} + τ \hat{w} |^{2} \\ = κ | ξ |^{2} Re ⟨ i ξ \cdot \hat{q}, \hat{v} + τ \hat{w} ⟩ + η | ξ |^{4} | \hat{θ} |^{2} \\ - \frac{τ_{0}}{κ} | ξ |^{2} Re ⟨ i ξ (\hat{v} + τ \hat{w})), \hat{q} ⟩ - \frac{1}{κ} | ξ |^{2} Re (⟨ i ξ \cdot \hat{q}, \hat{u} + τ \hat{v} ⟩) \\ = (\frac{τ_{0}}{κ} + κ) | ξ |^{2} Re ⟨ i ξ \cdot \hat{q}, \hat{v} + τ \hat{w} ⟩ + η | ξ |^{4} | \hat{θ} |^{2} + \frac{1}{κ} | ξ |^{2} Re (⟨ \hat{q}, i ξ (\hat{u} + τ \hat{v}) ⟩) . \end{array}$ Applying Young’s inequality, we obtain $\begin{array}{l} \frac{d}{d t} {\tilde{F}}_{3} (ξ, t) + (η - {\tilde{ε}}_{1}) | ξ |^{4} | \hat{v} + τ \hat{w} |^{2} \\ (4.8) & ⩽ {\tilde{ε}}_{2} | ξ |^{6} | \hat{u} + τ \hat{v} |^{2} + η | ξ |^{4} | \hat{θ} |^{2} + C ({\tilde{ε}}_{1}, {\tilde{ε}}_{2}) (1 + | ξ |^{2}) | \hat{q} |^{2}, \end{array}$ for all ${\tilde{ε}}_{1}, {\tilde{ε}}_{2} > 0$ . Now, define the functional $\begin{array}{rcl} {\tilde{F}}_{4} (ξ, t) = F_{3} (ξ, t) - | ξ |^{2} τ_{0} η Re ⟨ i ξ (\hat{u} + τ \hat{v})), \hat{q} ⟩ . \end{array}$ Then, we obtain from (3.16) and (4.7) $\begin{array}{rcl} \frac{d}{d t} {\tilde{F}}_{4} (ξ, t) + κ | ξ |^{2} | \hat{θ} |^{2} & = & τ_{0} κ | ξ \cdot \hat{q} |^{2} + Re ⟨ \hat{q}, i ξ \hat{θ} ⟩ - η τ_{0} | ξ |^{2} Re ⟨ i ξ (\hat{v} + τ \hat{w}), \hat{q} ⟩ . \end{array}$ Applying Young’s inequality, we find for any ${\tilde{ε}}_{3}, {\tilde{ε}}_{4} > 0$ , $\begin{array}{l} \frac{d}{d t} {\tilde{F}}_{4} (ξ, t) + (κ - {\tilde{ε}}_{3}) | ξ |^{2} | \hat{θ} |^{2} \\ (4.9) & ⩽ {\tilde{ε}}_{4} \frac{| ξ |^{2}}{1 + | ξ |^{2}} | \hat{v} + τ \hat{w} |^{2} + C ({\tilde{ε}}_{3}, {\tilde{ε}}_{4}) (1 + | ξ |^{2} + | ξ |^{4} + | ξ |^{6}) | \hat{q} |^{2} . \end{array}$ Now, we define the Lyapunov functional as $\begin{array}{rcl} L (ξ, t) & = & N_{0} (1 + | ξ |^{2} + | ξ |^{4} + | ξ |^{6}) \hat{E} (ξ, t) + N_{1} \frac{| ξ |^{4}}{{(1 + | ξ |^{2})}^{2}} F_{1} (ξ, t) \\ (4.10) & + N_{2} \frac{1}{{(1 + | ξ |^{2})}^{2}} {\tilde{F}}_{3} (ξ, t) + N_{3} \frac{| ξ |^{2}}{1 + | ξ |^{2}} {\tilde{F}}_{4} (ξ, t), \end{array}$ where $N_{i}$ , $i = 0, \dots, 3$ are positive constants that should be fixed later on. Hence, taking the time derivative of (4.10) and making use of (4.3), (4.2), (4.8) and (4.9) (and several inequalities on the fractions and powers of $| ξ |$ involved), we get $\begin{array}{rcl} (4.11) & \begin{matrix} \frac{d}{d t} L (ξ, t) + [N_{2} (η - {\tilde{ε}}_{1}) - N_{1} - N_{3} {\tilde{ε}}_{4}] \frac{| ξ |^{4}}{{(1 + | ξ |^{2})}^{2}} | \hat{v} + τ \hat{w} |^{2} \\ + [N_{1} (1 - ϵ_{0}) - N_{2} {\tilde{ε}}_{2}] \frac{| ξ |^{6}}{{(1 + | ξ |^{2})}^{2}} | \hat{u} + τ \hat{v} |^{2} \\ + [N_{3} (κ - {\tilde{ε}}_{3}) - N_{2} η - C (ϵ_{0}) N_{1}] \frac{| ξ |^{4}}{{(1 + | ξ |^{2})}^{2}} | \hat{θ} |^{2} \\ + [N_{0} - \tilde{Λ}] (1 + | ξ |^{2} + | ξ |^{4} + | ξ |^{6}) | \hat{q} |^{2} ⩽ 0, \forall t ⩾ 0, \end{matrix} \end{array}$ where $\tilde{Λ}$ is a generic positive constants that depends on $ϵ_{0}, \dots, N_{1}, \dots$ , but it does not depend on $N_{0}$ . We fix the above constants as follows: we take $ϵ_{0}$ , ${\tilde{ε}}_{1}$ and ${\tilde{ε}}_{3}$ small enough such that $\begin{array}{rcl} ϵ_{0} < 1, {\tilde{ε}}_{1} < η, and {\tilde{ε}}_{3} < κ . \end{array}$ Now, we fix $N_{1} = 1$ and choose $N_{2}$ large enough such that $\begin{array}{rcl} N_{2} > \frac{1}{η - {\tilde{ε}}_{1}} . \end{array}$ Then, we fix $N_{3}$ large enough such that $\begin{array}{rcl} N_{3} (κ - {\tilde{ε}}_{3}) > N_{2} η + C (ϵ_{0}) . \end{array}$ After that we pick ${\tilde{ε}}_{2}$ and ${\tilde{ε}}_{4}$ small enough such that $\begin{array}{rcl} {\tilde{ε}}_{2} < \frac{1 - ϵ_{0}}{N_{2}} and {\tilde{ε}}_{4} < \frac{N_{2} (η - {\tilde{ε}}_{1}) - 1}{N_{3}} \end{array}$ Once all the above constants are fixed, we take $N_{0}$ large enough such that $N_{0} > \tilde{Λ}$ . Consequently (4.11) becomes $\begin{array}{rcl} (4.12) & \frac{d}{d t} L (ξ, t) + \tilde{α} \frac{| ξ |^{4}}{{(1 + | ξ |^{2})}^{2}} \hat{E} (ξ, t) ⩽ 0, \forall t ⩾ 0 . \end{array}$ On the other hand, it is not hard to see that there exists two positive constants ${\tilde{c}}_{1}$ and ${\tilde{c}}_{2}$ such that for all $t ⩾ 0$ , $\begin{matrix} {\tilde{c}}_{1} (1 + | ξ |^{2} + | ξ |^{4} + | ξ |^{6}) \hat{E} (ξ, t) ⩽ L (ξ, t) ⩽ {\tilde{c}}_{2} (1 + | ξ |^{2} + | ξ |^{4} + | ξ |^{6}) \hat{E} (ξ, t) . \end{matrix}$ Consequently, this together with (4.12) leads to our desired result. □

4.2. Asymptotic expansion of the eigenvalues

In this section, we compute the asymptotic expansion of the eigenvalues in the Cattaneo model when $0 < τ = β$ using, as before, the eigenvalues expansion method. In this case, and as a difference from Section 3, this asymptotic behaviour will not be the same as the one that in the previous section gives us the decay and regularity of the norm related to the solution, as the following remark explains.

Remark 4.4.

The decay rate in Theorem 4.1 comes from the exponent $\begin{matrix} ϱ (ξ) = \frac{| ξ |^{4}}{{(1 + | ξ |^{2})}^{2} (1 + | ξ |^{2} + | ξ |^{4} + | ξ |^{6})}, \end{matrix}$ of Proposition 4.3. Observe that $ϱ (ξ) \sim | ξ |^{4}$ when $| ξ | \to 0$ , and $ϱ (ξ) \sim | ξ |^{- 6}$ when $| ξ | \to \infty$ . From Lemmas 4.5 and 4.6 below, we can see that the asymptotic behaviour of the real parts of the slowest eigenvalues when $| ξ | \to 0$ is also $| ξ |^{4}$ . But when $| ξ | \to \infty$ the slowest real parts of the eigenvalues behave as $| ξ |^{- 2}$ , which is a different rate than the one obtained in Proposition 4.3. That allows us to conclude that the decay exponent obtained in this Proposition 4.3 may not optimal. And, hence, the decay result of Theorem 4.1 may not optimal either. More concretely, while its decay rate is in agreement with the present results (as for $| ξ | \to 0$ both asymptotic behaviours agree), the regularity-loss of the solutions (which comes from the asymptotic behaviour when $| ξ | \to \infty$ ) may be improved.

First, observe that when $τ = β$ the characteristic equation (3.32) takes the form $\begin{array}{l} λ^{5} + (\frac{1}{τ_{0}} + \frac{1}{τ}) λ^{4} + \frac{η^{2} τ_{0} ζ^{4} - (κ^{2} + τ_{0}) ζ^{2} + 1}{τ_{0}} λ^{3} \\ (4.13) & + \frac{(τ + τ_{0}) η^{2} ζ^{2} - (τ + τ_{0} + κ^{2})}{τ τ_{0}} ζ^{2} λ^{2} + \frac{(η^{2} + τ κ^{2}) ζ^{2} - 1}{τ τ_{0}} ζ^{2} λ + \frac{κ^{2} ζ^{4}}{τ τ_{0}} = 0 \end{array}$ where $ζ = i | ξ |$ .

Lemma 4.5.

We assume that $0 < τ = β$ . In this case, we get the following expansion for $| ξ | \to 0$ : $\begin{array}{rcl} \{\begin{matrix} Re (λ_{1, 2}) (ξ) = - \frac{η^{2} κ^{2}}{2} | ξ |^{4} + O (| ξ |^{5}), \\ Re (λ_{3}) (ξ) = - κ^{2} | ξ |^{2} + O (| ξ |^{3}), \\ Re (λ_{4}) (ξ) = - \frac{1}{τ} + O (| ξ |^{2}), \\ Re (λ_{5}) (ξ) = - \frac{1}{τ_{0}} + O (| ξ |^{2}) . \end{matrix} \end{array}$ In particular, all these real parts are negative.

Proof.

As in the previous sections, it is easy to compute that for $ζ \to 0$ , we have the following asymptotic expansion for the roots of (4.13): $\begin{array}{l} λ_{1, 2} (ζ) = \pm ζ \pm (\frac{η^{2}}{2} + \frac{5 τ^{2}}{4}) ζ^{3} - \frac{η^{2} κ^{2}}{2} ζ^{4} + O (ζ^{5}), \\ λ_{3} (ζ) = κ^{2} ζ^{2} + O (ζ^{3}), \\ λ_{4} (ζ) = - \frac{1}{τ} + O (ζ^{2}), \\ λ_{5} (ζ) = - \frac{1}{τ_{0}} + O (ζ^{2}) . \end{array}$ Now, coming back to the variable ξ, we get the following expansion for $| ξ | \to 0$ : $\begin{array}{rcl} \{\begin{matrix} Re (λ_{1, 2}) (ξ) = - \frac{η^{2} κ^{2}}{2} | ξ |^{4} + O (| ξ |^{5}), \\ Re (λ_{3}) (ξ) = - κ^{2} | ξ |^{2} + O (| ξ |^{3}), \\ Re (λ_{4}) (ξ) = - \frac{1}{τ} + O (| ξ |^{2}), \\ Re (λ_{5}) (ξ) = - \frac{1}{τ_{0}} + O (| ξ |^{2}), \end{matrix} \end{array}$ which, indeed, are negative. □

Lemma 4.6.

We assume that $0 < τ = β$ . In this case, we get the following expansion for $| ξ | \to \infty$ : $\begin{matrix} \{\begin{matrix} Re (λ_{1, 2}) (ξ) = - \frac{κ^{2} τ^{2}}{2 τ^{2} η^{2} τ_{0}^{2}} | ξ |^{- 2} + O (| ξ |^{- 3}), \\ Re (λ_{3, 4, 5}) (ξ) = {\tilde{σ}}_{3, 4, 5} + O (| ξ |^{- 1}), \end{matrix} \end{matrix}$ with ${\tilde{σ}}_{3, 4, 5} < 0$ , as the rest of the real parts.

Proof.

We proceed as Lemma 3.10. Hence, applying the change of variables $ν = ζ^{- 1} = {(i | ξ |)}^{- 1}$ , computing the asymptotic expansion of $μ (ν)$ (the roots of the corresponding characteristic polynomial for ν) when $ν \to 0$ , and coming back to the eigenvalues $λ (ξ)$ , we can easily show that the following expansion holds $\begin{matrix} \{\begin{matrix} Re (λ_{1, 2}) (ξ) = - \frac{κ^{2} τ^{2}}{2 τ^{2} η^{2} τ_{0}^{2}} | ξ |^{- 2} + O (| ξ |^{- 3}), \\ Re (λ_{3, 4, 5}) (ξ) = {\tilde{σ}}_{3, 4, 5} + O (| ξ |^{- 1}), \end{matrix} \end{matrix}$ with ${\tilde{σ}}_{3, 4, 5}$ being the roots of (3.40) when $τ = β$ . In this case, these roots can be explicitly calculated and are $\begin{matrix} \tilde{σ_{3}} = - \frac{1}{τ}, \tilde{σ_{4, 5}} = - \frac{η \pm \sqrt{η^{2} - 4 κ^{2} τ_{0}}}{2 η τ_{0}} . \end{matrix}$ Hence, we can conclude all the eigenvalues have negative real parts in this case too. □

Footnotes

Acknowledgements

The authors would like to thank the anonymous referees for their helpful comments and suggestions, that have allowed us to definitely improve the present work. Also, the authors would like to thank Prof. Solà-Morales for his helpful discussions on the problem.

M. Pellicer is part of the Catalan Research group 2017 SGR 1392 and has been supported by the MINECO grant MTM2017-84214-C2-2-P (Spain), and also by MPC UdG 2016/047 (U. of Girona, Catalonia).

References

M.S.

Alves,

Buriol,

M.V.

Ferreira,

J.E.

Muñoz Rivera,

Sepúlveda and

Vera, Asymptotic behaviour for the vibrations modeled by the standard linear solid model with a thermal effect, J. Math. Anal. App. (2013).

M.S.

Alves,

Gamboa,

G.C.

Gorain,

Rambaud and

Vera, Asymptotic behavior of a flexible structure with Cattaneo type of thermal effect, Indagationes Mathematicae 27(3) (2016). doi:10.1016/j.indag.2016.03.001.

T.A.

Apalara,

S.A.

Messaoudi and

A.A.

Keddi, On the decay rates of Timoshenko system with second sound, Math. Methods Appl. Sci. 39(10) (2016), 2671–2684. doi:10.1002/mma.3720.

S.K.

Bose and

G.C.

Gorain, Stability of the boundary stabilised internally damped wave equation

y^{″} + λ y^{‴} = c^{2} (Δ y + μ Δ y^{'})

in a bounded domain in

R^{n}

, Indian J. Math. 40(1) (1998), 1–15.

Brézis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011.

D.E.

Carlson, Linear thermoelasticity, in: Handbook of Physics, 1972.

J.A.

Conejero,

Lizama and

Ródenas, Chaotic behaviour of the solutions of the Moore–Gibson–Thompson equation, App. Math. and Inf. Sciences 9(5) (2015), 1–6.

Conti,

Pata,

Pellicer and

Quintanilla, On the analyticity of the MGT-viscoelastic plate with heat conduction, J. of Differential Equations 269(10) (2020), 7862–7880. doi:10.1016/j.jde.2020.05.043.

Conti,

Pata,

Pellicer and

Quintanilla, A new approach to MGT-thermoviscoelasticity, 2020, Submitted.

10.

Conti,

Pata and

Quintanilla, Thermoelasticity of Moore–Gibson–Thompson type with history dependence in the temperature, Asymptotic Analysis, 120(1–2) (2020), 1–21. doi:10.3233/ASY-191576.

11.

Dell’Oro,

Lasiecka and

Pata, The Moore–Gibson–Thompson equation with memory in the critical case, J. Differential Equations 261(7) (2016), 4188–4222. doi:10.1016/j.jde.2016.06.025.

12.

Dell’Oro and

Pata, On the Moore–Gibson–Thompson equation and its relation to linear viscoelasticity, Appl. Math. Optim. 76 (2017), 641–655. doi:10.1007/s00245-016-9365-1.

13.

G.C.

Gorain, Stabilization for the vibrations modeled by the ‘standard linear model’ of viscoelasticity, Proc. Indian Acad. Sci. Math. Sci. 120(4) (2010), 495–506. doi:10.1007/s12044-010-0038-8.

14.

G.C.

Gorain and

S.K.

Bose, Exact controllability and boundary stabilization of torsional vibrations of an internally damped flexible space structure, J. Optim. Theory Appl. 99(2) (1998), 423–442. doi:10.1023/A:1021778428222.

15.

Hosono and

Kawashima, Decay property of regularity-loss type and application to some nonlinear hyperbolic-elliptic system, Math. Mod. Meth. Appl. Sci. 16 (2006), 1839–1859. doi:10.1142/S021820250600173X.

16.

Kaltenbacher,

Lasiecka and

Marchand, Wellposedness and exponential decay rates for the Moore–Gibson–Thompson equation arising in high intensity ultrasound, Control and Cybernetics 40 (2011), 971–988.

17.

Kaltenbacher,

Lasiecka and

M.K.

Pospieszalska, Well-posedness and exponential decay of the energy in the nonlinear Jordan–Moore–Gibson–Thompson equation arising in high intensity ultrasound, Math. Models Methods Appl. Sci. 22(11) (2012), 1250035, 34.

18.

Khader and

Said-Houari, On the decay rate of solutions of the Bresse system with Gurtin–Pipkin thermal law, Asymptotic Analysis 103(1–2) (2017), 1–32. doi:10.3233/ASY-171417.

19.

Lasiecka and

Wang, Moore–Gibson–Thompson equation with memory, part II: General decay of energy, J. Differential Equations 259 (2015), 7610–7635. doi:10.1016/j.jde.2015.08.052.

20.

Liu and

Zheng, Semigroups Associated with Dissipative Systems, Research Notes in Mathematics, Vol. 398, Chapman & Hall/CRC, 1999.

21.

Marchand,

McDevitt and

Triggiani, An abstract semigroup approach to the third-order Moore–Gibson–Thompson partial differential equation arising in high-intensity ultrasound: Structural decomposition, spectral analysis, exponential stability, Math. Methods. Appl. Sci. 35(15) (2012), 1896–1929. doi:10.1002/mma.1576.

22.

Misra,

Alves,

Gorain and

Vera, Stability of the vibrations of an inhomogeneous flexible structure with thermal effect, International Journal of Dynamics and Control (2014).

23.

Montanaro, On the constitutive relations for second sound in thermo-electroelasticity, Arch. Mech. (Arch. Mech. Stos.) 63(3) (2011), 225–254.

24.

Mori and

Kawashima, Decay property for the Timoshenko system with Fourier’s type heat conduction, Journal of Hyperbolic Differential Equations 11(01) (2014), 135–157. doi:10.1142/S0219891614500039.

25.

Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983.

26.

Pellicer and

Quintanilla, On uniqueness and instability for some thermoemechanical problems involving the Moore–Gibson–Thompson equation, Z. Angew. Math. Phys 71(3) (2020).

27.

Pellicer and

Said-Houari, Wellposedness and decay rates for the Cauchy problem of the Moore–Gibson–Thompson equation arising in high intensity ultrasound, Appl Math. Optim 80 (2019), 447–478. doi:10.1007/s00245-017-9471-8.

28.

Pellicer and

Said-Houari, On the Cauchy problem of the standard linear solid model with Fourier heat conduction, 2020, Submitted.

29.

Pellicer and

Solà-Morales, Optimal scalar products in the Moore–Gibson–Thompson equation, Evol. Eqs. and Control Theory 8(1) (2019).

30.

Quintanilla, Moore–Gibson–Thompson thermoelasticity, Math. Mech. Solids 24 (2019), 4020–4031. doi:10.1177/1081286519862007.

31.

Racke, Thermoelasticity with second sound – exponential stability in linear and non-linear 1-d, Math. Meth. Appl. Sci. 25 (2002), 409–441. doi:10.1002/mma.298.

32.

Racke, Thermoelasticity, in: Handbook of Differential Equations: Evolutionary Equations, Handb. Diff. Equ., Vol. V, Elsevier/North-Holland, Amsterdam, 2009, pp. 315–420.

33.

Racke and

Said-Houari, Global existence and decay property of the Timoshenko system in thermoelasticity with second sound, Nonlinear Anal. 75(13) (2012), 4957–4973. doi:10.1016/j.na.2012.04.011.

34.

Racke and

Said-Houari, Decay rates and global existence for semilinear dissipative Timoshenko systems, Quart. Appl. Math. 72(2) (2013), 229–266. doi:10.1090/S0033-569X-2012-01280-8.

35.

Racke and

Ueda, Dissipative structures for thermoelastic plate equations in

R^{n}

, Adv. Differential Equations 21(7–8) (2016), 601–630.

36.

J.E.M.

Rivera and

Racke, Large solutions and smoothing properties for nonlinear thermoelastic systems, J. Differential Equations 127(2) (1996), 454–483. doi:10.1006/jdeq.1996.0078.

37.

Said-Houari, Decay properties of linear thermoelastic plates: Cattaneo versus Fourier law, Appl. Anal. 92(2) (2013), 424–440. doi:10.1080/00036811.2011.625015.

38.

Said-Houari and

Kasimov, Decay property of Timoshenko system in thermoelasticity, Math. Methods. Appl. Sci. 35(3) (2012), 314–333. doi:10.1002/mma.1569.

39.

Said-Houari and

Kasimov, Damping by heat conduction in the Timoshenko system: Fourier and Cattaneo are the same, J. of Differential Equations 255 (2013), 611–632. doi:10.1016/j.jde.2013.04.026.

40.

P.A.

Thompson, Compressible-Fluid Dynamics, McGraw-Hill, New York, 1972.

41.

Wan and

Liu, Well-posedness and decay property of regularity-loss type for the Cauchy problem of the standard linear solid model with Gurtin–Pipkin thermal law, Asymptot. Anal., (2020), 1–21. doi:10.3233/ASY-201631.