Thermoelasticity of Moore–Gibson–Thompson type with history dependence in the temperature

Abstract

In this paper, we consider a thermoelastic model where heat conduction is described by the history dependent version of the Moore–Gibson–Thompson equation, arising via the introduction of a relaxation parameter in the Green-Naghdi type III theory. The well-posedness of the resulting integro-differential system is discussed. In the one-dimensional case, the exponential decay of the energy is proved.

Keywords

Moore–Gibson–Thompson equation relaxation parameter memory kernel thermoelasticity solution semigroup exponential stability

1. Introduction

The Moore–Gibson–Thompson (MGT) equation $\begin{matrix} (1.1) & u_{t t t} + α u_{t t} + β A u_{t} + γ A u = 0, \end{matrix}$ where A is a strictly positive operator on some Hilbert space H and $α, β, γ > 0$ are given parameters, has deserved much attention in recent years, with several papers appeared in the literature on the argument (see [3,11–13,23,25,31,39,40], among others). The model has been originally introduced in connection with fluids mechanics [50]. In this work, instead, we propose a different interpretation within the theory of thermoelasticity, where the heat transfer is ruled by an integro-differential equation. We will see how the history dependent version of the MGT equation is obtained in a natural way from the Green-Naghdi heat conduction model, through the introduction of a relaxation parameter.

1.1. Notation

Along this work, we will denote a vector indifferently by $v$ or by its generic $i^{th}$ -component $v_{i}$ . Given any function $f = f (x, t)$ , we will write $f_{, i}$ to mean its derivative with respect to the space variable $x_{i}$ , and $f_{t}$ to mean the derivative in time. Whenever confusion may occur, we will write $\partial_{t} f$ instead. We will also employ the Einstein notation, where $v_{i, i} = div v$ .

1.2. Classical heat conduction

The classical theory of heat conduction, for a heat conductor occupying a volume Ω, is based on the Fourier law $\begin{matrix} q_{i} = k θ_{, i}, \end{matrix}$ where $q_{i}$ is the flux vector and θ is the relative temperature, both depending on $x \in Ω$ and on time t, while $k > 0$ is the thermal conductivity of the material.1

¹
Several authors include a sign “−” at the right hand side of the relation, however here we adopt the approach of [21] (see page 41).

Substituting

q_{i}

into the energy equation

\begin{matrix} (1.2) & q_{i, i} = c θ_{t}, \end{matrix}

where

c > 0

is the thermal capacity, we obtain the classical heat equation

\begin{matrix} c θ_{t} - k Δ θ = 0, \end{matrix}

predicting the instantaneous propagation of thermal waves. A fact which is known to be incompatible with the causality principle. This is the reason why many scientists have suggested alternative approaches in the description of heat conduction. A well-established theory is the one introduced by Maxwell and Cattaneo [1], where the Fourier law is replaced by the constitutive equation, containing a relaxation parameter

τ > 0

\begin{matrix} (1.3) & τ {\partial_{t} q}_{i} + q_{i} = k θ_{, i} . \end{matrix}

The combination of (1.2) and (1.3) leads to the damped hyperbolic equation

\begin{matrix} τ c θ_{t t} + c θ_{t} - k Δ θ = 0, \end{matrix}

in which thermal waves propagate indeed with finite speed. This setting has been extended to cover thermoelasticity in the Lord–Shulman theory [29]. The system of equations obtained in this way has been widely investigated (see e.g. [2,19,20,22,46,47]).

1.3. Green–Naghdi heat conduction

Three theories for heat conduction have been proposed by Green and Naghdi at the end of the last century [16–18], nowadays known as type I, II and III, respectively. Type I heat conduction is nothing but the Fourier law. Type II concerns with another hyperbolic equation where there is no dissipation. In this case, the heat flux vector is a linear expression of the thermal displacement α, satisfying the relation $\begin{matrix} α_{t} (x, t) = θ (x, t), \end{matrix}$ of the form $\begin{matrix} q_{i} = k^{*} α_{, i}, \end{matrix}$ where $k^{*} > 0$ is the conductivity rate parameter. The constitutive equation for the heat flux vector of type III theory reads $\begin{matrix} (1.4) & q_{i} = k^{*} α_{, i} + k θ_{, i} . \end{matrix}$ In particular, when $k^{*} = 0$ or $k = 0$ we boil down to type I or type II, respectively. Type III theory attracted a lot of interest, witnessed by a flurry of publications appeared on the argument (e.g. [5,15,24,26,27,30,32–36,42,43,45,49]). In the nontrivial case when both k and $k^{*}$ are positive, and we substitute $q_{i}$ into (1.2), we obtain a generalization of the Fourier classical heat equation, namely, $\begin{matrix} c α_{t t} - k Δ α_{t} - k^{*} Δ α = 0 . \end{matrix}$ Unfortunately, the equation above (sometimes called strongly damped wave equation) suffers from the same drawback of the Fourier one, as instantaneous propagation of thermal waves is still present (see [48, p.39]; see also [15]). To be more precise, in this model one observes an instantaneous regularization of the temperature $θ = α_{t}$ . A natural way to overcome this problem is to modify the constitutive equation (1.4) by introducing a (small) relaxation parameter.

1.4. MGT heat conduction

Following the approach of Maxwell and Cattaneo, we correct the constitutive law (1.4) in the following manner: $\begin{matrix} (1.5) & τ \partial_{t} q_{i} + q_{i} = k^{*} α_{, i} + k θ_{, i}, \end{matrix}$ where the relaxation parameter $τ > 0$ is thought to be small compared to the other constants. Collecting (1.2) and (1.5), we find $\begin{matrix} c θ_{t t} = div q_{t} = τ^{- 1} (- div q + k^{*} Δ α + k Δ θ) = τ^{- 1} (- c θ_{t} + k^{*} Δ α + k Δ θ) . \end{matrix}$ As a result, the corresponding heat equation becomes the MGT equation $\begin{matrix} (1.6) & τ c α_{t t t} + c α_{t t} - k Δ α_{t} - k^{*} Δ α = 0 . \end{matrix}$ It is well-known that the asymptotic behavior of the abstract MGT equation (1.1) strongly depends on the stability number $\begin{matrix} χ = β - \frac{γ}{α} . \end{matrix}$ In particular, the associated semigroup on the natural weak energy space is exponentially stable if and only if we are in the subcritical regime, corresponding to $χ > 0$ (see e.g. [12,23,31,39,40]). Since, in absence of external heat sources, it is natural to assume that a reasonable heat model is exponentially stable, the new heat equation (1.6), understood to comply with the Dirichlet boundary conditions, is physically meaningful if and only if its stability number is strictly positive. In other words, it must be $\begin{matrix} χ = \frac{1}{c} (τ^{- 1} k - k^{*}) > 0, \end{matrix}$ which is clearly implied by a choice of a sufficiently small relaxation parameter $τ > 0$ .

The goal of the present paper is to develop a thermoelastic theory based on heat conduction of MGT type. This strategy has been first devised in [44]. However, we will consider here, rather than the pure MGT, an integro-differential equation of which the MGT heat law (1.6) is just a particular instance, corresponding to the choice of a (negative) exponential convolution kernel. Accordingly, following the lines of [12], we will view (1.6) as an integro-differential equation, sharing the same mathematical structure of the one of linear viscoelasticity (see e.g. [4,6–10,14,37,48]).

1.5. Plan of the paper

In the next Section 2, we introduce a nonhomogeneous version of the MGT equation, and we translate it into an integro-differential one. After that, in Section 3, we define the corresponding thermoelastic system. The well-posedness of the problem in space-dimension 3, and under quite general assumptions on the memory kernel, is studied in Section 4. In the final Section 5, we discuss the exponential decay of the solutions in the one-dimensional case.

2. The MGT heat equation with history

We consider a nonhomogeneous heat conductor occupying a bounded domain $Ω \subset R^{3}$ . According to linear type III theory with relaxation parameter, the general constitutive equation for centrosymmetric materials is given by $\begin{matrix} (2.1) & τ \partial_{t} q_{i} + q_{i} = k_{i j} θ_{, j} + k_{i j}^{*} α_{, j}, \end{matrix}$ where, as before, $\begin{matrix} α_{t} = θ . \end{matrix}$ Here, $τ = τ (x)$ is the relaxation function, supposed to be strictly positive and bounded, $k_{i j} = k_{i j} (x)$ is the thermal conductivity tensor, and $k_{i j}^{*} = k_{i j}^{*} (x)$ is the conductivity rate tensor. Both $k_{i j}$ and $k_{i j}^{*}$ are assumed to be symmetric. Note that (2.1) can be rewritten as $\begin{matrix} \partial_{t} [q_{i} e^{τ^{- 1} t}] = τ^{- 1} [k_{i j} θ_{, j} + k_{i j}^{*} α_{, j}] e^{τ^{- 1} t} . \end{matrix}$ We also make the reasonable assumptions that $\begin{matrix} (2.2) & \underset{s \to - \infty}{lim sup} q_{i} (x, s) e^{τ^{- 1} (x) s} = 0 and \underset{s \to - \infty}{lim sup} α_{, i} (x, s) e^{τ^{- 1} (x) s} = 0 . \end{matrix}$ Then, an integration on $(- \infty, t)$ yields, omitting the dependence on $x$ , $\begin{matrix} q_{i} (t) = τ^{- 1} \int_{- \infty}^{t} e^{- τ^{- 1} (t - s)} [k_{i j} θ_{, j} (s) + k_{i j}^{*} α_{, j} (s)] d s . \end{matrix}$ In light of the second limit in (2.2), we observe that $\begin{matrix} \int_{- \infty}^{t} e^{- τ^{- 1} (t - s)} θ_{, j} (s) d s = α_{, j} (t) - τ^{- 1} \int_{- \infty}^{t} e^{- τ^{- 1} (t - s)} α_{, j} (s) d s . \end{matrix}$ Therefore, we end up with $\begin{matrix} q_{i} = τ^{- 1} k_{i j} α_{, j} (t) - τ^{- 1} \int_{- \infty}^{t} e^{- τ^{- 1} (t - s)} [τ^{- 1} k_{i j} - k_{i j}^{*}] α_{, j} (s) d s . \end{matrix}$ Defining the kernel $\begin{matrix} (2.3) & g_{i j} (x, s) = k_{i j}^{*} (x) + e^{- τ^{- 1} (x) s} [τ^{- 1} (x) k_{i j} (x) - k_{i j}^{*} (x)], \end{matrix}$ we are led to the constitutive equation (2.1) in the integro-differential form (again, omitting the dependence on $x$ ) $\begin{matrix} (2.4) & q_{i} (t) = g_{i j} (0) α_{, j} (t) + \int_{0}^{\infty} g_{i j}^{'} (s) α_{, j} (t - s) d s . \end{matrix}$ As a final step, in order to write the corresponding heat equation, we need the nonhomogeneous version of the energy equality. Assuming that the thermal capacity of the material depends on the point, we have $\begin{matrix} (2.5) & q_{i, i} = c θ_{t}, \end{matrix}$ where $c = c (x)$ is strictly positive and bounded. Coupling now (2.4) and (2.5), we arrive at the following heat conduction equation: $\begin{matrix} (2.6) & c \partial_{t t} α - {(g_{i j} (0) α_{, j} (t) + \int_{0}^{\infty} g_{i j}^{'} (s) α_{, j} (t - s) d s)}_{, i} = 0, \end{matrix}$ having the form of an equation of viscoelasticity in the variable α, with a memory kernel $g_{i j}$ that is allowed to depend on the spatial variable.

Remark 2.1.
When τ is independent of $x$ , equation (2.6) can be rewritten in a generalized MGT form. Indeed, adding (2.6) and its time-derivative times τ, and keeping in mind the explicit form of $g_{i j}$ given by (2.3), we obtain (see [44]) $\begin{matrix} τ c \partial_{t t t} α + c \partial_{t t} α + A \partial_{t} α + A^{} α = 0, \end{matrix}$ where A and $A^{}$ are the second order differential operators defined by $\begin{matrix} A v = {(k_{i j} v_{, j})}_{, i}, and A^{} v = {(k_{i j}^{} v_{, j})}_{, i}, \end{matrix}$ which are strictly positive whenever the same is true for the (symmetric) tensors $k_{i j}$ and $k_{i j}^{*}$ .

3. Thermoelasticity of MGT type

Along the same lines of [44], we now propose a thermoelasticity model, where heat conduction is ruled by the MGT law (2.4). Actually, in more generality, we consider a kernel $g_{i j}$ of which (2.4) is only a particular case, and whose properties will be specified shortly. We start from the constitutive equations $\begin{array}{l} t_{i j} = C_{i j k l} u_{k, l} - β_{i j} θ, \\ e = c θ + β_{i j} u_{i, j} . \end{array}$ Here, $u_{i} = u_{i} (x, t)$ is the displacement vector, $t_{i j} = t_{i j} (x, t)$ is the stress tensor, $e = e (x, t)$ is the entropy, $C_{i j k l} = C_{i j k l} (x)$ is the elasticity tensor, satisfying the symmetry condition $\begin{matrix} C_{i j k l} = C_{k l i j}, \end{matrix}$ and $β_{i j} = β_{i j} (x)$ is the coupling tensor. The evolution of the displacement and the entropy is described by the equations $\begin{array}{l} ρ \partial_{t t} u_{i} = t_{i j, j}, \\ T_{0} \partial_{t} e = q_{i, i}, \end{array}$ where $T_{0}$ is the reference temperature, assumed to be constant ( $T_{0} = 1$ in the sequel), and $ρ = ρ (x)$ is the mass density. Substituting the constitutive equations into the evolution, we find the system $\begin{matrix} (3.1) & \{\begin{matrix} ρ \partial_{t t} u_{i} (t) = {(C_{i j k l} u_{k, l} (t) - β_{i j} \partial_{t} α (t))}_{, j}, \\ c \partial_{t t} α (t) = - β_{i j} \partial_{t} u_{i, j} (t) + {(g_{i j} (0) α_{, j} (t) + \int_{0}^{\infty} g_{i j}^{'} (s) α_{, j} (t - s) d s)}_{, i} . \end{matrix} \end{matrix}$ In this section, we study (3.1) in a three-dimensional domain Ω, whose boundary $\partial Ω$ is smooth enough to apply the divergence theorem. The system is supplemented with the Dirichlet boundary conditions $\begin{matrix} (3.2) & u_{i} {(x, t)}_{| x \in \partial Ω} = 0 and α {(x, t)}_{| x \in \partial Ω} = 0 . \end{matrix}$

3.1. General assumptions

We assume that the constitutive tensors are bounded above, and that there exist strictly positive constants $c_{0}$ , $ρ_{0}$ , $C_{0}$ such that $\begin{matrix} (3.3) & c (x) ⩾ c_{0}, ρ (x) ⩾ ρ_{0}, \end{matrix}$ and $\begin{matrix} (3.4) & \int_{Ω} C_{i j k l} ξ_{i, j} ξ_{k, l} d x ⩾ C_{0} \int_{Ω} ξ_{i, j} ξ_{i, j} d x, \end{matrix}$ for every vector $ξ_{i}$ vanishing on $\partial Ω$ . The memory kernel $\begin{matrix} g_{i j} = g_{i j} (s) \end{matrix}$ is supposed to be independent of the variable $x$ . This assumption, albeit not essential, simplifies the exposition. The precise hypotheses on $g_{i j}$ read as follows (cf. [28]):

The tensor $g_{i j}$ is symmetric, i.e. $g_{i j} = g_{j i}$ .

The tensor $g_{i j}$ is twice differentiable with respect to s, and $g_{i j}^{'}$ is summable on $R^{+}$ .

There exists a positive constant $k_{0}$ such that $\begin{matrix} g_{i j} (\infty) ξ_{i} ξ_{j} ⩾ k_{0} ξ_{i} ξ_{i}, \end{matrix}$ where $g_{i j} (\infty) = {lim}_{s \to \infty} g_{i j} (s)$ .

There exists a positive, decreasing, scalar function $μ \in L^{1} (R^{+}) \cap C ([0, \infty))$ and a constant $k_{1} ⩾ 1$ such that $\begin{matrix} μ (s) ξ_{i} ξ_{i} ⩽ - g_{i j}^{'} (s) ξ_{i} ξ_{j} ⩽ k_{1} μ (s) ξ_{i} ξ_{i}, \forall s > 0 . \end{matrix}$

The tensor $g_{i j}^{″}$ is nonnegative definite, i.e. $\begin{matrix} g_{i j}^{″} (s) ξ_{i} ξ_{j} ⩾ 0, \forall s > 0 . \end{matrix}$

In (iii)-(v) above,

ξ_{i}

is any vector of

R^{3}

Remark 3.1.
Condition (iii) is natural to guarantee the stability of the solutions. For the MGT kernel $g_{i j}$ previously defined, that is $\begin{matrix} (3.5) & g_{i j} (s) = k_{i j}^{} + e^{- τ^{- 1} s} (τ^{- 1} k_{i j} - k_{i j}^{}), \end{matrix}$ this is the same as taking $k_{i j}^{}$ positive definite. The fact that the derivative of $g_{i j}$ is negative definite, as in (iv), corresponds to assume that $τ^{- 1} k_{i j} - k_{i j}^{}$ is positive definite, which, as we saw, arises in a natural way in connection with the (dissipative) MGT-equation. In particular it implies that $k_{i j}$ is also positive definite, a consequence of the second principle of Thermodynamics (see [16]). Condition (3.4) falls in the realm of the elastic stability theory. The meaning of the conditions on the heat capacity and the mass density is obvious.

3.2. Functional setting

We denote by $(H, ⟨ \cdot, \cdot ⟩, ‖ \cdot ‖)$ the usual Hilbert space $L^{2} (Ω)$ , and by $(V, {⟨ \cdot, \cdot ⟩}_{1}, ‖ \cdot ‖_{1})$ the standard Sobolev space $H_{0}^{1} (Ω)$ of $H^{1}$ -functions vanishing on $\partial Ω$ . We denote by $\begin{matrix} H = {[L^{2} (Ω)]}^{3}, V = {[H_{0}^{1} (Ω)]}^{3} \end{matrix}$ the corresponding vectorial versions, keeping the same scalar notation for their norms. In order to translate system (3.1) into an ODE in the so-called past history framework of Dafermos [9,10], we introduce the Hilbert space $\begin{matrix} M = L_{μ}^{2} (R^{+}; V) \end{matrix}$ of square summable function with respect to the measure $μ (s) d s$ , endowed with the product and norm $\begin{matrix} {⟨ η, η^{*} ⟩}_{M} = \int_{0}^{\infty} μ (s) {⟨ η (s), η^{*} (s) ⟩}_{1} d s and ‖ η ‖_{M}^{2} = \int_{0}^{\infty} μ (s) {‖ η (s) ‖}_{1}^{2} d s . \end{matrix}$ Note that, in light of assumption (iv), $\begin{matrix} | η |_{M}^{2} = - \int_{0}^{\infty} \int_{Ω} g_{i j}^{'} (s) η_{, i} (x, s) η_{, j} (x, s) d x d s \end{matrix}$ is an equivalent norm on $M$ , with corresponding scalar product $\begin{matrix} {(η, η^{*})}_{M} = - \int_{0}^{\infty} \int_{Ω} g_{i j}^{'} (s) η_{, i} (x, s) η_{, j}^{*} (x, s) d x d s . \end{matrix}$ Finally, we define the product Hilbert space $\begin{matrix} H = V \times H \times V \times H \times M, \end{matrix}$ endowed with the norm $\begin{matrix} | U |_{H}^{2} = \int_{Ω} C_{i j k l} u_{i, j} u_{k, l} d x + {‖ ρ^{1 / 2} v ‖}^{2} + \int_{Ω} g_{i j} (\infty) α_{, i} α_{, j} d x + {‖ c^{1 / 2} θ ‖}^{2} + | η |_{M}^{2}, \end{matrix}$ where $U = (u_{i}, v_{i}, α, θ, η)$ . Thanks to (3.3), (3.4) and (iii), this is equivalent to the standard product norm $\begin{matrix} ‖ U ‖_{H}^{2} = ‖ u ‖_{1}^{2} + ‖ v ‖^{2} + ‖ α ‖_{1}^{2} + ‖ θ ‖^{2} + ‖ η ‖_{M}^{2} . \end{matrix}$ We will also consider the infinitesimal generator of the right-translation semigroup on $M$ , i.e. the linear operator T given by $\begin{matrix} T η = - η^{'} with domain D (T) = {η \in M : η^{'} \in M, η (0) = 0}, \end{matrix}$ the prime standing for the distributional derivative with respect to the variable $s > 0$ . In light of the assumptions on the kernel one has the dissipative estimate (see e.g. [37]) $\begin{matrix} (3.6) & {(T η, η)}_{M} = - \frac{1}{2} \int_{0}^{\infty} \int_{Ω} g_{i j}^{″} (s) η_{, i} (x, s) η_{, j} (x, s) d x d s ⩽ 0, \forall η \in D (T) . \end{matrix}$

3.3. The system in the past history framework

In the same spirit of [9,10], we consider the auxiliary variable $η = η^{t} (s)$ , containing all the information on the past history of α, and formally defined as (omitting the dependence on $x$ ) $\begin{matrix} η^{t} (s) = α (t) - α (t - s) . \end{matrix}$ Then, (3.1) becomes the evolution system $\begin{matrix} (3.7) & \{\begin{matrix} ρ \partial_{t t} u_{i} = {(C_{i j k l} u_{k, l} - β_{i j} \partial_{t} α)}_{, j}, \\ c \partial_{t t} α = - β_{i j} \partial_{t} u_{i, j} + {(g_{i j} (\infty) α_{, j} (t) - \int_{0}^{\infty} g_{i j}^{'} (s) η_{, j}^{t} (s) d s)}_{, i}, \\ \partial_{t} η = T η + \partial_{t} α . \end{matrix} \end{matrix}$ Introducing the state vector $\begin{matrix} U (t) = (u_{i} (t), \partial_{t} u_{i} (t), α (t), \partial_{t} α (t), η^{t}), \end{matrix}$ we view (3.7) as the ODE in $H$ $\begin{matrix} (3.8) & \frac{d}{d t} U (t) = A U (t), \end{matrix}$ where $A$ is the linear operator given by $\begin{matrix} (3.9) & A (\begin{matrix} u_{i} \\ v_{i} \\ α \\ θ \\ η \end{matrix}) = (\begin{matrix} v_{i} \\ ρ^{- 1} {(C_{i j k l} u_{k, l} - β_{i j} θ)}_{, j} \\ θ \\ c^{- 1} [- β_{i j} v_{i, j} + {(g_{i j} (\infty) α_{, j} (t) - \int_{0}^{\infty} g_{i j}^{'} (s) η_{, j} (s) d s)}_{, i}] \\ T η + θ \end{matrix}), \end{matrix}$ with (dense) domain $\begin{matrix} D (A) = \{U \in H |\begin{matrix} v_{i} \in V \\ θ \in V \\ η \in D (T) \\ ρ^{- 1} {(C_{i j k l} u_{k, l} - β_{i j} θ)}_{, j} \in H \\ c^{- 1} [- β_{i j} v_{i, j} + {(g_{i j} (\infty) α_{, j} (t) - \int_{0}^{\infty} g_{i j}^{'} (s) η_{, j} (s) d s)}_{, i}] \in H \end{matrix}\} . \end{matrix}$

4. The solution semigroup

The main result of this section concerns with the generation of the solution semigroup.

Theorem 4.1.
The operator $A$ is the infinitesimal generator of a strongly continuous linear semigroup $S (t)$ on the phase space $H$ . Besides, $S (t)$ is a contraction with respect to the $| \cdot |_{H}$ -norm of $H$ .

Accordingly, for every initial datum $U_{0} \in H$ , equation (3.8) admits a unique mild solution (in the sense of [38]) $U (t)$ given by $\begin{matrix} U (t) = S (t) U_{0}, \forall t ⩾ 0, \end{matrix}$ and $\begin{matrix} {| S (t) U_{0} |}_{H} ⩽ | U_{0} |_{H} . \end{matrix}$

The proof of Theorem 4.1 is carried out via the Lumer–Phillips theorem [38], which amounts to prove the next two lemmas. Lemma 4.2.
The operator $A$ is dissipative, that is, $\begin{matrix} Re {(A U, U)}_{H} ⩽ 0, \forall U \in D (A) . \end{matrix}$ Here, ${(\cdot, \cdot)}_{H}$ denotes the scalar product associated with the norm $| \cdot |_{H}$ .
Proof.
By direct computations, $\begin{matrix} Re {(A U, U)}_{H} = {(T η, η)}_{M} . \end{matrix}$ The claim then follows from (3.6). □
Lemma 4.3.
There exists $ω > 0$ such that $\begin{matrix} Range (A - ω I) = H . \end{matrix}$
Proof.
Given any $F = (f_{1}, f_{2}, f_{3}, f_{4}, f_{5}) \in H$ , we look for a solution $\begin{matrix} (u, v, α, θ, η) \in D (A) \end{matrix}$ to the system $\begin{matrix} \{\begin{matrix} (1 - ω) v = f_{1}, \\ {(C_{i j k l} u_{k, l} - β_{i j} θ)}_{, j} - ω u = ρ f_{2}, \\ (1 - ω) θ = f_{3}, \\ - β_{i j} v_{i, j} + {(g_{i j} (\infty) α_{, j} - \int_{0}^{\infty} g_{i j}^{'} (s) η_{, j} d s)}_{, i} - ω α = c f_{4}, \\ - η^{'} - ω η + θ = f_{5} . \end{matrix} \end{matrix}$ The solutions $v$ and θ are obviously found whenever $ω \neq 1$ . By assumptions (3.3)–(3.4), the equation $\begin{matrix} {(C_{i j k l} u_{k, l})}_{, j} - ω u_{i} = ρ f_{2 i} + {(1 - ω)}^{- 1} {(β_{i j} f_{3})}_{, j} \end{matrix}$ is elliptic for ω small enough (independent of $F$ ). Since the right-hand side belongs to $H$ , we obtain a unique solution $u \in V$ . Let us come to the last equation $\begin{matrix} η^{'} + ω η = h, with h = - f_{5} + {(1 - ω)}^{- 1} f_{3} \in M . \end{matrix}$ A straightforward integration, along with the condition $η (0) = 0$ , entail $\begin{matrix} η (s) = \int_{0}^{s} e^{- ω (s - y)} h (y) d y . \end{matrix}$ In order to show that $η \in M$ , we compute as follows, exploiting the fact that μ is decreasing: $\begin{matrix} \sqrt{μ (s)} {‖ η (s) ‖}_{1} ⩽ \sqrt{μ (s)} \int_{0}^{s} e^{- ω (s - y)} {‖ h (y) ‖}_{1} d y ⩽ \int_{0}^{\infty} e^{- ω (s - y)} \sqrt{μ (y)} {‖ h (y) ‖}_{1} d y . \end{matrix}$ Hence $\begin{matrix} ‖ η ‖_{M}^{2} ⩽ ‖ E * ϕ ‖_{L^{2} (R^{+})}, \end{matrix}$ having set $\begin{matrix} E (s) = e^{- ω s} and ϕ (s) = \sqrt{μ (s)} {‖ h (s) ‖}_{1}, \end{matrix}$ the star standing for convolution on $R^{+}$ . But $\begin{matrix} ‖ E * ϕ ‖_{L^{2} (R^{+}} ⩽ ‖ E ‖_{L^{1} (R^{+})} ‖ ϕ ‖_{L^{2} (R^{+})} = \frac{1}{ω} ‖ h ‖_{M}, \end{matrix}$ proving that $η \in M$ . By comparison, $η^{'} \in M$ if and only if $η \in M$ , hence $η \in D (T)$ . Finally, after substitution we have $\begin{matrix} {(g_{i j} (\infty) α_{, j})}_{, i} - ω α = {(\int_{0}^{\infty} g_{i j}^{'} (s) η_{, j} d s)}_{, i} + c f_{4} + β_{i j} f_{1 i, j} . \end{matrix}$ In light of (ii), this is an elliptic equation provided that ω is small enough. We thus obtain a unique solution $α \in V$ , and the proof is complete. □

5. Exponential stability: The one-dimensional case

5.1. Statement of the result

For the one-dimensional case, where we take $Ω = (0, ℓ)$ , and assuming that the constitutive tensors do not depend on the point $x \in (0, ℓ)$ , we find the system $\begin{matrix} \{\begin{matrix} ρ u_{t t} - ϰ u_{x x} + β α_{t x} = 0, \\ c α_{t t} - g (\infty) α_{x x} (t) + \int_{0}^{\infty} g^{'} (s) [α_{x x} (t) - α_{x x} (t - s)] d s + β u_{t x} = 0, \end{matrix} \end{matrix}$ where $\begin{matrix} g (s) = k^{*} + (τ^{- 1} k - k^{*}) e^{- τ^{- 1} s} . \end{matrix}$ The parameters appearing in the equations are related to the properties of the material, and have to satisfy some thermomechanical restrictions. In particular, it is assumed that all the constants above are strictly positive, except β that is only required to be nonzero, and $\begin{matrix} τ^{- 1} k > k^{*} . \end{matrix}$ These assumptions are in agreement with the thermomechanical axioms and the empirical experiments. The assumptions concerning the mass density and the thermal capacity are obvious. The condition on ϰ can be understood by invoking the elastic stability. The conditions on k, $k^{*}$ and τ are the natural ones to have dissipation. Finally, $β \neq 0$ is needed in order to guarantee the coupling between the mechanical and the thermal parts.

In this case, we adopt different boundary conditions, namely, the Neumann boundary conditions for u $\begin{matrix} u_{x} (0, t) = u_{x} (ℓ, t) = 0, \end{matrix}$ and the Dirichlet ones for α $\begin{matrix} α (0, t) = α (ℓ, t) = 0 . \end{matrix}$ Setting for simplicity all the constants but ϰ equal to 1, and calling $\begin{matrix} μ (s) = - g^{'} (s), \end{matrix}$ we introduce as before the auxiliary variable $η = η^{t} (s)$ , formally defined as $\begin{matrix} η^{t} (s) = α (t) - α (t - s), \end{matrix}$ and we rewrite the system above in the form $\begin{matrix} (5.1) & \{\begin{matrix} u_{t t} - ϰ u_{x x} + α_{t x} = 0, \\ α_{t t} - α_{x x} - \int_{0}^{\infty} μ (s) η_{x x} (s) d s + u_{t x} = 0, \\ η_{t} = T η + α_{t} . \end{matrix} \end{matrix}$

In what follows, $⟨ \cdot, \cdot ⟩$ and $‖ \cdot ‖$ are the standard inner product and norm on the Hilbert space $H = L^{2} (0, ℓ)$ . We define the Hilbert subspace $\begin{matrix} H_{*} = {v \in H : \int_{0}^{ℓ} v (x) d x = 0} \end{matrix}$ of zero-mean functions, along with the Sobolev spaces $\begin{matrix} V = H_{0}^{1} (0, ℓ) and V_{*} = H^{1} (0, ℓ) \cap H_{*}, \end{matrix}$ both endowed with the gradient norm, due to the Poincaré inequality. Then, introducing the phase space $\begin{matrix} H = V_{*} \times H_{*} \times V \times H \times M, \end{matrix}$ where $\begin{matrix} M = L_{μ}^{2} (R^{+}; V), \end{matrix}$ by the same arguments of the previous sections, system (5.1) generates a (linear) contraction semigroup $S (t)$ on $H$ .

Remark 5.1.
The choice of the spaces of zero-mean functions for the variable u and its derivative is consistent. Indeed, calling $\begin{matrix} m (t) = \int_{0}^{ℓ} u (x, t) d x, \end{matrix}$ and integrating the first equation of system (5.1) on $(0, ℓ)$ , we obtain the differential equation $m_{t t} (t) = 0$ . Hence, if $m (0) = m_{t} (0) = 0$ it follows that $m (t)$ is zero for all times.

In fact, as we saw, the generation of the semigroup holds true also if we consider a kernel μ of a more general form.
5.2. Assumptions on the memory kernel

In greater generality, we will assume in the sequel that $μ : [0, \infty) \to R^{+}$ is an absolutely continuous summable function satisfying the inequality $\begin{matrix} (5.2) & μ^{'} (s) + δ μ (s) ⩽ 0, \end{matrix}$ for some $δ > 0$ and almost every $s > 0$ . In particular, $μ^{'}$ is negative. For further use, we denote $\begin{matrix} κ = \int_{0}^{\infty} μ (s) d s \in (0, \infty) . \end{matrix}$

Then the following theorem holds.

Theorem 5.2.
Let μ satisfy ( 5.2 ). Then, the semigroup $S (t)$ is exponentially stable. Namely, $\begin{matrix} ‖ S (t) ‖ ⩽ M e^{- ω t}, \end{matrix}$ for some $M ⩾ 1$ and $ω > 0$ , the norm above being the operator one on the Hilbert space $H$ .

Although it is possible to prove the result via linear semigroup techniques (via the Prüss theorem [41]), we will choose here to provide a more direct proof via energy estimates. This approach has the advantage that can be successfully exported to study nonlinear versions of the problem. However, before going to the proof, some comments are in order.
The choice of the spaces of zero-mean functions is essential, with the above boundary conditions. Indeed, if we relax this assumption, every triple $(u (t), α (t), η^{t})$ of the form $(k, 0, 0)$ , for a fixed constant k, solves system (5.1). In this case, we clearly do not have any decay.

In the assumptions above, we did not take $ϰ = 1$ to stress that the exponential decay occurs independently of the respective wave speeds of the two equations. In fact, when $ϰ = 1$ the dissipation is stronger (see the forthcoming Remark 5.5), and the proof of the exponential decay becomes simpler.

Actually, Theorem 5.2 holds true also for different boundary conditions, such as the Dirichlet-Dirichlet considered in the previous part of this work. In that case, the semigroup $S (t)$ acts (and is exponentially stable) on the space $V \times H \times V \times H \times M$ . This can be shown by semigroup techniques, whereas an energy-estimate based proof seems to be more difficult to obtain.

5.3. Proof of Theorem 5.2

Along the proof, $C > 0$ will stand for a generic constant. We will use several times the estimate, obtained via the Hölder inequality, $\begin{matrix} (5.3) & \int_{0}^{\infty} μ (s) ‖ η_{x} (s) ‖ d s = \int_{0}^{\infty} \sqrt{μ (s)} \sqrt{μ (s)} ‖ η_{x} (s) ‖ d s ⩽ \sqrt{κ} ‖ η ‖_{M}^{2} . \end{matrix}$ Also, we will often apply without mention the Young inequality.

By density, it is clearly enough to obtain the desired decay for initial data in the domain of the infinitesimal generator $A$ of $S (t)$ . For any given such initial datum, let us define (twice) the energy of the system as $\begin{matrix} E (t) = ϰ {‖ u_{x} (t) ‖}^{2} + {‖ u_{t} (t) ‖}^{2} + {‖ α_{x} (t) ‖}^{2} + {‖ α_{t} (t) ‖}^{2} + {‖ η^{t} ‖}_{M}^{2} . \end{matrix}$ As before, the basic multiplication in (5.1) yields $\begin{matrix} (5.4) & \frac{d}{d t} E = \int_{0}^{\infty} μ^{'} (s) {‖ η_{x} (s) ‖}^{2} d s ⩽ 0 . \end{matrix}$ We now need to reconstruct the energy with the “good” sign. This will be done through the introduction of suitable energy functionals.

Lemma 5.3.

The functional $\begin{matrix} Φ (t) = 2 ⟨ u_{t} (t), u (t) ⟩ + 2 ⟨ α_{t} (t), α (t) ⟩ + 2 ⟨ u_{x} (t), α (t) ⟩ \end{matrix}$ fulfills the estimate $\begin{matrix} \frac{d}{d t} Φ + ϰ ‖ u_{x} ‖^{2} + ‖ α_{x} ‖^{2} + {‖ η^{t} ‖}_{M}^{2} ⩽ C_{Φ} ‖ u_{t} ‖^{2} + C_{Φ} ‖ α_{t} ‖^{2} - C_{Φ} \int_{0}^{\infty} μ^{'} (s) {‖ η_{x} (s) ‖}^{2} d s, \end{matrix}$ for some $C_{Φ} > 0$ .

Proof.

The time-derivative of Φ fulfills the equality $\begin{array}{l} \frac{d}{d t} Φ + 2 ϰ ‖ u_{x} ‖^{2} + 2 ‖ α_{x} ‖^{2} + {‖ η^{t} ‖}_{M}^{2} \\ = 2 ‖ u_{t} ‖^{2} + 2 ‖ α_{t} ‖^{2} + 4 ⟨ α_{t}, u_{x} ⟩ + {‖ η^{t} ‖}_{M}^{2} - 2 \int_{0}^{\infty} μ (s) ⟨ η_{x} (s), α_{x} ⟩ d s . \end{array}$ Clearly, $\begin{matrix} 4 ⟨ α_{t}, u_{x} ⟩ ⩽ ϰ ‖ u_{x} ‖^{2} + C ‖ α_{t} ‖^{2}, \end{matrix}$ and, using (5.3), $\begin{array}{l} {‖ η^{t} ‖}_{M}^{2} - 2 \int_{0}^{\infty} μ (s) ⟨ η_{x} (s), α_{x} ⟩ d s & ⩽ {‖ η^{t} ‖}_{M}^{2} + ‖ α_{x} ‖ \int_{0}^{\infty} μ (s) ‖ η_{x} (s) ‖ d s \\ ⩽ ‖ α_{x} ‖^{2} + C ‖ η ‖_{M}^{2} . \end{array}$ Since by (5.2) $\begin{matrix} ‖ η ‖_{M}^{2} ⩽ - \frac{1}{δ} \int_{0}^{\infty} μ^{'} (s) {‖ η_{x} (s) ‖}^{2} d s, \end{matrix}$ we are done. □

In order to introduce the next functional, we define $\begin{matrix} \hat{u} (x, t) = \int_{0}^{x} u (y, t) d y, x \in (0, ℓ) . \end{matrix}$ Note that $\hat{u} \in V$ and $\begin{matrix} ‖ {\hat{u}}_{t} ‖ ⩽ \sqrt{ℓ} ‖ u_{t} ‖ . \end{matrix}$

Lemma 5.4.

The functional $\begin{matrix} Ψ (t) = - 2 ⟨ α_{t} (t), {\hat{u}}_{t} (t) ⟩ - 2 ⟨ α_{x} (t), u (t) ⟩ \end{matrix}$ fulfills the estimate $\begin{matrix} \frac{d}{d t} Ψ + ‖ u_{t} ‖^{2} ⩽ ε_{1} ‖ u_{x} ‖^{2} + \frac{C_{Ψ}}{ε_{1}} ‖ α_{t} ‖^{2} - C_{Ψ} \int_{0}^{\infty} μ^{'} (s) {‖ η_{x} (s) ‖}^{2} d s, \end{matrix}$ for every $ε_{1} > 0$ small and some $C_{Ψ} > 0$ .

Proof.

The derivative of the first term in the right-hand side reads $\begin{matrix} - 2 \frac{d}{d t} ⟨ α_{t}, {\hat{u}}_{t} ⟩ = - 2 ⟨ α_{t t}, {\hat{u}}_{t} ⟩ - 2 ⟨ α_{t}, {\hat{u}}_{t t} ⟩ . \end{matrix}$ Integrating on $(0, x)$ the first equation of (5.1), and using the boundary conditions, we find the relation $\begin{matrix} {\hat{u}}_{t t} = ϰ u_{x} - α_{t} . \end{matrix}$ Hence, $\begin{matrix} - 2 \frac{d}{d t} ⟨ α_{t}, {\hat{u}}_{t} ⟩ = 2 ⟨ α_{x}, u_{t} ⟩ + 2 \int_{0}^{\infty} μ (s) ⟨ η_{x} (s), u_{t} ⟩ d s - 2 ‖ u_{t} ‖^{2} - 2 ϰ ⟨ α_{t}, u_{x} ⟩ + 2 ‖ α_{t} ‖^{2} . \end{matrix}$ Concerning the second term, we readily obtain $\begin{matrix} - 2 \frac{d}{d t} ⟨ α_{x}, u ⟩ = 2 ⟨ α_{t}, u_{x} ⟩ - 2 ⟨ α_{x}, u_{t} ⟩ . \end{matrix}$ Adding the two identities gives $\begin{matrix} \frac{d}{d t} Ψ + 2 ‖ u_{t} ‖^{2} = 2 ‖ α_{t} ‖^{2} + 2 (1 - ϰ) ⟨ α_{t}, u_{x} ⟩ + 2 \int_{0}^{\infty} μ (s) ⟨ η_{x} (s), u_{t} ⟩ d s . \end{matrix}$ Estimating $\begin{matrix} 2 (1 - ϰ) ⟨ α_{t}, u_{x} ⟩ ⩽ ε_{1} ‖ u_{x} ‖^{2} + \frac{C}{ε_{1}} ‖ α_{t} ‖^{2}, \end{matrix}$ and the latter integral via (5.3) as $\begin{matrix} 2 \int_{0}^{\infty} μ (s) ⟨ η_{x} (s), u_{t} ⟩ d s ⩽ ‖ u_{t} ‖^{2} + C ‖ η ‖_{M}^{2}, \end{matrix}$ the claim follows by (5.2). □

Remark 5.5.

It is clear from the proof above that, in the particular case when $ϰ = 1$ , one obtains the better estimate $\begin{matrix} \frac{d}{d t} Ψ + ‖ u_{t} ‖^{2} ⩽ 2 ‖ α_{t} ‖^{2} - C_{Ψ} \int_{0}^{\infty} μ^{'} (s) {‖ η_{x} (s) ‖}^{2} d s . \end{matrix}$ This would simplify the final argument, providing also a faster decay rate.

Lemma 5.6.

The functional $\begin{matrix} Θ (t) = - \frac{2}{κ} \int_{0}^{\infty} μ (s) ⟨ η^{t} (s), α_{t} (t) ⟩ d s, \end{matrix}$ fulfills the estimate $\begin{matrix} \frac{d}{d t} Θ + ‖ α_{t} ‖^{2} ⩽ ε_{2} ‖ α_{x} ‖^{2} + ε_{2} ‖ u_{t} ‖^{2} - \frac{C_{Θ}}{ε_{2}} \int_{0}^{\infty} μ^{'} (s) {‖ η_{x} (s) ‖}^{2} d s, \end{matrix}$ for every $ε_{2} > 0$ small and some $C_{Θ} > 0$ .

Proof.

By direct calculations we have $\begin{matrix} \frac{d}{d t} Θ = - \frac{2}{κ} \int_{0}^{\infty} μ (s) ⟨ η_{t} (s), α_{t} ⟩ d s - \frac{2}{κ} \int_{0}^{\infty} μ (s) ⟨ η (s), α_{t t} ⟩ d s . \end{matrix}$ Concerning the first term in the right-hand side, $\begin{matrix} - \frac{2}{κ} \int_{0}^{\infty} μ (s) ⟨ η_{t} (s), α_{t} ⟩ d s = - \frac{2}{κ} \int_{0}^{\infty} μ (s) ⟨ T η (s), α_{t} ⟩ d s - 2 ‖ α_{t} ‖^{2} . \end{matrix}$ Integrating by parts with respect to s, in light of the decay of μ and of the equality $η (0) = 0$ , and using the Poincaré inequality, we get $\begin{array}{l} - \frac{2}{κ} \int_{0}^{\infty} μ (s) ⟨ T η (s), α_{t} ⟩ d s & = - \frac{2}{κ} \int_{0}^{\infty} μ^{'} (s) ⟨ η (s), α_{t} ⟩ d s \\ ⩽ \frac{2}{κ} ‖ α_{t} ‖ \int_{0}^{\infty} \sqrt{- μ^{'} (s)} \sqrt{- μ^{'} (s)} ‖ η (s) ‖ d s \\ ⩽ \frac{2 \sqrt{μ (0)}}{κ} ‖ α_{t} ‖ {(- \int_{0}^{\infty} μ^{'} (s) {‖ η (s) ‖}^{2} d s)}^{1 / 2} \\ ⩽ ‖ α_{t} ‖^{2} - C \int_{0}^{\infty} μ^{'} (s) {‖ η_{x} (s) ‖}^{2} d s . \end{array}$ Here, we treated the integral as in (5.3). In summary, $\begin{matrix} - \frac{2}{κ} \int_{0}^{\infty} μ (s) ⟨ η_{t} (s), α_{t} ⟩ d s = - ‖ α_{t} ‖^{2} - C \int_{0}^{\infty} μ^{'} (s) {‖ η_{x} (s) ‖}^{2} d s . \end{matrix}$ Coming to the second term, we write $\begin{array}{l} - \frac{2}{κ} \int_{0}^{\infty} μ (s) ⟨ η (s), α_{t t} ⟩ d s \\ = \frac{2}{κ} \int_{0}^{\infty} μ (s) ⟨ η_{x} (s), α_{x} ⟩ d s + \frac{2}{κ} {‖ \int_{0}^{\infty} μ (s) η_{x} (s) d s ‖}^{2} - \frac{2}{κ} \int_{0}^{\infty} μ (s) ⟨ η_{x} (s), u_{t} ⟩ d s . \end{array}$ By means of (5.3), for every $ε_{2} > 0$ small we easily find $\begin{matrix} - \frac{2}{κ} \int_{0}^{\infty} μ (s) ⟨ η (s), α_{t t} ⟩ d s ⩽ ε_{2} ‖ α_{x} ‖^{2} + ε_{2} ‖ u_{t} ‖^{2} + \frac{C}{ε_{2}} ‖ η ‖_{M}^{2} . \end{matrix}$ Recalling (5.2), the conclusion follows. □

At this point, for $a, b > 0$ to be fixed (as well as $ε_{1}$ , $ε_{2}$ ), we define $\begin{matrix} Λ (t) = Φ (t) + a Ψ (t) + b Θ (t) . \end{matrix}$ Collecting the estimates of the three previous lemmas, we get $\begin{array}{l} \frac{d}{d t} Λ + (ϰ - a ε_{1}) ‖ u_{x} ‖^{2} + (1 - b ε_{2}) ‖ α_{x} ‖^{2} + (a - C_{Φ} - b ε_{2}) ‖ u_{t} ‖^{2} \\ + (b - C_{Φ} - \frac{a C_{Ψ}}{ε_{1}}) ‖ α_{t} ‖^{2} + {‖ η^{t} ‖}_{M}^{2} \\ ⩽ - C_{Λ} \int_{0}^{\infty} μ^{'} (s) {‖ η_{x} (s) ‖}^{2} d s, \end{array}$ where $\begin{matrix} C_{Λ} = C_{Φ} + a C_{Ψ} + \frac{b C_{Θ}}{ε_{2}} . \end{matrix}$ Now it is necessary a subtle balance of the constants. We first fix $\begin{matrix} a = 1 + C_{Φ} . \end{matrix}$ Then we choose $ε_{1}$ small enough that $\begin{matrix} ϰ - a ε_{1} ⩾ \frac{ϰ}{2} . \end{matrix}$ Once a and $ε_{1}$ are chosen, we take b large enough that $\begin{matrix} b - C_{Φ} - \frac{a C_{Ψ}}{ε_{1}} ⩾ \frac{1}{2} . \end{matrix}$ We are left to fix $ε_{2}$ . We choose it small enough in such a way that $\begin{matrix} 1 - b ε_{2} ⩾ \frac{1}{2} . \end{matrix}$ Accordingly, $\begin{matrix} a - C_{Φ} - b ε_{2} = 1 - b ε_{2} ⩾ \frac{1}{2} . \end{matrix}$ With this selection of the constants (which fixes also $C_{Λ}$ ), we arrive at the inequality $\begin{matrix} (5.5) & \frac{d}{d t} Λ + \frac{1}{2} E ⩽ - C_{Λ} \int_{0}^{\infty} μ^{'} (s) {‖ η_{x} (s) ‖}^{2} d s . \end{matrix}$ Finally, for $ε > 0$ , we introduce the last energy functional $\begin{matrix} W (t) = E (t) + 2 ε Λ (t) . \end{matrix}$ Up to choosing ε sufficiently small (in particular, $ε ⩽ 1 / 2 C_{Λ}$ ), it is clear that $\begin{matrix} \frac{1}{2} E (t) ⩽ W (t) ⩽ 2 E (t), \end{matrix}$ and, adding (5.4)–(5.5), $\begin{matrix} \frac{d}{d t} W + ε E ⩽ 0 . \end{matrix}$ An application of the standard Gronwall lemma completes the argument. The proof of Theorem 5.2 is finished. □

Remark 5.7.

As a final comment, we dwell on the hypotheses on μ adopted in this section, as well as in the previous one. In particular, we assumed that $μ (0) < \infty$ . This assumption is actually not needed in Section 4, where the kernel μ can be (weakly) singular at zero. For instance, we can handle a kernel of the form $\begin{matrix} μ (s) = \frac{e^{- s}}{\sqrt{s}} . \end{matrix}$ The restriction $μ (0) < \infty$ is instead used in connection with the estimate of the functional Θ in Lemma 5.6. In fact, also in this case, the problem could be circumvented via a suitable cut-off technique, which however would render the computations much more involved (see e.g. [37]).

Footnotes

Acknowledgements

The authors thank the anonymous referee her/his useful comments. Work supported by the Project “Análisis Matemático de Problemas de la Termomecánica” (MTM2016-74934-P) (AEI/FEDER, UE) of the Spanish Ministry of Economy and Competitiveness.

References

Cattaneo, On a form of heat equation which eliminates the paradox of instantaneous propagation, C. R. Acad. Sci. Paris 247 (1958), 431–433.

D.S.

Chandrasekharaiah, Hyperbolic thermoelasticity: A review of recent literature, Appl. Mech. Rev. 51 (1998), 705–729. doi:10.1115/1.3098984.

J.A.

Conejero,

Lizama and

Ródenas, Chaotic behaviour of the solutions of the Moore–Gibson–Thompson equation, Appl. Math. Inf. Sci. 9 (2015), 2233–2238.

Conti,

Danese,

Giorgi and

Pata, A model of viscoelasticity with time-dependent memory kernels, Amer. J. Math. 140 (2019), 349–389. doi:10.1353/ajm.2018.0008.

Conti,

Gatti,

Miranville and

Quintanilla, On a Caginalp phase-field system with two temperatures and memory, Milan J. Math. 85 (2017), 1–27. doi:10.1007/s00032-017-0263-z.

Conti,

Gatti and

Pata, Uniform decay properties of linear Volterra integro-differential equations, Math. Models Methods Appl. Sci. 18 (2008), 21–45. doi:10.1142/S0218202508002590.

Conti,

E.M.

Marchini and

Pata, Semilinear wave equations of viscoelasticity in the minimal state framework, Discrete Contin. Dyn. Syst. 27 (2010), 1535–1552. doi:10.3934/dcds.2010.27.1535.

Conti and

Pata, Weakly dissipative semilinear equations of viscoelasticity, Commun. Pure Appl. Anal. 4 (2005), 705. doi:10.3934/cpaa.2005.4.705.

C.M.

Dafermos, An abstract Volterra equation with applications to linear viscoelasticity, J. Differential Equations 7 (1970), 554–569. doi:10.1016/0022-0396(70)90101-4.

10.

C.M.

Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal. 37 (1970), 554–569.

11.

Dell’Oro,

Lasiecka and

Pata, The Moore–Gibson–Thompson equation with memory in the critical case, J. Differential Equations 261 (2016), 4188–4222.

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.

Dell’Oro and

Pata, On a fourth-order equation of Moore–Gibson–Thompson type, Milan J. Math. 85 (2017), 215–234. doi:10.1007/s00032-017-0270-0.

14.

A.D.

Drozdov and

V.B.

Kolmanovskii, Stability in Viscoelasticity, North-Holland, Amsterdam, 1994.

15.

Giorgi,

Grandi and

Pata, On the Green–Naghdi type III heat conduction model, Discrete Contin. Dyn. Syst. Ser. B 19 (2014), 2133–2143. doi:10.3934/dcdsb.2014.19.2133.

16.

A.E.

Green and

P.M.

Naghdi, On undamped heat waves in an elastic solid, J. Thermal Stresses 15 (1992), 253–264. doi:10.1080/01495739208946136.

17.

A.E.

Green and

P.M.

Naghdi, Thermoelasticity without energy dissipation, J. Elasticity 31 (1993), 189–208. doi:10.1007/BF00044969.

18.

A.E.

Green and

P.M.

Naghdi, A unified procedure for construction of theories of deformable media. I. Classical continuum physics, II. Generalized continua, III. Mixtures of interacting continua, Proc. Royal Society London A 448 (1995), 335–356, 357–377, 379–388. doi:10.1098/rspa.1995.0021.

19.

R.B.

Hetnarski and

Ignaczak, Generalized thermoelasticity, J. Thermal Stresses 22 (1999), 451–470. doi:10.1080/014957399280832.

20.

R.B.

Hetnarski and

Ignaczak, Nonclassical dynamical thermoelasticity, Internat. J. Solids Structures 37 (2000), 215–224. doi:10.1016/S0020-7683(99)00089-X.

21.

Iesan and

Scalia, Thermoelastic Deformations, Kluwer Academic Publishers, Dordrecht, 1996.

22.

Ignaczak and

Ostoja-Starzewski, Thermoelasticity with Finite Wave Speeds, Oxford Mathematical Monographs, Oxford, 2010.

23.

Kaltenbacher,

Lasiecka and

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

24.

Kothari and

Mukhopadhyay, Study of harmonic plane waves in rotating thermoelastic media of type III, Math. Mech. Solids 17 (2012), 824–839. doi:10.1177/1081286511432021.

25.

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.

26.

Y.Y.

Liu,

Lin and

Li, Convergence result for the thermoelasticity of type III, Appl. Math. Letters 26 (2013), 97–102. doi:10.1016/j.aml.2012.04.001.

27.

Liu and

Quintanilla, Analiticity of solutions in type III thermoelastic plates, IMA J. Appl. Math. 75 (2010), 637–646. doi:10.1093/imamat/hxq041.

28.

Liu and

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

29.

H.W.

Lord and

Shulman, A generalized dynamical theory of thermoelasticity, J. Mech. Phys. Solids 15 (1967), 299–309. doi:10.1016/0022-5096(67)90024-5.

30.

Magaña and

Quintanilla, Exponential stability in type III thermoelasticity with microtemperatures, Z. Angew. Math. Phys. (ZAMP) 69(5) (2018), 129(1)–129(8).

31.

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 (2012), 1896–1929. doi:10.1002/mma.1576.

32.

Miranville and

Quintanilla, Some generalizations of the Caginalp phase-field system, Appl. Anal. 88 (2009), 877–894. doi:10.1080/00036810903042182.

33.

Miranville and

Quintanilla, A type III phase-field system with logarithmic potential, Appl. Math. Letters 24 (2011), 1003–1008. doi:10.1016/j.aml.2011.01.016.

34.

Miranville and

Quintanilla, Exponential stability in type III thermoelasticity with voids, Appl. Math. Letters 94 (2019), 30–37. doi:10.1016/j.aml.2019.02.014.

35.

Mukhopadhyay and

Kumar, A problem of thermoelastic interactions in an infinite medium with a cylindrical hole in generalized thermoelasticity III, J. Thermal Stress 31 (2008), 455–475. doi:10.1080/01495730801912561.

36.

Mukhopadhyay,

Prasad and

Kumar, Variational and reciprocal principles in linear theory of type-III thermoelasticity, Math. Mech. Solids 16 (2011), 435–444. doi:10.1177/1081286511399015.

37.

Pata, Stability and exponential stability in linear viscoelasticity, Milan J. Math. 77 (2009), 333–360. doi:10.1007/s00032-009-0098-3.

38.

Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, Vol. 44, Springer-Verlag, New York, 1983.

39.

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. (2017), 1–32.

40.

Pellicer and

Sola-Morales, Optimal scalar products in the Moore–Gibson–Thompson equation, Evol. Equ. Control Theory 8 (2019), 203–220. doi:10.3934/eect.2019011.

41.

Prüss, On the spectrum of

C_{0}

-semigroups, Trans. Amer. Math. Soc. 284 (1984), 847–857. doi:10.2307/1999112.

42.

Quintanilla, Structural stability and continuous dependence of solutions in thermoelasticity of type III, Discrete Contin. Dyn. Syst. Ser. B 1 (2001), 463–470. doi:10.3934/dcdsb.2001.1.463.

43.

Quintanilla, Convergence and structural stability in thermoelasticity, Appl. Math. Comput. 135 (2003), 287–300.

44.

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

45.

Quintanilla and

Straughan, Growth and uniqueness in thermoelasticity, Proc. Roy. Soc. London A 455 (2000), 1429–1429.

46.

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.

47.

Racke, Asymptotic behavior of solutions in linear 2- or 3-D thermoelasticity with second sound, Quart. Appl. Math. 61 (2003), 315–328. doi:10.1090/qam/1976372.

48.

Renardy,

W.J.

Hrusa and

J.A.

Nohel, Mathematical Problems in Viscoelasticity, John Wiley & Sons, Inc., New York, 1987.

49.

Straughan, Heat Waves, Springer, New York, 2011.

50.

P.A.

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

Thermoelasticity of Moore–Gibson–Thompson type with history dependence in the temperature

Abstract

Keywords

1. Introduction

1.1. Notation

1.2. Classical heat conduction

1 Several authors include a sign “−” at the right hand side of the relation, however here we adopt the approach of [21] (see page 41).

1.4. MGT heat conduction

1.5. Plan of the paper

2. The MGT heat equation with history

3.1. General assumptions

3.3. The system in the past history framework

4. The solution semigroup

5.1. Statement of the result

Footnotes

Acknowledgements

References

¹
Several authors include a sign “−” at the right hand side of the relation, however here we adopt the approach of [21] (see page 41).