1 Introduction
Let \(\Omega \subset {\mathbb {R}}^n\) be a bounded domain with smooth boundary \(\partial \Omega\). On the space-time cylinder \(\Omega _+=\Omega \times [0,\infty )\), we consider the linear evolution system of partial differential equationsin the unknown variablesHere, \(\alpha ,\beta ,\gamma ,\kappa >0\) and \(\eta \ne 0\) are fixed structural constants, while \(\Delta\) is the Laplace operator with Dirichlet boundary conditions. In the uncoupled case when \(\eta =0\), the first equation readsThis is the so-called Moore-Gibson-Thompson (MGT) equation appearing in the context of acoustic wave propagation in viscous thermally relaxing fluids [18, 23], although it has been originally introduced in a very old paper of Stokes [21]. Such an equation has received a considerable attention in recent years, mainly due to the large number of possible applications not only in acoustics but also in lithotripsy, high intensity focused ultrasounds and other physical phenomena (see for instance [14] and references therein). Quite interestingly, it can be used as a model for the vibrations in a standard linear viscoelastic solid, for it can obtained by differentiating in time the equation of viscoelasticity with an exponential kernel (see [8] for more details). The same equation also arises as a model for the temperature evolution in a type III heat conduction with a relaxation parameter (see [20]). In the system (1.1), the MGT equation is coupled with the classical Fourier heat equation by means of the coupling constant \(\eta\). Such a model pops up for instance in the description of the vibrations of a viscoelastic heat conductor obeying the Fourier thermal law and governed by the so-called standard linear solid model (see [1, 10‐12]).
$$\begin{aligned} {\left\{ \begin{array}{ll} u_{ttt}+\alpha u_{tt} - \beta \Delta u_t - \gamma \Delta u =- \eta \Delta \theta , \\ \theta _t - \kappa \Delta \theta = \eta \Delta u_{tt} + \eta \alpha \Delta u_t, \end{array}\right. } \end{aligned}$$
(1.1)
$$\begin{aligned} u=u(\varvec{x},t):\Omega _+\rightarrow {\mathbb {R}}\qquad \text {and}\qquad \theta =\theta (\varvec{x},t):\Omega _+\rightarrow {\mathbb {R}}. \end{aligned}$$
$$\begin{aligned} u_{ttt}+\alpha u_{tt} - \beta \Delta u_t - \gamma \Delta u =0. \end{aligned}$$
From the mathematical viewpoint, system (1.1) generates a solution semigroup S(t) acting on the natural weak energy spacewhere with standard notation \(L^2(\Omega )\) is the Lebesgue space of square summable functions on \(\Omega\), while \(H_0^1(\Omega )\) is the space of functions belonging to \(L^2(\Omega )\) along with their first derivatives, and having null trace on the boundary \(\partial \Omega\). The asymptotic properties of S(t) depend on choice of the constitutive parameters \(\alpha ,\beta ,\gamma ,\kappa ,\eta\). Namely, introducing the stability number \(\varkappa\) defined asthe following three regimes occur:As shown in [1], in the subcritical case the semigroup is exponentially stable, namely, there exist \(\omega >0\) and \(M\ge 1\) such thatwhere \(\Vert \cdot \Vert\) denotes the operator norm. Moreover, it has been recently proved in [6] that there exists a structural threshold \({\mathsf t}\) independent of \(\eta\), of the formsuch that exponential stability occurs wheneverIn particular, S(t) turns out to be exponentially stable (for all \(\eta \ne 0\)) also in the critical case, whereas in the supercritical case exponential stability takes place as soon as \(|\eta |\) is sufficiently large.
$$\begin{aligned} H_0^1(\Omega ) \times H_0^1(\Omega ) \times L^2(\Omega ) \times L^2(\Omega ), \end{aligned}$$
$$\begin{aligned} \varkappa = \beta - \frac{\gamma }{\alpha }, \end{aligned}$$
(1.2)
-
the subcritical case \(\varkappa >0\);
-
the critical case \(\varkappa =0\);
-
the supercritical case \(\varkappa <0\).
$$\begin{aligned} \Vert S(t)\Vert \le M \mathrm{e}^{-\omega t},\quad \forall t\ge 0, \end{aligned}$$
$$\begin{aligned} {\mathsf t}=0&\quad \text {if } \varkappa \ge 0,\\ {\mathsf t}>0&\quad \text {if } \varkappa <0, \end{aligned}$$
$$\begin{aligned} |\eta |> {\mathsf t}. \end{aligned}$$
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Another meaningful physical model can be obtained by replacing the operator \(-\Delta\) in the MGT equation with the Bilaplacian \(\Delta ^2\) (i.e. the square of \(-\Delta\)). In this situation, one has the evolution systemmodeling thermoviscoelastic plates of MGT type. Such a system has been introduced and analyzed in [7], where it has been proved that in the subcritical case the associated solution semigroup S(t), acting now on the spaceis not only exponentially stable but also analytic (see [9, 19] for the definition of an analytic semigroup). This means that the coupling with the (parabolic) heat equation is still able to stabilize uniformly the dynamics to zero (in the subcritical case) and, in addition, it produces strong regularizing effects on the solutions. The analyticity of S(t) has been recently shown to occur also in the critical and in the supercritical cases [5].
$$\begin{aligned} {\left\{ \begin{array}{ll} u_{ttt}+\alpha u_{tt} + \beta \Delta ^2 u_t + \gamma \Delta ^2 u =- \eta \Delta \theta , \\ \theta _t - \kappa \Delta \theta = \eta \Delta u_{tt} + \eta \alpha \Delta u_t, \end{array}\right. } \end{aligned}$$
(1.3)
$$\begin{aligned} \big[H^2(\Omega )\cap H_0^1(\Omega )\big] \times \big [H^2(\Omega )\cap H_0^1(\Omega )\big] \times L^2(\Omega ) \times L^2(\Omega ), \end{aligned}$$
In light of the discussion above, it is then natural to wonder whether the semigroup associated to (1.1) is analytic or not. Even more so, one may ask what happens when the operator \(-\Delta\) is replaced by the fractional Laplace-Dirichlet operator \((-\Delta )^\varrho\) with \(\varrho \in [1,2]\). The aim of the present article is to address these issues.
2 The Abstract System
Let \((H,\langle \cdot ,\cdot \rangle ,\Vert \cdot \Vert )\) be a complex Hilbert space, and letbe a strictly positive selfadjoint unbounded linear operator. For \(\varrho \in [1,2]\), we consider the abstract evolution systemin the unknowns \(u:[0,\infty )\rightarrow H\) and \(\theta :[0,\infty )\rightarrow H\), where as before \(\alpha ,\beta ,\gamma ,\kappa >0\) and \(\eta \ne 0\) are fixed constants. Unless explicitly stated otherwise, we shall always work in the subcritical regimewhere the stability number \(\varkappa\) has been defined in (1.2).
$$\begin{aligned} A:\mathrm{dom}(A)\subset H\rightarrow H \end{aligned}$$
$$\begin{aligned} {\left\{ \begin{array}{ll} u_{ttt}+\alpha u_{tt}+\beta A^\varrho u_t+\gamma A^\varrho u = \eta A \theta , \\ \theta _t+\kappa A \theta = -\eta A u_{tt} - \eta \alpha A u_t, \end{array}\right. } \end{aligned}$$
(2.1)
$$\begin{aligned} \varkappa >0, \end{aligned}$$
Remark 2.1
The reason why we work with a complex Hilbert space is that in this paper we use semigroup techniques based on the spectral properties of certain operators. Such techniques are meaningful only in the complex setting. Nonetheless, since the operators appearing in (2.1) are selfadjoint, all the results hold for a real Hilbert space as well.
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With an eye to the remark above, let us choose H as the (real) Hilbert space \(L^2(\Omega )\), and A as the Laplace-Dirichlet operatorIn this situation, the physical systems (1.1) and (1.3) turn out to be the concrete realizations of (2.1) corresponding to the choice \(\varrho =1\) and \(\varrho =2\), respectively.
$$\begin{aligned} A=-\Delta \,\,\, \text {with}\,\,\, \mathrm{dom}(A)=H^2(\Omega )\cap H^1_0(\Omega ). \end{aligned}$$
As our main result, we prove that the solution semigroup S(t) generated by system (2.1) is analytic and exponentially stable for every \(\varrho \in [1,2]\).
2.1 Plan of the paper
In the forthcoming Sect. 3 we introduce the functional setting and the notation. The subsequent Sect. 4 deals with the existence of the solution semigroup S(t) and the analysis of some spectral properties of its infinitesimal generator. In Sect. 5 we state and prove the main results of the article concerning the analyticity and the exponential stability of S(t) in the subcritical case. Possible extensions to the critical case are briefly discussed in Sect. 6, while the final Sect. 7 is devoted to some concluding remarks.
3 Functional setting
For \(p \in {\mathbb {R}}\), we consider the scale of continuously nested Hilbert spaces (the index p will be always omitted whenever zero)If \(p>0\) it is understood that \(H^{-p}\) denotes the completion of the domain. Denoting by \(\sigma (A)\) the spectrum of A, and recalling that \(\sigma (A)\) is a nonempty closed subset of \((0,\infty )\), we setThen, for \(p>0\), we have the generalized Poincaré inequalityand, for \(0\le p\le q\), we have the interpolation inequalityFor a fixed \(\varrho \in [1,2]\), the phase space associated to our abstract system (2.1) is the product Hilbert spaceendowed with the inner productSince \(\varkappa >0\), using the Poincaré and Young inequalities one can readily check that the associated (square of the) normis equivalent to the standard (square of the) product norm
$$\begin{aligned} H^p = \mathrm{dom}(A^{p/2}),\,\, \langle u,v\rangle _{p} =\langle A^{p/2}u,A^{p/2}v\rangle ,\,\, \Vert u\Vert _p=\Vert A^{p/2}u\Vert . \end{aligned}$$
$$\begin{aligned} s_0 = \min \{s : s\in \sigma (A)\} >0. \end{aligned}$$
$$\begin{aligned} \Vert u\Vert ^2 \le s_0^{-p} \Vert u\Vert _p^2, \quad \,\, \forall u \in H^p, \end{aligned}$$
$$\begin{aligned} \Vert u\Vert _{p} \le \Vert u\Vert _q^{1/2}\Vert u\Vert _{2p-q}^{1/2},\quad \,\, \forall u \in H^q. \end{aligned}$$
$$\begin{aligned} {\mathcal H}= H^\varrho \times H^\varrho \times H \times H \end{aligned}$$
$$\begin{aligned} \langle (u,v,w,\theta ) , (u_1 , v_1, w_1,\theta_1)\rangle _{{\mathcal H}}= & {} \frac{\gamma }{\alpha } \langle v + \alpha u , v_1 + \alpha u_1\rangle _\varrho \\&+\langle w+\alpha v,w_1 + \alpha v_1\rangle +\varkappa \langle v,v_1\rangle _\varrho + \langle \theta , \theta_1\rangle . \end{aligned}$$
$$\begin{aligned} \Vert (u,v,w,\theta )\Vert _{{\mathcal H}}^2= \,& {} \frac{\gamma }{\alpha } \Vert v + \alpha u \Vert ^2_\varrho + \Vert w+\alpha v\Vert ^2 +\varkappa \Vert v\Vert ^2_\varrho +\Vert \theta \Vert ^2 \end{aligned}$$
$$\begin{aligned} \Vert u \Vert ^2_\varrho + \Vert v\Vert ^2_\varrho + \Vert w\Vert ^2 +\Vert \theta \Vert ^2. \end{aligned}$$
3.1 General agreements
Along the paper, the Poincaré and Young inequalities will be tacitly used in several occasions. Moreover, as customary, we shall regard \(A^p\) as an isometric isomorphism from \(H^q\) onto \(H^{q-2p}\) for all \(p,q\in {\mathbb {R}}\).
4 The infinitesimal generator
Introducing the state vector \({\varvec{u}}= (u,v,w,\theta ) \in {\mathcal H}\), we view system (2.1) as the abstract ordinary differential equationwhere \({{\mathbb {A}}}: \mathrm{dom}({{\mathbb {A}}}) \subset {\mathcal H}\rightarrow {\mathcal H}\) is the linear operator defined aswith (dense) domainRecalling that \(\varrho \in [1,2]\), the third condition above yieldsIn turn, for every \({\varvec{u}}=(u,v,w,\theta ) \in \mathrm{dom}({{\mathbb {A}}})\), a straightforward computation entailsHence, the operator \({{\mathbb {A}}}\) is dissipative. By means of standard arguments, it is also possible to prove that the operator \(1-{{\mathbb {A}}}\) is surjective (the details are left to the interested reader). As a consequence, appealing to the Lumer-Phillips theorem (see e.g. [19]), we infer that the operator \({{\mathbb {A}}}\) is the infinitesimal generator of a contraction semigroupWe conclude the section by showing that the spectrum \(\sigma ({{\mathbb {A}}})\) of \({{\mathbb {A}}}\) is contained in the open left half-planeSuch a result will play an important role in the sequel.
$$\begin{aligned} {\varvec{u}}' = {{\mathbb {A}}}{\varvec{u}}, \end{aligned}$$
$$\begin{aligned} {\mathbb {A}} \begin{pmatrix} u \\ v \\ w \\ \theta \end{pmatrix} = \begin{pmatrix} v \\ w \\ -\alpha w + A(\eta \theta - A^{\varrho -1}(\beta v + \gamma u) )\\ - A (\kappa \theta +\eta ( w + \alpha v)) \end{pmatrix} \end{aligned}$$
(4.1)
$$\begin{aligned} \mathrm{dom}({{\mathbb {A}}}) = \left\{ (u,v,w,\theta ) \in {\mathcal H}\, \bigg | \, \begin{array}{c} w \in H^\varrho \\ \eta \theta - A^{\varrho -1}(\beta v + \gamma u) \in H^2\\ \kappa \theta +\eta (w + \alpha v) \in H^2 \end{array} \right\} . \end{aligned}$$
$$\begin{aligned} \theta \in H^{\varrho }\subset H^1. \end{aligned}$$
$$\begin{aligned} \mathfrak {Re\,}\langle {\mathbb {A}} {\varvec{u}}, {\varvec{u}}\rangle _{{\mathcal H}} = - \alpha \varkappa \Vert v\Vert ^2_\varrho - \kappa \Vert \theta \Vert ^2_1\le 0. \end{aligned}$$
(4.2)
$$\begin{aligned} S(t) : {\mathcal H}\rightarrow {\mathcal H}. \end{aligned}$$
$$\begin{aligned} {{\mathbb {C}}}^{-} = \{ z \in {{\mathbb {C}}}:\, \mathfrak {Re\,}(z) < 0\}. \end{aligned}$$
Proposition 4.1
We have the inclusion \(\sigma ({{\mathbb {A}}}) \subset {{\mathbb {C}}}^{-}\).
Proof
Being \({{\mathbb {A}}}\) the infinitesimal generator of a contraction semigroup, the Hille-Yosida theorem (see e.g. [19]) implies that \(\sigma ({{\mathbb {A}}})\) is contained in the closed left half-plane. Accordingly, we only need to show the equality \(\sigma ({{\mathbb {A}}}) \cap \mathrm{i}{\mathbb {R}}= \emptyset\). To this end, let us assume by contradiction that \(\mathrm{i}\lambda \in \sigma ({{\mathbb {A}}})\) for some \(\lambda \in {\mathbb {R}}\). Since \(\mathrm{i}{\mathbb {R}}\) is contained in the topological boundary of \(\sigma ({{\mathbb {A}}})\), it follows that \(\mathrm{i}\lambda\) is an approximate eigenvalue of \({{\mathbb {A}}}\) (see e.g. [3, Proposition B.2], but see also [22, Theorem 5.1-D]). Thus, there exists a sequence of vectors \({\varvec{u}}_n =(u_n,v_n,w_n,\theta _n)\in \mathrm{dom}({{\mathbb {A}}})\) of unit norm (in \({\mathcal H}\)) such thatIn components, we obtain the following systemMultiplying (4.3) by \({\varvec{u}}_n\) in \({\mathcal H}\) and invoking (4.2), we see immediately thatRelation (4.5) now tells that \(w_n\rightarrow 0\) in \(H^\varrho\), from whichFinally, we take the inner product in H of (4.6) with \(u_n\) to getRecalling that \(\varrho \ge 1\), we estimatefor some \(c>0\) (depending on \(\lambda\)). Since \(u_n\) is bounded in \(H^\varrho\), from (4.8)–(4.10) we learn that the right-hand side converges to zero. Therefore, we end up withCollecting (4.8)–(4.11) we conclude that \({\varvec{u}}_n\rightarrow 0\) in \({\mathcal H}\), against the assumption that \({\varvec{u}}_n\) has unit norm. \(\square\)
$$\begin{aligned} \mathrm{i}\lambda {\varvec{u}}_n - {{\mathbb {A}}}{\varvec{u}}_n \rightarrow 0 \quad \text {in } {\mathcal H}. \end{aligned}$$
(4.3)
$$\begin{aligned}&\mathrm{i}\lambda u_n - v_n \rightarrow 0 \quad \text {in } H^\varrho , \end{aligned}$$
(4.4)
$$\begin{aligned}&\mathrm{i}\lambda v_n - w_n \rightarrow 0 \quad \text {in } H^\varrho , \end{aligned}$$
(4.5)
$$\begin{aligned}&\mathrm{i}\lambda w_n + \alpha w_n - A(\eta \theta _n - A^{\varrho -1}(\beta v_n + \gamma u_n)) \rightarrow 0 \quad \text {in } H, \end{aligned}$$
(4.6)
$$\begin{aligned}&\mathrm{i}\lambda \theta _n + A (\kappa \theta _n +\eta (w_n + \alpha v_n)) \rightarrow 0 \quad \text {in } H. \end{aligned}$$
(4.7)
$$\begin{aligned}&v_n \rightarrow 0 \quad \text {in } H^\varrho , \end{aligned}$$
(4.8)
$$\begin{aligned}&\theta _n \rightarrow 0 \quad \text {in } H^1 \quad \Rightarrow \quad \theta _n \rightarrow 0 \quad \text {in } H. \end{aligned}$$
(4.9)
$$\begin{aligned} w_n \rightarrow 0 \quad \text {in } H. \end{aligned}$$
(4.10)
$$\begin{aligned}&\gamma \Vert u_n\Vert ^2_\varrho + \mathrm{i}\lambda \langle w_n , u_n\rangle + \alpha \langle w_n , u_n \rangle \\&\quad - \eta \langle \theta _n, u_n\rangle _1 + \beta \langle v_n, u_n\rangle _\varrho \rightarrow 0. \end{aligned}$$
$$\begin{aligned}&|\mathrm{i}\lambda \langle w_n , u_n\rangle + \alpha \langle w_n , u_n \rangle - \eta \langle \theta _n, u_n\rangle _1 + \beta \langle v_n, u_n\rangle _\varrho | \\&\quad \le c \Vert u_n\Vert _{\varrho } \big [\Vert w_n\Vert +\Vert \theta _n\Vert _1+\Vert v_n\Vert _\varrho \big ] \end{aligned}$$
$$\begin{aligned} u_n \rightarrow 0 \quad \text {in } H^\varrho . \end{aligned}$$
(4.11)
5 The main result
5.1 The theorem
In this section, we show that the contraction semigroup S(t) generated by the operator \({{\mathbb {A}}}\) is analytic and exponentially stable.
Theorem 5.1
The semigroup S(t) is analytic.
Once the analyticity of S(t) has been established, the exponential stability follows from the spectral inclusion \(\sigma ({{\mathbb {A}}}) \subset {{\mathbb {C}}}^{-}\) ensured by Proposition 4.1. Indeed, an analytic semigroup whose infinitesimal generator satisfies such a spectral inclusion has a strictly negative growth bound. This is shown, for instance, in the proof of [2, Proposition 2.7].
Corollary 5.2
The semigroup S(t) is exponentially stable.
The proof of Theorem 5.1 is based on the following well-known abstract result (see for instance [15, Chapter 3, Theorem 3E.3] and [16, Chapter 1, Theorem 1.3.3]).
Lemma 5.3
Assume that \(\sigma ({{\mathbb {A}}}) \cap \mathrm{i}{\mathbb {R}}= \emptyset\). Then the (bounded) semigroup S(t) is analytic if and only ifwhere \(\Vert \cdot \Vert\) denotes the operator norm.
$$\begin{aligned} \limsup _{|\lambda |\rightarrow \infty } \Vert \lambda (\mathrm{i}\lambda - {{\mathbb {A}}})^{-1}\Vert <\infty , \end{aligned}$$
(5.1)
5.2 Proof of Theorem 5.1
Since Proposition 4.1 tells that \(\sigma ({{\mathbb {A}}}) \cap \mathrm{i}{\mathbb {R}}= \emptyset\), we only need to prove (5.1). To this end, assuming by contradiction that (5.1) fails to hold, it is readily seen that there exist a real sequence \(\lambda _n\) with \(|\lambda _n|\rightarrow \infty\) and a sequence of vectors \({\varvec{u}}_n =(u_n,v_n,w_n,\theta _n)\in \mathrm{dom}({{\mathbb {A}}})\) of unit norm such thatOur aim is to reach a contradiction by proving that every component of \({\varvec{u}}_n\) goes to zero in its own norm. In what follows, \(c>0\) will denote a generic constant, whose value might change from line to line, or even within the same line. We will also tacitly employ the boundedness of \(u_n\) and \(v_n\) in \(H^\varrho\) (and consequently in \(H^1\), as \(\varrho \ge 1\)) as well as the boundedness of \(w_n\) and \(\theta _n\) in H.
$$\begin{aligned} \mathrm{i}{\varvec{u}}_n - \lambda _{n}^{-1}{{\mathbb {A}}}{\varvec{u}}_n \rightarrow 0 \quad \text {in } {\mathcal H}. \end{aligned}$$
(5.2)
First, we multiply relation (5.2) by \({\varvec{u}}_n\) in \({\mathcal H}\). Exploiting (4.2), we infer thatNext, writing (5.2) componentwise, we obtain the systemRecalling that \(|\lambda _n|\rightarrow \infty\), it follows immediately from (5.4) thatWe now defineMultiplying (5.7) by \(w_n\) in H we deduce that \(\varepsilon _n\rightarrow 0\), hence \(|\varepsilon _n|\le c\). Since (5.3) ensures that \(|\lambda _{n}|^{-1/2}\Vert \theta _n\Vert _1\le c\), we can estimateAccordingly, we are led to the boundNext, we take the inner product in H of (5.7) and \(\theta _n\) to getThe second term converges to zero due to (5.3). Recalling (5.9) and exploiting once more (5.3), we also infer thatTherefore, we end up with the convergenceIn turn, as \(\varrho \le 2\), we learn from (5.7) together with the Poincaré inequality thatInvoking now (5.4)-(5.5) together with (5.8), the relation above implies thatIn particular, we findOn the other hand, we readily learn from (5.6) that the sequenceis bounded in \(H^2\), hence in \(H^\varrho\) appealing again on \(\varrho \le 2\). Thus,By interpolation, since \(\varrho \ge 1\), we now obtain the boundNext, a multiplication of (5.11) by \(\beta v_n + \gamma u_n\) in \(H^\varrho\) givesRelation (5.3) together with the boundedness of \(|\lambda _n|^{-1/2} \Vert \beta v_n + \gamma u_n\Vert _{2\varrho -1}\) ensure thatyielding the convergence \(\beta v_n + \gamma u_n\rightarrow 0\) in \(H^\varrho\). In the light of (5.8), the latter entailsAt this point, we take the inner product in H of (5.6) and \(w_n\) to getSince \(|\lambda _n|\rightarrow \infty\), the second term converges to zero. Owing to (5.3) and (5.9), we also see thatFinally, collecting (5.5) and (5.12), we learn that \(\lambda _{n}^{-1} w_n\rightarrow 0\) in \(H^\varrho\), which leads toIn conclusion, we have proved the convergence \(w_n\rightarrow 0\) in H. Together with (5.8), (5.10) and (5.12), the latter leads to the desired contradiction. \(\square\)
$$\begin{aligned} |\lambda _{n}|^{-1/2}\theta _n \rightarrow 0 \quad \text {in } H^1. \end{aligned}$$
(5.3)
$$\begin{aligned}&\mathrm{i}u_n - \lambda _{n}^{-1} v_n \rightarrow 0 \quad \text {in } H^\varrho , \end{aligned}$$
(5.4)
$$\begin{aligned}&\mathrm{i}v_n - \lambda _{n}^{-1} w_n \rightarrow 0 \quad \text {in } H^\varrho , \end{aligned}$$
(5.5)
$$\begin{aligned}&\mathrm{i}w_n + \lambda _{n}^{-1} (\alpha w_n - A(\eta \theta _n - A^{\varrho -1}(\beta v_n + \gamma u_n))) \rightarrow 0 \quad \text {in } H, \end{aligned}$$
(5.6)
$$\begin{aligned}&\mathrm{i}\theta _n + \lambda _{n}^{-1} A (\kappa \theta _n +\eta (w_n + \alpha v_n)) \rightarrow 0 \quad \text {in } H. \end{aligned}$$
(5.7)
$$\begin{aligned} u_n \rightarrow 0 \quad \text {in } H^\varrho . \end{aligned}$$
(5.8)
$$\begin{aligned} \varepsilon _n= \mathrm{i}\langle \theta _n , w_n \rangle + \kappa \lambda _{n}^{-1}\langle \theta _n, w_n \rangle _1 +\eta \lambda _{n}^{-1} \Vert w_n\Vert _1^2 + \eta \alpha \lambda _{n}^{-1} \langle v_n , w_n\rangle _1. \end{aligned}$$
$$\begin{aligned} |\lambda _{n}|^{-1} \Vert w_n\Vert _1^2&\le c |\varepsilon _n| + c\Vert \theta _n\Vert \Vert w_n\Vert + c \big [|\lambda _{n}|^{-1/2}\Vert w_n \Vert _1\big ] \big [|\lambda _{n}|^{-1/2} \Vert \theta _n\Vert _{1} + |\lambda _{n}|^{-1/2} \Vert v_n\Vert _1\big ]\\&\le \frac{1}{2} |\lambda _{n}|^{-1} \Vert w_n\Vert _1^2 + c. \end{aligned}$$
$$\begin{aligned} |\lambda _{n}|^{-1/2} \Vert w_n\Vert _1\le c. \end{aligned}$$
(5.9)
$$\begin{aligned} \mathrm{i}\Vert \theta _n\Vert ^2 + \kappa \lambda _n^{-1}\Vert \theta _n\Vert _1^2 + \eta \lambda _n^{-1}\langle w_n +\alpha v_n, \theta _n \rangle _1 \rightarrow 0. \end{aligned}$$
$$\begin{aligned} |\eta \lambda _n^{-1}\langle w_n +\alpha v_n, \theta _n \rangle _1 | \le c \big [|\lambda _{n}|^{-1/2} \Vert \theta _n\Vert _{1}\big ]\big [|\lambda _{n}|^{-1/2}\Vert w_n\Vert _1 + |\lambda _{n}|^{-1/2}\Vert v_n \Vert _1\big ] \rightarrow 0. \end{aligned}$$
$$\begin{aligned} \theta _n \rightarrow 0 \quad \text {in } H. \end{aligned}$$
(5.10)
$$\begin{aligned} \lambda _{n}^{-1} (\kappa \theta _n +\eta (w_n + \alpha v_n) )\rightarrow 0 \quad \text {in } H^\varrho . \end{aligned}$$
$$\begin{aligned} \lambda _{n}^{-1} \beta \kappa \theta _n +\mathrm{i}\eta (\beta v_n + \gamma u_n) \rightarrow 0 \quad \text {in } H^\varrho . \end{aligned}$$
(5.11)
$$\begin{aligned} |\lambda _n|^{-1} \Vert \theta _n\Vert _{\varrho }\le c. \end{aligned}$$
$$\begin{aligned} \lambda _{n}^{-1}(\eta \theta _n - A^{\varrho -1}(\beta v_n + \gamma u_n)) \end{aligned}$$
$$\begin{aligned} |\lambda _{n}|^{-1} \Vert A^{\varrho -1}(\beta v_n + \gamma u_n)\Vert _{\varrho } =|\lambda _n|^{-1} \Vert \beta v_n + \gamma u_n\Vert _{3\varrho -2} \le c. \end{aligned}$$
$$\begin{aligned} |\lambda _n|^{-1/2} \Vert \beta v_n + \gamma u_n\Vert _{2\varrho -1} \le |\lambda _n|^{-1/2} \Vert \beta v_n + \gamma u_n\Vert _{3\varrho -2}^{1/2} \, \Vert \beta v_n + \gamma u_n\Vert _\varrho ^{1/2} \le c. \end{aligned}$$
$$\begin{aligned} \mathrm{i}\eta \Vert \beta v_n + \gamma u_n\Vert _{\varrho }^2 + \lambda _{n}^{-1} \beta \kappa \langle \theta _n , \beta v_n + \gamma u_n\rangle _\varrho \rightarrow 0. \end{aligned}$$
$$\begin{aligned} |\lambda _{n}^{-1} \beta \kappa \langle \theta _n , \beta v_n + \gamma u_n\rangle _\varrho | \le c\big [|\lambda _{n}|^{-1/2} \Vert \theta _n\Vert _{1}\big ] \big [|\lambda _{n}|^{-1/2} \Vert \beta v_n + \gamma u_n\Vert _{2\varrho -1}\big ]\rightarrow 0, \end{aligned}$$
$$\begin{aligned} v_n \rightarrow 0 \quad \text {in } H^\varrho . \end{aligned}$$
(5.12)
$$\begin{aligned} \mathrm{i}\Vert w_n\Vert ^2 + \alpha \lambda _{n}^{-1} \Vert w_n\Vert ^2 - \eta \lambda _{n}^{-1} \langle \theta _n , w_n\rangle _1 + \lambda _{n}^{-1} \langle \beta v_n +\gamma u_n, w_n\rangle _\varrho \rightarrow 0. \end{aligned}$$
$$\begin{aligned} |\eta \lambda _{n}^{-1} \langle \theta _n , w_n\rangle _1| \le c \big [|\lambda _{n}|^{-1/2} \Vert \theta _n\Vert _{1}\big ]\big [|\lambda _{n}|^{-1/2}\Vert w_n\Vert _1\big ]\rightarrow 0. \end{aligned}$$
$$\begin{aligned} \lambda _{n}^{-1} \langle \beta v_n +\gamma u_n, w_n\rangle _\varrho \rightarrow 0. \end{aligned}$$
6 The critical case
In the critical case \(\varkappa =0\), it is readily seen that \(\Vert \cdot \Vert _{{\mathcal H}}\) defines only a seminorm on the space \({\mathcal H}\). Indeed, when \(\varkappa =0\), we havehence there exist nonzero vectors \({\varvec{u}}\in {\mathcal H}\) with \(\Vert {\varvec{u}}\Vert _{{\mathcal H}}=0\). Exploiting (for instance) a “pumping” technique firstly devised in [8], it is still possible to show that the operator \({{\mathbb {A}}}\) is the infinitesimal generator of a \(C_0\)-semigroup S(t) on \({\mathcal H}\). The only difference is that now we do not know that S(t) is bounded. However, it has been proved in [6] that S(t) is exponentially stable if \(\varrho =1\). In this situation, S(t) is certainly bounded and, from the general theory of \(C_0\)-semigroups (see for instance [9, 19]), we also know that Proposition 4.1 holds true. Moreover, a close inspection to the proof of Theorem 5.1 reveals that the argument used to show the validity of (5.1) applies verbatim to the case \(\varkappa =0\) as well (one just has to notice that the seminorm is controlled by the norm). We are thus in a position to employ Lemma 5.3 and conclude that S(t) is analytic when \(\varrho =1\). For \(\varrho =2\), the analyticity of S(t) in the critical (and actually in the supercritical) case has been recently proved in [5] by means of a suitable perturbation argument. It is reasonably expected that the semigroup S(t) is analytic also for \(\varrho \in (1,2)\), but a detailed proof of this fact is beyond our scopes.
$$\begin{aligned} \Vert (u,v,w,\theta )\Vert _{{\mathcal H}}^2= \frac{\gamma }{\alpha } \Vert v + \alpha u \Vert ^2_\varrho + \Vert w+\alpha v\Vert ^2 +\Vert \theta \Vert ^2, \end{aligned}$$
7 Further remarks
Despite being less relevant from the physical viewpoint, one may wonder what happens when \(\varkappa >0\) but \(\varrho \not \in [1,2]\). It can be shown that the operator \({{\mathbb {A}}}\) generates a contraction semigroup S(t) on \({\mathcal H}\) provided that \(\varrho \ge 0\), but in general Proposition 4.1 is no longer valid. More precisely, the following hold:To see that, we preliminary observe that \(\sigma ({{\mathbb {A}}}) \cap \mathrm{i}{\mathbb {R}}\subset \{ 0\}\) whenever \(\varrho \ge 0\). Indeed, with reference to the proof of Proposition 4.1, it is immediate to check that the convergencescan be obtained under the sole condition \(\varrho \ge 0\). Moreover, when \(\lambda \ne 0\), the convergencefollows immediately from (4.4) and (4.8), again under the sole condition \(\varrho \ge 0\).
$$\begin{aligned}&\sigma ({{\mathbb {A}}}) \cap \mathrm{i}{\mathbb {R}}= \{ 0\} \qquad \text {if } \varrho <1, \end{aligned}$$
(7.1)
$$\begin{aligned}&\sigma ({{\mathbb {A}}}) \cap \mathrm{i}{\mathbb {R}}= \emptyset \qquad \quad \text {if } \varrho >2. \end{aligned}$$
(7.2)
$$\begin{aligned} v_n \rightarrow 0 \quad&\text {in } H^\varrho \\ \theta _n \rightarrow 0 \quad&\text {in } H\\ w_n \rightarrow 0 \quad&\text {in } H \end{aligned}$$
$$\begin{aligned} u_n \rightarrow 0 \quad \text {in } H^\varrho \end{aligned}$$
(7.3)
We are left to prove that \(0\in \sigma ({{\mathbb {A}}})\) if and only if \(\varrho <1\). To this end, a closer look at the proof of Proposition 4.1 reveals that in order to obtain (7.3) one needs to use the constraint \(\varrho \ge 1\), but not the constraint \(\varrho \le 2\). To wit, equality (7.2) holds true. On the other hand, the operator \({{\mathbb {A}}}\) is not surjective for \(\varrho <1\). To see that, let us choose a nonzero vector \(u_1 \in H^\varrho \setminus H^1\). Callingwe consider the equation \({{\mathbb {A}}}{\varvec{u}}= {{\varvec{u}}_1}\) in the unknown \({\varvec{u}}=(u,v,w,\theta )\in \mathrm{dom}({{\mathbb {A}}})\). In components, it readsSubstituting the first, the second and the fourth equalities into the third one, we findSince \({{u_1}} \notin H^1\), it is readily seen that the right-hand side does not belong to \(H^\varrho\), meaning that the system has no solutions \({\varvec{u}}\in \mathrm{dom}({{\mathbb {A}}})\). In conclusion, equality (7.1) holds true.
$$\begin{aligned} {\varvec{u}_1} = (u_1, 0,0,0)\in {\mathcal H}, \end{aligned}$$
$$\begin{aligned} {\left\{ \begin{array}{ll} v = {{u_1}},\\ w = 0 ,\\ \alpha w - A(\eta \theta - A^{\varrho -1}(\beta v + \gamma u)) = 0, \\ A (\kappa \theta +\eta (w + \alpha v)) =0. \end{array}\right. } \end{aligned}$$
(7.4)
$$\begin{aligned} u = -\frac{1}{\gamma } \Big [\beta u_1 + \frac{\eta ^2 \alpha }{\kappa } A^{1-\varrho } u_1\Big ]. \end{aligned}$$
The spectral identity (7.1) immediately tells that S(t) cannot be exponentially stable when \(\varrho <1\) (see for instance [9, 19]). In this situation, S(t) is not even semiuniformly stable, meaning that there exists no function h(t) vanishing at infinity and such thatSee e.g. [2, 4]. Nonetheless, it is immediate to check that 0 is not an eigenvalue of \({{\mathbb {A}}}\). Just note that system (7.4) with \(u_1 =0\) has only the trivial solution. We are thus in a position to exploit the classical Arendt-Batty-Lyubich-Vũ theorem [2, 17] and conclude that S(t) is stable, i.e.for every fixed \({\varvec{u}}\in {\mathcal H}\). Moreover, it is worth mentioning that the arguments carried out in the proof of Theorem 5.1 can be adapted also to the case \(\varrho <1\) (the details are left to the interested reader). Hence, appealing to a slightly generalized version of Lemma 5.3 (see e.g. [13, Lemma 2.1]), the semigroup S(t) turns out to be analytic also for \(\varrho <1\). On the contrary, it is possible to show that condition (5.1) does not hold when \(\varrho >2\) (again, the details are left to the interested reader). Therefore, in this case, the semigroup S(t) is not analytic.
$$\begin{aligned} \Vert S(t){\varvec{u}}\Vert _{{\mathcal H}} \le h(t) \Vert {{\mathbb {A}}}{\varvec{u}}\Vert _{{\mathcal H}}, \quad \forall {\varvec{u}}\in \mathrm{dom}({{\mathbb {A}}}). \end{aligned}$$
$$\begin{aligned} \lim _{t\rightarrow \infty }\Vert S(t){\varvec{u}}\Vert _{{\mathcal H}} =0, \end{aligned}$$
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