Correction to "Thermoelasticity with antidissipation" (volume 15, number 8, 2022, 2173–2188)

In the present correction we add the missing sections 7, 8 and 9 to the original paper [1]. Such sections were present and peer-reviewed in the original submission, but they were mistakenly omitted during the preparation of the final version with the AIMS template. 7. Exponential Stability: Proof of Theorem 4.5. The proofs rely on the energy estimates obtained in Section 5. 7.1. The case (α, β) ∈ R'0 and n > 1. Having in mind the proof of Proposition 5.1, we consider the functional G defined in (5.2), with ν as in (5.3). Besides, on account of (5.6), we fix p sufficiently large to guarantee the equivalence G ∼ E, together with the differential inequality d/dt G + σ||ut||^2 + σ||θ||^2 β ≤ 0. Here σ is defined by (5.4), and is positive since n > 1. Then we introduce the functional L(t) = G(t) + ε^2Υ(t), with Υ(t) as in (5.8) and ε > 0 to be chosen later. In view of (5.9), we see that d/dt L + ε^2||u||^2 1 + (σ - ε^2)||ut||^2 + σ||θ||^2 β = ε^2 p〈θ, u〉α. The right-hand side can be controlled as in the proof of Proposition 5.4. Hence, an application of the Young inequality yields the estimate ε^2p〈θ, u〉α ≤ cpε3||u||^2 1 + pε||θ||^2 β, for some c > 0. Thus we arrive at d/dt L + ε^2(1 - cpε)||u||^2 1 + (σ - ε^2)||ut||^2 + (σ - pε)||θ||^2 β ≤ 0. Up to choosing ε > 0 small enough, we obtain d/dt L + ωE ≤ 0, for some ω > 0. Since L ∼ E, an application of the Gronwall lemma completes the argument. 7.2. The case (α, β) ∈ R''0 and n > 2. The idea of the proof is the same, but now the starting point is Proposition 5.4. Accordingly, the only difference here is that we take G as in (5.10) with ν as in (5.11), instead of the ones above. The assumption n > 2 is needed to obtain a positive σ (see (5.12)) for which d/dt G + σ||ut||^2 + σ||θ||^2 β ≤ 0. At this point we proceed exactly as before, via the introduction of Υ and of the energy-like functional L. The details are left to the interested reader. 8. Exponential Blow Up: Proofs of Theorems 4.6 and 4.7. Recall that if u is an eigenvector of A corresponding to an eigenvalue ζ, then S(t)u = e^ζt u. If ζ has strictly positive real part, then the semigroup S(t) blows up exponentially, in compliance with Definition 4.3. We also know from Lemma 3.3 that if ζ is a root of the polynomial Pn defined in (3.3), then ζ is an eigenvalue of A. Accordingly, the strategy to prove Theorems 4.6 and 4.7 consists in finding a root ζ of Pn, for some integer n, such that Re ζ > 0. To this end, we will make use of the well known Hurwitz criterion, written here in a version suitably tailored to our scopes; see, e.g., [31]. Lemma 8.1. Let P(z) = z^3 + az^2 + bz + c be a third order, monic polynomial, with real coefficients. If one of the coefficients a, b or c is strictly negative, or if ∆ = ab - c < 0, then there exists a (complex) root of P with strictly positive real part. With reference to this lemma, the coefficients of Pn read an = κλ^β n - γ, bn = λn + p^2 λ^(2α) n - γκλ^β n, cn = κλ1+βn. Hence, ∆n = anbn - cn = (κλ^β n - γ)p^2 λ^(2α) n - (κλ^β n - γ)γκλ^β n - γλn. We are now in position to prove Theorems 4.6 and 4.7. Proof of Theorem 4.6. We distinguish three cases: - If n < 1, then an < 0 for every n. - If n = 1, then ∆n = -γλn < 0 for every n. - If n > 1, since (α, β) ∈ R2 it is straightforward to check that, as n → ∞, ∆n ∼ -γκ^2 λ^(2β) n if β ≥ 1/2, while ∆n ∼ -γλn if β < 1/2. Thus ∆n < 0 for n large enough. In all cases, we fall within the situation depicted in Lemma 8.1. Proof of Theorem 4.7. If n < 1, then an < 0. On the other hand, if n ≥ 1, we immediately see that ∆n < 0 whenever p is small. 9. Final Remarks. Somehow, the longterm analysis of system (2.1) carried out in this work establishes that the antidamping of the wave equation (W) and the structural dissipation mechanism of the Fourier equation (F) are comparable from a qualitative point of view. We also recalled that the situation is quite different for system (2.3), where an antidissipative MGT equation is coupled with the Fourier one. In that case, as shown in [10], the MGT antidamping turns out to be substantially weaker. Indeed, the whole system decays to zero, provided that the coupling parameter is sufficiently large, independently of the strength of the antidamping of the MGT equation, measured in terms of the size of its (negative) stability number. We can subsume these considerations by writing MGT ≪ F ≈ W, where the order relation above is understood to affect the dissipative properties of the equations. Here, the symbol ≪ means that no quantitative balance between the two mechanisms is necessary for the associated coupled system to decay exponentially fast, up to choosing a coupling parameter large enough. It is then natural to wonder whether this order is coherent, namely, if MGT ≪ W. To this end, we performed numerical simulations in which a subcritical (hence dissipative) MGT equation has been coupled with a weakly antidamped wave equation. The indication is that, even for a very small antidamping parameter, no amount of coupling can be enough to drive the system to stability, suggesting that our guess is correct. To a certain extent, this alters the way one should think about the MGT equation. In the literature, the subcritical MGT equation has often been perceived as a weakly damped wave-type equation, but its dissipation mechanism seems to be fundamentally different, and much weaker, than the structural damping of a wave equation. All authors have confirmed this correction to the original paper [1] and sincerely apologize to the readers for any confusion.