Thermoelasticity with antidissipation

Monica Conti; Lorenzo Liverani; Vittorino Pata

doi:10.3934/dcdss.2022040

Article Contents

2022, Volume 15, Issue 8: 2173-2188. Doi: 10.3934/dcdss.2022040

This issue Previous Article Well-posedness and optimal control for a Cahn–Hilliard–Oono system with control in the mass term Next Article Exponential stability of Timoshenko-Gurtin-Pipkin systems with full thermal coupling

Thermoelasticity with antidissipation

Politecnico di Milano - Dipartimento di Matematica, Via Bonardi 9, 20133 Milano, Italy

^*Corresponding author: Vittorino Pata
^*Corresponding author: Vittorino Pata

To Professor Maurizio Grasselli on his 60th birthday

Received: December 2021

Early access: February 2022

Published: August 2022

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Abstract

We provide a complete stability analysis for the abstract differential system made by an antidamped wave-type equation, coupled with a dissipative heat-type equation

$ \begin{cases} u_{tt} + A u -\gamma u_t = p A^{\alpha} \theta \\ \theta_{t} + \kappa A^{\beta} \theta = - p A^{\alpha} u_t \end{cases} $

where $ A $ is a strictly positive selfadjoint operator on a Hilbert space, $ \gamma, \kappa>0 $, and both the parameters $ \alpha $ and $ \beta $ can vary between $ 0 $ and $ 1 $. The asymptotic properties of the associated solution semigroup are determined by the strength of the coupling, as well as the quantitative balance between the antidamping $ \gamma $ and the damping $ \kappa $. Depending on the value of $ (\alpha, \beta) $ in the unit square, one of the following mutually disjoint situations can occur: either the related semigroup decays exponentially fast, or all the solutions vanish but not uniformly, or there exists a trajectory whose norm blows up exponentially fast as $ t\to\infty $.

Correction: Sections 7, 8 and 9 are missing from this article. 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. They are added in Correction to “Thermoelasticity with antidissipation” (volume 15, number 8, 2022, 2173-2188).

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Mathematics Subject Classification: Primary: 35B40, 35Q79; Secondary: 74B05, 74F05.

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Figure 1. Regions of stability and instability of system (1.1)

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Figure 2. Regions $ \mathsf R_0' $ and $ \mathsf R_0'' $

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