文摘
We are interested in the Moore–Gibson–Thompson equation with memory $$\tau{u}_{ttt}+ \alpha u_{tt}+c^{2}\mathcal{A}u+b\mathcal{A}u_t -\int_0^{t}g(t-s)\mathcal{A} w(s){\rm {d}}s=0.$$This model arises in high-frequency ultrasound applications accounting for thermal flux and molecular relaxation times. According to revisited extended irreversible thermodynamics, thermal flux relaxation leads to the third-order derivative in time while molecular relaxation leads to non-local effects governed by memory terms. The resulting model is of hyperbolic type with viscous effects. We first classify the memory into three types. Then, we study how a memory term creates damping mechanism and how the memory causes energy decay even in the cases when the original memoryless system is unstable. Keywords Moore–Gibson–Thompson (MGT) equation High-frequency ultrasound waves Memory Damping Multipliers Energy estimate Exponential decay Mathematics Subject Classification 35Q70 35L05 74D99 Page %P Close Plain text Look Inside Reference tools Export citation EndNote (.ENW) JabRef (.BIB) Mendeley (.BIB) Papers (.RIS) Zotero (.RIS) BibTeX (.BIB) Add to Papers Other actions Register for Journal Updates About This Journal Reprints and Permissions Share Share this content on Facebook Share this content on Twitter Share this content on LinkedIn Related Content Supplementary Material (0) References (36) References1.Adhikari S.: Structural Dynamic Analysis with Generalized Damping Models: Analysis, pp. 384. Wiley-ISTE, New York (2013)CrossRef2.Alabau-Boussouira F., Cannarsa P., Sforza D.: Decay estimates for second order evolution equations with memory. J Funct. 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Phys., Erice, 1977), pp. 197249, Lecture Notes in Phys., 73, Springer, Berlin-New York, 1978 About this Article Title Moore–Gibson–Thompson equation with memory, part I: exponential decay of energy Journal Zeitschrift für angewandte Mathematik und Physik 67:17 Online DateApril 2016 DOI 10.1007/s00033-015-0597-8 Print ISSN 0044-2275 Online ISSN 1420-9039 Publisher Springer International Publishing Additional Links Register for Journal Updates Editorial Board About This Journal Manuscript Submission Topics Theoretical and Applied Mechanics Mathematical Methods in Physics Keywords 35Q70 35L05 74D99 Moore–Gibson–Thompson (MGT) equation High-frequency ultrasound waves Memory Damping Multipliers Energy estimate Exponential decay Industry Sectors Aerospace Engineering Oil, Gas & Geosciences Authors Irena Lasiecka (1) (2) Xiaojun Wang (1) Author Affiliations 1. Department of Mathematical Sciences, University of Memphis, Memphis, TN, 38152, USA 2. 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