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On the regularity of solutions to the Moore–Gibson–Thompson equation: a perspective via wave equations with memory

  • Francesca BucciEmail author
  • Luciano Pandolfi
Article
  • 19 Downloads

Abstract

We undertake a study of the initial/boundary value problem for the (third order in time) Moore–Gibson–Thompson (MGT) equation. The key to the present investigation is that the MGT equation falls within a large class of systems with memory, with affine term depending on a parameter. For this model equation, a regularity theory is provided, which is also of independent interest; it is shown in particular that the effect of boundary data that are square integrable (in time and space) is the same displayed by the wave equation. Then, a general picture of the (interior) regularity of solutions corresponding to homogeneous boundary conditions is specifically derived for the MGT equation in various functional settings. This confirms the gain of one unity in space regularity for the time derivative of the unknown, a feature that sets the MGT equation apart from other partial differential equation models for wave propagation. The adopted perspective and method of proof enable us to attain as well boundary regularity results for both the integro-differential equation and the MGT equation.

Keywords

Interior regularity Boundary regularity Moore–Gibson–Thompson equation Wave equations with memory Volterra integro-differential equations Ultrasound propagation 

Mathematics Subject Classification

35L35 35B65 45K05 47D09 

Notes

Acknowledgements

The authors would like to thank Irena Lasiecka for stimulating mathematical discussions on well-posedness for the Moore–Gibson–Thompson equation with non-trivial boundary data and her insightful comments on the former version of the manuscript. The research of F. Bucci was partially supported by the Università degli Studi di Firenze under the Project Analisi e controllo di sistemi di Equazioni a Derivate Parziali di evoluzione, and by the GDRE (Groupement de Recherche Européen) ConEDP (Control of PDEs). F. Bucci is a member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM), whose partial support is acknowledged. The research of L. Pandolfi was partially supported by the Politecnico di Torino and by the GDRE ConEDP. L. Pandolfi is a member of the GNAMPA of INdAM, whose partial support is acknowledged.

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Copyright information

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Authors and Affiliations

  1. 1.Dipartimento di Matematica e InformaticaUniversità degli Studi di FirenzeFirenzeItaly
  2. 2.Dipartimento di Scienze Matematiche “Giuseppe Luigi Lagrange” (retired)Politecnico di TorinoTorinoItaly

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