Hölder regularity for the Moore-Gibson-Thompson equation with infinite delay

Luciano Abadías; Carlos Lizama; Marina Murillo-Arcila

doi:10.3934/cpaa.2018015

Article Contents

2018, Volume 17, Issue 1: 243-265. Doi: 10.3934/cpaa.2018015

This issue Previous Article Nodal solutions for the Robin p-Laplacian plus an indefinite potential and a general reaction term Next Article On stability of functional differential equations with rapidly oscillating coefficients

Hölder regularity for the Moore-Gibson-Thompson equation with infinite delay

1.
Universidad de Santiago de Chile, Departamento de Matemáticay Ciencia de la Computación, Las Sophoras 173, Estación Central, Santiago, Chile
2.
BCAM-Basque Center for Applied Mathematics, Mazarredo, 14, E48009 Bilbao, Basque Country, Spain

* Corresponding author
* Corresponding author

Received: December 2016

Revised: June 2017

Published: January 2018

The first author is supported by the Project POSTDOC DICYT-041633LY at the USACH. The second author is partially supported by CONICYT, under Fondecyt Grant number 1140258 and and CONICYT - PIA - Anillo ACT1416. The third author is supported by the Basque Government through the BERC 2014-2017 program and by Spanish Ministry of Economy and Competitiveness MINECO: BCAM Severo Ochoa excellence accreditation SEV-2013-0323 and GEAGAM, 644202 H2020-MSCA-RISE-2014.

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Abstract

We characterize the well-posedness of a third order in time equation with infinite delay in Hölder spaces, solely in terms of spectral properties concerning the data of the problem. Our analysis includes the case of the linearized Kuznetzov and Westerwelt equations. We show in case of the Laplacian operator the new and surprising fact that for the standard memory kernel $g(t)=\frac{t^{ν-1}}{Γ(ν)}e^{-at}$ the third order problem is ill-posed whenever $0<ν ≤q 1$ and $a$ is inversely proportional to one of the terms of the given model.

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Mathematics Subject Classification: Primary:45N05;Secondary:45D05, 43A15.

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Figure 1. Example of a parametric plot (${\mathfrak R}{\mathfrak e}\, \beta_2(\eta), {\mathfrak I}{\mathfrak m}\,\beta_2(\eta)$)

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Figure 2. Example of a parametric plot (${\mathfrak R}{\mathfrak e}\, \beta_3(\eta), {\mathfrak I}{\mathfrak m}\,\beta_3(\eta)$)

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