Global well-posedness of the Cauchy problem for the Jordan–Moore–Gibson–Thompson equation with arbitrarily large higher-order Sobolev norms

Belkacem Said-Houari

doi:10.3934/dcds.2022066

Article Contents

2022, Volume 42, Issue 9: 4615-4635. Doi: 10.3934/dcds.2022066

This issue Previous Article Bismut formula for intrinsic/Lions derivatives of distribution dependent SDEs with singular coefficients Next Article Multiple ergodic averages for variable polynomials

Global well-posedness of the Cauchy problem for the Jordan–Moore–Gibson–Thompson equation with arbitrarily large higher-order Sobolev norms

Belkacem Said-Houari

Department of Mathematics, College of Sciences, University of Sharjah, P. O. Box: 27272, Sharjah, United Arab Emirates

Received: September 30, 2021

Revised: March 31, 2022

Early access: May 2022

Published: August 2022

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Abstract

In this paper, we consider the 3D Jordan–Moore–Gibson–Thompson equation arising in nonlinear acoustics. First, we prove that the solution exists globally in time provided that the lower order Sobolev norms of the initial data are small, while the higher-order norms can be arbitrarily large. This improves some available results in the literature. Second, we prove a new decay estimate for the linearized model removing the $ L^1 $-assumption on the initial data. The proof of this decay estimate is based on the high-frequency and low-frequency decomposition of the solution together with an interpolation inequality related to Sobolev spaces with negative order.

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Mathematics Subject Classification: Primary: 35L75, 35G25; Secondary: 35L70, 35G20.

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