Abstract
In this paper, we study the Moore–Gibson–Thompson equation in \(\mathbb {R}^N\), which is a third order in time equation that arises in viscous thermally relaxing fluids and also in viscoelastic materials (then under the name of standard linear viscoelastic model). First, we use some Lyapunov functionals in the Fourier space to show that, under certain assumptions on some parameters in the equation, a norm related to the solution decays with a rate \((1+t)^{-N/4}\). Since the decay of the previous norm does not give the decay rate of the solution itself then, in the second part of the paper, we show an explicit representation of the solution in the frequency domain by analyzing the eigenvalues of the Fourier image of the solution and writing the solution accordingly. We use this eigenvalues expansion method to give the decay rate of the solution (and also of its derivatives), which results in \((1+t)^{1-N/4}\) for \(N=1,2\) and \((1+t)^{1/2-N/4}\) when \(N\ge 3\).
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Acknowledgements
The authors would like to thank Prof. Dr. R. Racke and also Prof. Dr. J. Solà-Morales for their helpful discussions on the problem. This work is partially supported by the Grants MTM2014-52402-C3-3-P (Spain) and MPC UdG 2016/047 (U. de Girona, Catalonia). Also, M. Pellicer is part of the Catalan research group 2014 SGR 1083.
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Pellicer, M., Said-Houari, B. Wellposedness and Decay Rates for the Cauchy Problem of the Moore–Gibson–Thompson Equation Arising in High Intensity Ultrasound. Appl Math Optim 80, 447–478 (2019). https://doi.org/10.1007/s00245-017-9471-8
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DOI: https://doi.org/10.1007/s00245-017-9471-8