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Wellposedness and Decay Rates for the Cauchy Problem of the Moore–Gibson–Thompson Equation Arising in High Intensity Ultrasound

Wellposedness and Decay Rates for the Cauchy Problem of the Moore–Gibson–Thompson Equation... In this paper, we study the Moore–Gibson–Thompson equation in $$\mathbb {R}^N$$ 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 $$(1+t)^{-N/4}$$ ( 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 $$(1+t)^{1-N/4}$$ ( 1 + t ) 1 - N / 4 for $$N=1,2$$ N = 1 , 2 and $$(1+t)^{1/2-N/4}$$ ( 1 + t ) 1 / 2 - N / 4 when $$N\ge 3$$ N ≥ 3 . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

Wellposedness and Decay Rates for the Cauchy Problem of the Moore–Gibson–Thompson Equation Arising in High Intensity Ultrasound

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References (24)

Publisher
Springer Journals
Copyright
Copyright © 2017 by Springer Science+Business Media, LLC, part of Springer Nature
Subject
Mathematics; Calculus of Variations and Optimal Control; Optimization; Systems Theory, Control; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Physics, Simulation
ISSN
0095-4616
eISSN
1432-0606
DOI
10.1007/s00245-017-9471-8
Publisher site
See Article on Publisher Site

Abstract

In this paper, we study the Moore–Gibson–Thompson equation in $$\mathbb {R}^N$$ 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 $$(1+t)^{-N/4}$$ ( 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 $$(1+t)^{1-N/4}$$ ( 1 + t ) 1 - N / 4 for $$N=1,2$$ N = 1 , 2 and $$(1+t)^{1/2-N/4}$$ ( 1 + t ) 1 / 2 - N / 4 when $$N\ge 3$$ N ≥ 3 .

Journal

Applied Mathematics and OptimizationSpringer Journals

Published: Dec 30, 2017

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