Well-posedness and asymptotic behavior of solutions for theBlackstock-Crighton-Westervelt equation

Rainer Brunnhuber; Barbara Kaltenbacher

doi:10.3934/dcds.2014.34.4515

Article Contents

2014, Volume 34, Issue 11: 4515-4535. Doi: 10.3934/dcds.2014.34.4515

This issue Previous Article Renormalizations of circle hoemomorphisms with a single break point Next Article Localization, smoothness, and convergence to equilibrium for a thin film equation

Well-posedness and asymptotic behavior of solutions for the Blackstock-Crighton-Westervelt equation

Rainer Brunnhuber^1, and
Barbara Kaltenbacher^1,

1.
Institut für Mathematik, Universität Klagenfurt, Universitätsstraße 65-67, 9020 Klagenfurt am Wörthersee

Received: October 31, 2013

Revised: February 28, 2014

Published: October 2014

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Abstract

We consider a nonlinear fourth order in space partial differential equation arising in the context of the modeling of nonlinear acoustic wave propagation in thermally relaxing viscous fluids.
We use the theory of operator semigroups in order to investigate the linearization of the underlying model and see that the underlying semigroup is analytic. This leads to exponential decay results for the linear homogeneous equation.
Moreover, we prove local in time well-posedness of the model under the assumption that initial data are sufficiently small by employing a fixed point argument. Global in time well-posedness is obtained by performing energy estimates and using the classical barrier method, again for sufficiently small initial data.
Additionally, we provide results concerning exponential decay of solutions of the nonlinear equation.

Keywords:

Mathematics Subject Classification: Primary: 35L75, 35Q35; Secondary: 35B40, 35B65.

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