November  2014, 34(11): 4515-4535. doi: 10.3934/dcds.2014.34.4515

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

1. 

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

Received  November 2013 Revised  March 2014 Published  May 2014

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.
Citation: Rainer Brunnhuber, Barbara Kaltenbacher. Well-posedness and asymptotic behavior of solutions for the Blackstock-Crighton-Westervelt equation. Discrete & Continuous Dynamical Systems, 2014, 34 (11) : 4515-4535. doi: 10.3934/dcds.2014.34.4515
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show all references

References:
[1]

Elsevier/Academic Press, Amsterdam, 2003.  Google Scholar

[2]

G. Chen and D. L. Russell, A mathematical model for linear elastic systems with structural damping,, Quarterly of Applied Mathematics, 39 (): 433.   Google Scholar

[3]

Pacific Journal of Mathematics, 136 (1989), 15-55. doi: 10.2140/pjm.1989.136.15.  Google Scholar

[4]

Journal d'Acoustique, 5 (1992), 321-359. Google Scholar

[5]

Annual Review of Fluid Mechanics, 11 (1979), 11-33. doi: 10.1146/annurev.fl.11.010179.000303.  Google Scholar

[6]

Springer, New York, 2000.  Google Scholar

[7]

American Mathematical Society, Providence, 2010.  Google Scholar

[8]

Addison-Wesley, Massachusetts, 1983.  Google Scholar

[9]

Academic Press, New York, 1997. doi: 10.1121/1.426968.  Google Scholar

[10]

Physics Letters A, 326 (2004), 77-84. doi: 10.1016/j.physleta.2004.03.067.  Google Scholar

[11]

Discrete and Continuous Dynamical Systems Series S, 2 (2009), 503-523. doi: 10.3934/dcdss.2009.2.503.  Google Scholar

[12]

Mathematische Nachrichten, 285 (2012), 295-321. doi: 10.1002/mana.201000007.  Google Scholar

[13]

DCDS Supplement, Proceedings of the 8th AIMS Conference, II (2011), 763-773.  Google Scholar

[14]

Control and Cybernetics, 40 (2011), 971-988.  Google Scholar

[15]

Mathematical Models and Methods in Applied Sciences, 22 (2012), 1250035, 34 pages. doi: 10.1142/S0218202512500352.  Google Scholar

[16]

Springer, Berlin, 2004. doi: 10.1007/978-3-662-05358-4.  Google Scholar

[17]

Masson-John Wiley, Paris-Chicester, 1994.  Google Scholar

[18]

Soviet physics. Acoustics, 16 (1971), 467-470. Google Scholar

[19]

Computers and Mathematics with Applications, 33 (1997), 1-9. doi: 10.1016/S0898-1221(97)00072-2.  Google Scholar

[20]

Bulletin des Sciences Mathématiques, 136 (2012), 521-573. Google Scholar

[21]

Springer, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[22]

Comptes Rendus Mathematique, 344 (2007), 337-342. doi: 10.1016/j.crma.2007.01.010.  Google Scholar

[23]

Acta Acustica united with Acustica, 87 (2001), 316-321. Google Scholar

[24]

Journal of the Acoustical Society of America, 35 (1963), 535-537. doi: 10.1121/1.1918525.  Google Scholar

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