# American Institute of Mathematical Sciences

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 - A, 2014, 34 (11) : 4515-4535. doi: 10.3934/dcds.2014.34.4515
##### References:
 [1] R. A. Adams and J. J. F. Fournier, Sobolev Spaces, Second Edition,, Elsevier/Academic Press, (2003). [2] G. Chen and D. L. Russell, A mathematical model for linear elastic systems with structural damping,, Quarterly of Applied Mathematics, 39 (): 433. [3] S. Chen and R. Triggiani, Proof of extensions of two conjectures on structural damping for elastic systems,, Pacific Journal of Mathematics, 136 (1989), 15. doi: 10.2140/pjm.1989.136.15. [4] F. Coulouvrat, On the equations of nonlinear acoustics,, Journal d'Acoustique, 5 (1992), 321. [5] D. G. Crighton, Model equations of nonlinear acoustics,, Annual Review of Fluid Mechanics, 11 (1979), 11. doi: 10.1146/annurev.fl.11.010179.000303. [6] K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations,, Springer, (2000). [7] L. C. Evans, Partial Differential Equations, Second Edition,, American Mathematical Society, (2010). [8] H. O. Fattorini, The Cauchy Problem,, Addison-Wesley, (1983). [9] M. F. Hamilton and D. T. Blackstock, Nonlinear Acoustics,, Academic Press, (1997). doi: 10.1121/1.426968. [10] P. M. Jordan, An analytical study of Kuznetsov's equation: Diffusive solutions, shock formation and solution bifurcation,, Physics Letters A, 326 (2004), 77. doi: 10.1016/j.physleta.2004.03.067. [11] B. Kaltenbacher and I. Lasiecka, Global existence and exponential decay rates for the Westervelt equation,, Discrete and Continuous Dynamical Systems Series S, 2 (2009), 503. doi: 10.3934/dcdss.2009.2.503. [12] B. Kaltenbacher and I. Lasiecka, An analysis of nonhomogeneous Kuznetsov's equation: Local and global well-posedness; exponential decay,, Mathematische Nachrichten, 285 (2012), 295. doi: 10.1002/mana.201000007. [13] B. Kaltenbacher and I. Lasiecka, Well-posedness of the Westervelt and the Kuznetsov equation with nonhomogeneous Neumann boundary conditions,, DCDS Supplement, II (2011), 763. [14] B. Kaltenbacher, I. Lasiecka and R. Marchand, Well-posedness and exponential decay rates for the Moore-Gibson-Thompson equation,, Control and Cybernetics, 40 (2011), 971. [15] B. Kaltenbacher, I. Lasiecka and M. K. Pospieszahlska, Well-posedness and exponential decay of the energy in the nonlinear Jordan-Moore-Gibson-Thompson equation arising in high intensity ultrasound,, Mathematical Models and Methods in Applied Sciences, 22 (2012). doi: 10.1142/S0218202512500352. [16] M. Kaltenbacher, Numerical Simulation of Mechatronic Sensors and Actuators,, Springer, (2004). doi: 10.1007/978-3-662-05358-4. [17] V. Komornik, Exact Controllability and Stabilization. The Multiplier Method,, Masson-John Wiley, (1994). [18] V. P. Kuznetsov, Equations of nonlinear acoustics,, Soviet physics. Acoustics, 16 (1971), 467. [19] J. Liang and T. Xiao, Semigroups arising from elastic systems with dissipation,, Computers and Mathematics with Applications, 33 (1997), 1. doi: 10.1016/S0898-1221(97)00072-2. [20] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces,, Bulletin des Sciences Mathématiques, 136 (2012), 521. [21] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Springer, (1983). doi: 10.1007/978-1-4612-5561-1. [22] A. Rozanova, The Khokhlov-Zabolotskaya-Kuznetsov equation,, Comptes Rendus Mathematique, 344 (2007), 337. doi: 10.1016/j.crma.2007.01.010. [23] S. Tjøtta, Higher order model equations in nonlinear acoustics,, Acta Acustica united with Acustica, 87 (2001), 316. [24] P. J. Westervelt, Parametric acoustic array,, Journal of the Acoustical Society of America, 35 (1963), 535. doi: 10.1121/1.1918525.

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##### References:
 [1] R. A. Adams and J. J. F. Fournier, Sobolev Spaces, Second Edition,, Elsevier/Academic Press, (2003). [2] G. Chen and D. L. Russell, A mathematical model for linear elastic systems with structural damping,, Quarterly of Applied Mathematics, 39 (): 433. [3] S. Chen and R. Triggiani, Proof of extensions of two conjectures on structural damping for elastic systems,, Pacific Journal of Mathematics, 136 (1989), 15. doi: 10.2140/pjm.1989.136.15. [4] F. Coulouvrat, On the equations of nonlinear acoustics,, Journal d'Acoustique, 5 (1992), 321. [5] D. G. Crighton, Model equations of nonlinear acoustics,, Annual Review of Fluid Mechanics, 11 (1979), 11. doi: 10.1146/annurev.fl.11.010179.000303. [6] K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations,, Springer, (2000). [7] L. C. Evans, Partial Differential Equations, Second Edition,, American Mathematical Society, (2010). [8] H. O. Fattorini, The Cauchy Problem,, Addison-Wesley, (1983). [9] M. F. Hamilton and D. T. Blackstock, Nonlinear Acoustics,, Academic Press, (1997). doi: 10.1121/1.426968. [10] P. M. Jordan, An analytical study of Kuznetsov's equation: Diffusive solutions, shock formation and solution bifurcation,, Physics Letters A, 326 (2004), 77. doi: 10.1016/j.physleta.2004.03.067. [11] B. Kaltenbacher and I. Lasiecka, Global existence and exponential decay rates for the Westervelt equation,, Discrete and Continuous Dynamical Systems Series S, 2 (2009), 503. doi: 10.3934/dcdss.2009.2.503. [12] B. Kaltenbacher and I. Lasiecka, An analysis of nonhomogeneous Kuznetsov's equation: Local and global well-posedness; exponential decay,, Mathematische Nachrichten, 285 (2012), 295. doi: 10.1002/mana.201000007. [13] B. Kaltenbacher and I. Lasiecka, Well-posedness of the Westervelt and the Kuznetsov equation with nonhomogeneous Neumann boundary conditions,, DCDS Supplement, II (2011), 763. [14] B. Kaltenbacher, I. Lasiecka and R. Marchand, Well-posedness and exponential decay rates for the Moore-Gibson-Thompson equation,, Control and Cybernetics, 40 (2011), 971. [15] B. Kaltenbacher, I. Lasiecka and M. K. Pospieszahlska, Well-posedness and exponential decay of the energy in the nonlinear Jordan-Moore-Gibson-Thompson equation arising in high intensity ultrasound,, Mathematical Models and Methods in Applied Sciences, 22 (2012). doi: 10.1142/S0218202512500352. [16] M. Kaltenbacher, Numerical Simulation of Mechatronic Sensors and Actuators,, Springer, (2004). doi: 10.1007/978-3-662-05358-4. [17] V. Komornik, Exact Controllability and Stabilization. The Multiplier Method,, Masson-John Wiley, (1994). [18] V. P. Kuznetsov, Equations of nonlinear acoustics,, Soviet physics. Acoustics, 16 (1971), 467. [19] J. Liang and T. Xiao, Semigroups arising from elastic systems with dissipation,, Computers and Mathematics with Applications, 33 (1997), 1. doi: 10.1016/S0898-1221(97)00072-2. [20] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces,, Bulletin des Sciences Mathématiques, 136 (2012), 521. [21] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Springer, (1983). doi: 10.1007/978-1-4612-5561-1. [22] A. Rozanova, The Khokhlov-Zabolotskaya-Kuznetsov equation,, Comptes Rendus Mathematique, 344 (2007), 337. doi: 10.1016/j.crma.2007.01.010. [23] S. Tjøtta, Higher order model equations in nonlinear acoustics,, Acta Acustica united with Acustica, 87 (2001), 316. [24] P. J. Westervelt, Parametric acoustic array,, Journal of the Acoustical Society of America, 35 (1963), 535. doi: 10.1121/1.1918525.
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