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June  2013, 2(2): 365-378. doi: 10.3934/eect.2013.2.365

Global well-posedness and exponential stability for Kuznetsov's equation in $L_p$-spaces

1. 

Martin Luther University Halle-Wittenberg, NWF II - Institute of Mathematics, D - 06099 Halle (Saale), Germany, Germany

Received  November 2012 Revised  February 2013 Published  March 2013

We investigate a quasilinear initial-boundary value problem for Kuznetsov's equation with non-homogeneous Dirichlet boundary conditions. This is a model in nonlinear acoustics which describes the propagation of sound in fluidic media with applications in medical ultrasound. We prove that there exists a unique global solution which depends continuously on the sufficiently small data and that the solution and its temporal derivatives converge at an exponential rate as time tends to infinity. Compared to the analysis of Kaltenbacher & Lasiecka, we require optimal regularity conditions on the data and give simplified proofs which are based on maximal $L_p$-regularity for parabolic equations and the implicit function theorem.
Citation: Stefan Meyer, Mathias Wilke. Global well-posedness and exponential stability for Kuznetsov's equation in $L_p$-spaces. Evolution Equations and Control Theory, 2013, 2 (2) : 365-378. doi: 10.3934/eect.2013.2.365
References:
[1]

Robert A. Adams and John J. F. Fournier, "Sobolev Spaces," $2^{nd}$ edition, Pure and Applied Mathematics (Amsterdam), 140, Elsevier/Academic Press, Amsterdam, 2003.

[2]

Herbert Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, in "Function Spaces, Differential Operators and Nonlinear Analysis" (Friedrichroda, 1992), Teubner-Texte Math., 133, Teubner, Stuttgart, (1993), 9-126.

[3]

Herbert Amann, "Linear and Quasilinear Parabolic Problems. Vol. I. Abstract Linear Theory," Monographs in Mathematics, 89, Birkhäuser Boston, Inc., Boston, MA, 1995. doi: 10.1007/978-3-0348-9221-6.

[4]

Sigurd B. Angenent, Nonlinear analytic semiflows, Proc. Roy. Soc. Edinburgh Sect. A, 115 (1990), 91-107. doi: 10.1017/S0308210500024598.

[5]

Klaus Deimling, "Nonlinear Functional Analysis," Springer-Verlag, Berlin, 1985.

[6]

Robert Denk, Jürgen Saal and Jörg Seiler, Inhomogeneous symbols, the Newton polygon, and maximal $L^p$-regularity, Russ. J. Math. Phys., 15 (2008), 171-191. doi: 10.1134/S1061920808020040.

[7]

Robert Denk, Matthias Hieber and Jan Prüss, $\mathcal R$-boundedness, Fourier multipliers and problems of elliptic and parabolic type, Mem. Amer. Math. Soc., 166 (2003).

[8]

Robert Denk, Matthias Hieber and Jan Prüss, Optimal $L^p$- $L^q$-estimates for parabolic boundary value problems with inhomogeneous data, Math. Z., 257 (2007), 193-224. doi: 10.1007/s00209-007-0120-9.

[9]

Gabriella Di Blasio, Linear parabolic evolution equations in $L^p$-spaces, Ann. Mat. Pura Appl. (4), 138 (1984), 55-104. doi: 10.1007/BF01762539.

[10]

Barbara Kaltenbacher and Irena Lasiecka, An analysis of nonhomogeneous Kuznetsov's equation: Local and global well-posedness; exponential decay, Math. Nachr., 285 (2012), 295-321. doi: 10.1002/mana.201000007.

[11]

Manfred Kaltenbacher, "Numerical Simulation of Mechatronic Sensors and Actuators," Springer, 2007. Available from: http://dx.doi.org/10.1007/978-3-540-71360-9.

[12]

V. P. Kuznetsov, Equations of nonlinear acoustics, Sov. Phys. Acoust., 16 (1971), 467-470.

[13]

Yuri Latushkin, Jan Prüss and Roland Schnaubelt, Stable and unstable manifolds for quasilinear parabolic systems with fully nonlinear boundary conditions, J. Evol. Equ., 6 (2006), 537-576. doi: 10.1007/s00028-006-0272-9.

[14]

Alessandra Lunardi, "Analytic Semigroups and Optimal Regularity in Parabolic Problems," Progress in Nonlinear Differential Equations and Their Applications, 16, Birkhäuser Verlag, Basel, 1995.

[15]

Stefan Meyer and Mathias Wilke, Optimal regularity and long-time behavior of solutions for the Westervelt equation, Appl. Math. Optim., 64 (2011), 257-271. doi: 10.1007/s00245-011-9138-9.

[16]

Martin Meyries and Roland Schnaubelt, Interpolation, embeddings and traces of anisotropic fractional Sobolev spaces with temporal weights, J. Funct. Anal., 262 (2012), 1200-1229. doi: 10.1016/j.jfa.2011.11.001.

[17]

Hans Triebel, "Theory of Function Spaces," Monographs in Mathematics, 78, Birkhäuser Verlag, Basel, 1983. doi: 10.1007/978-3-0346-0416-1.

[18]

Hans Triebel, "Interpolation Theory, Function Spaces, Differential Operators," $2^{nd}$ edition, Johann Ambrosius Barth, Heidelberg, 1995.

[19]

Eberhard Zeidler, "Nonlinear Functional Analysis and its Applications. I. Fixed-Point Theorems," Springer-Verlag, New York, 1986. doi: 10.1007/978-1-4612-4838-5.

show all references

References:
[1]

Robert A. Adams and John J. F. Fournier, "Sobolev Spaces," $2^{nd}$ edition, Pure and Applied Mathematics (Amsterdam), 140, Elsevier/Academic Press, Amsterdam, 2003.

[2]

Herbert Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, in "Function Spaces, Differential Operators and Nonlinear Analysis" (Friedrichroda, 1992), Teubner-Texte Math., 133, Teubner, Stuttgart, (1993), 9-126.

[3]

Herbert Amann, "Linear and Quasilinear Parabolic Problems. Vol. I. Abstract Linear Theory," Monographs in Mathematics, 89, Birkhäuser Boston, Inc., Boston, MA, 1995. doi: 10.1007/978-3-0348-9221-6.

[4]

Sigurd B. Angenent, Nonlinear analytic semiflows, Proc. Roy. Soc. Edinburgh Sect. A, 115 (1990), 91-107. doi: 10.1017/S0308210500024598.

[5]

Klaus Deimling, "Nonlinear Functional Analysis," Springer-Verlag, Berlin, 1985.

[6]

Robert Denk, Jürgen Saal and Jörg Seiler, Inhomogeneous symbols, the Newton polygon, and maximal $L^p$-regularity, Russ. J. Math. Phys., 15 (2008), 171-191. doi: 10.1134/S1061920808020040.

[7]

Robert Denk, Matthias Hieber and Jan Prüss, $\mathcal R$-boundedness, Fourier multipliers and problems of elliptic and parabolic type, Mem. Amer. Math. Soc., 166 (2003).

[8]

Robert Denk, Matthias Hieber and Jan Prüss, Optimal $L^p$- $L^q$-estimates for parabolic boundary value problems with inhomogeneous data, Math. Z., 257 (2007), 193-224. doi: 10.1007/s00209-007-0120-9.

[9]

Gabriella Di Blasio, Linear parabolic evolution equations in $L^p$-spaces, Ann. Mat. Pura Appl. (4), 138 (1984), 55-104. doi: 10.1007/BF01762539.

[10]

Barbara Kaltenbacher and Irena Lasiecka, An analysis of nonhomogeneous Kuznetsov's equation: Local and global well-posedness; exponential decay, Math. Nachr., 285 (2012), 295-321. doi: 10.1002/mana.201000007.

[11]

Manfred Kaltenbacher, "Numerical Simulation of Mechatronic Sensors and Actuators," Springer, 2007. Available from: http://dx.doi.org/10.1007/978-3-540-71360-9.

[12]

V. P. Kuznetsov, Equations of nonlinear acoustics, Sov. Phys. Acoust., 16 (1971), 467-470.

[13]

Yuri Latushkin, Jan Prüss and Roland Schnaubelt, Stable and unstable manifolds for quasilinear parabolic systems with fully nonlinear boundary conditions, J. Evol. Equ., 6 (2006), 537-576. doi: 10.1007/s00028-006-0272-9.

[14]

Alessandra Lunardi, "Analytic Semigroups and Optimal Regularity in Parabolic Problems," Progress in Nonlinear Differential Equations and Their Applications, 16, Birkhäuser Verlag, Basel, 1995.

[15]

Stefan Meyer and Mathias Wilke, Optimal regularity and long-time behavior of solutions for the Westervelt equation, Appl. Math. Optim., 64 (2011), 257-271. doi: 10.1007/s00245-011-9138-9.

[16]

Martin Meyries and Roland Schnaubelt, Interpolation, embeddings and traces of anisotropic fractional Sobolev spaces with temporal weights, J. Funct. Anal., 262 (2012), 1200-1229. doi: 10.1016/j.jfa.2011.11.001.

[17]

Hans Triebel, "Theory of Function Spaces," Monographs in Mathematics, 78, Birkhäuser Verlag, Basel, 1983. doi: 10.1007/978-3-0346-0416-1.

[18]

Hans Triebel, "Interpolation Theory, Function Spaces, Differential Operators," $2^{nd}$ edition, Johann Ambrosius Barth, Heidelberg, 1995.

[19]

Eberhard Zeidler, "Nonlinear Functional Analysis and its Applications. I. Fixed-Point Theorems," Springer-Verlag, New York, 1986. doi: 10.1007/978-1-4612-4838-5.

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