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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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[5]

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

[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.  Google Scholar

[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).  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[12]

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

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[18]

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

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[5]

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

[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.  Google Scholar

[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).  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[12]

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

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[18]

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

[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.  Google Scholar

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