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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 |
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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