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Generation of semigroups for the thermoelastic plate equation with free boundary conditions
| 1. | University of Konstanz, Department of Mathematics and Statistics, 78457 Konstanz, Germany |
| 2. | Department of Mathematical Sciences, School of Science and Engineering, Waseda University, Ohkobu 3-4-1, Shinjuku-ku, Tokyo 169-8555, Japan |
We consider the linear thermoelastic plate equations with free boundary conditions in uniform $ C^4 $-domains, which includes the half-space, bounded and exterior domains. We show that the corresponding operator generates an analytic semigroup in $ L^p $-spaces for all $ p\in(1, \infty) $ and has maximal $ L^q $-$ L^p $-regularity on finite time intervals. On bounded $ C^4 $-domains, we obtain exponential stability.
References:
| [1] |
I. Chueshov and I. Lasiecka, Von Karman Evolution Equations, Springer Monographs in Mathematics, Springer, New York, 2010, Well-posedness and long-time dynamics.
doi: 10.1007/978-0-387-87712-9. |
| [2] |
R. Denk, M. Hieber and J. Prüss, $ \mathcal R $-boundedness, Fourier multipliers and problems of elliptic and parabolic type, Mem. Amer. Math. Soc., 166 (2003), viii+114pp.
doi: 10.1090/memo/0788. |
| [3] |
R. Denk and R. Racke, $ L^p $-resolvent estimates and time decay for generalized thermoelastic plate equations, Electron. J. Differential Equations, No. 48, 16 pp(electronic). |
| [4] |
R. Denk and R. Schnaubelt,
A structurally damped plate equation with Dirichlet–Neumann boundary conditions, J. Differential Equations, 259 (2015), 1323-1353.
doi: 10.1016/j.jde.2015.02.043. |
| [5] |
R. Denk and Y. Shibata,
Maximal regularity for the thermoelastic plate equations with free boundary conditions, J. Evol. Equ., 17 (2017), 215-261.
doi: 10.1007/s00028-016-0367-x. |
| [6] |
Y. Enomoto and Y. Shibata,
On the $ \mathcal R $-sectoriality and the initial boundary value problem for the viscous compressible fluid flow, Funkcial. Ekvac., 56 (2013), 441-505.
doi: 10.1619/fesi.56.441. |
| [7] |
M. Girardi and L. Weis, Criteria for R-boundedness of operator families, in Evolution Equations, vol. 234 of Lecture Notes in Pure and Appl. Math., Dekker, New York, 2003,203–221. |
| [8] |
J. U. Kim,
On the energy decay of a linear thermoelastic bar and plate, SIAM J. Math. Anal., 23 (1992), 889-899.
doi: 10.1137/0523047. |
| [9] |
P. C. Kunstmann and L. Weis, Maximal $ L_p $-regularity for parabolic equations, Fourier multiplier theorems and $ H^\infty $-functional calculus, in Functional Analytic Methods for Evolution Equations, vol. 1855 of Lecture Notes in Math., Springer, Berlin, 2004, 65–311.
doi: 10.1007/978-3-540-44653-8_2. |
| [10] |
J. E. Lagnese, Boundary Stabilization of Thin Plates, vol. 10 of SIAM Studies in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1989.
doi: 10.1137/1.9781611970821. |
| [11] |
I. Lasiecka and R. Triggiani, Analyticity, and lack thereof, of thermo-elastic semigroups, in Control and Partial Differential Equations (Marseille-Luminy, 1997), vol. 4 of ESAIM Proc., Soc. Math. Appl. Indust., Paris, 1998,199–222(electronic).
doi: 10.1051/proc:1998029. |
| [12] |
I. Lasiecka and R. Triggiani, Analyticity of thermo-elastic semigroups with free boundary conditions, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 27 (1998), 457–482(1999), URL http://www.numdam.org/item?id=ASNSP_1998_4_27_3-4_457_0. |
| [13] |
I. Lasiecka and M. Wilke,
Maximal regularity and global existence of solutions to a quasilinear thermoelastic plate system, Discrete Contin. Dyn. Syst., 33 (2013), 5189-5202.
doi: 10.3934/dcds.2013.33.5189. |
| [14] |
K. Liu and Z. Liu,
Exponential stability and analyticity of abstract linear thermoelastic systems, Z. Angew. Math. Phys., 48 (1997), 885-904.
doi: 10.1007/s000330050071. |
| [15] |
Z.-Y. Liu and M. Renardy,
A note on the equations of a thermoelastic plate, Appl. Math. Lett., 8 (1995), 1-6.
doi: 10.1016/0893-9659(95)00020-Q. |
| [16] |
Z. Liu and J. Yong,
Qualitative properties of certain $ C_0 $ semigroups arising in elastic systems with various dampings, Adv. Differential Equations, 3 (1998), 643-686.
|
| [17] |
Z. Liu and S. Zheng,
Exponential stability of the Kirchhoff plate with thermal or viscoelastic damping, Quart. Appl. Math., 55 (1997), 551-564.
doi: 10.1090/qam/1466148. |
| [18] |
J. E. Muñoz Rivera and R. Racke,
Smoothing properties, decay, and global existence of solutions to nonlinear coupled systems of thermoelastic type, SIAM J. Math. Anal., 26 (1995), 1547-1563.
doi: 10.1137/S0036142993255058. |
| [19] |
Y. Naito,
On the $ L_p $-$ L_q $ maximal regularity for the linear thermoelastic plate equation in a bounded domain, Math. Methods Appl. Sci., 32 (2009), 1609-1637.
doi: 10.1002/mma.1100. |
| [20] |
Y. Naito and Y. Shibata, On the $ L_p $ analytic semigroup associated with the linear thermoelastic plate equations in the half-space, J. Math. Soc. Japan, 61 (2009), 971–1011, URL http://projecteuclid.org/euclid.jmsj/1257520498.
doi: 10.2969/jmsj/06140971. |
| [21] |
K. Schade and Y. Shibata,
On strong dynamics of compressible nematic liquid crystals, SIAM J. Math. Anal., 47 (2015), 3963-3992.
doi: 10.1137/140970628. |
| [22] |
Y. Shibata,
On the exponential decay of the energy of a linear thermoelastic plate, Mat. Apl. Comput., 13 (1994), 81-102.
|
| [23] |
L. Weis,
Operator-valued Fourier multiplier theorems and maximal $ L_p $-regularity, Math. Ann., 319 (2001), 735-758.
doi: 10.1007/PL00004457. |
show all references
References:
| [1] |
I. Chueshov and I. Lasiecka, Von Karman Evolution Equations, Springer Monographs in Mathematics, Springer, New York, 2010, Well-posedness and long-time dynamics.
doi: 10.1007/978-0-387-87712-9. |
| [2] |
R. Denk, M. Hieber and J. Prüss, $ \mathcal R $-boundedness, Fourier multipliers and problems of elliptic and parabolic type, Mem. Amer. Math. Soc., 166 (2003), viii+114pp.
doi: 10.1090/memo/0788. |
| [3] |
R. Denk and R. Racke, $ L^p $-resolvent estimates and time decay for generalized thermoelastic plate equations, Electron. J. Differential Equations, No. 48, 16 pp(electronic). |
| [4] |
R. Denk and R. Schnaubelt,
A structurally damped plate equation with Dirichlet–Neumann boundary conditions, J. Differential Equations, 259 (2015), 1323-1353.
doi: 10.1016/j.jde.2015.02.043. |
| [5] |
R. Denk and Y. Shibata,
Maximal regularity for the thermoelastic plate equations with free boundary conditions, J. Evol. Equ., 17 (2017), 215-261.
doi: 10.1007/s00028-016-0367-x. |
| [6] |
Y. Enomoto and Y. Shibata,
On the $ \mathcal R $-sectoriality and the initial boundary value problem for the viscous compressible fluid flow, Funkcial. Ekvac., 56 (2013), 441-505.
doi: 10.1619/fesi.56.441. |
| [7] |
M. Girardi and L. Weis, Criteria for R-boundedness of operator families, in Evolution Equations, vol. 234 of Lecture Notes in Pure and Appl. Math., Dekker, New York, 2003,203–221. |
| [8] |
J. U. Kim,
On the energy decay of a linear thermoelastic bar and plate, SIAM J. Math. Anal., 23 (1992), 889-899.
doi: 10.1137/0523047. |
| [9] |
P. C. Kunstmann and L. Weis, Maximal $ L_p $-regularity for parabolic equations, Fourier multiplier theorems and $ H^\infty $-functional calculus, in Functional Analytic Methods for Evolution Equations, vol. 1855 of Lecture Notes in Math., Springer, Berlin, 2004, 65–311.
doi: 10.1007/978-3-540-44653-8_2. |
| [10] |
J. E. Lagnese, Boundary Stabilization of Thin Plates, vol. 10 of SIAM Studies in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1989.
doi: 10.1137/1.9781611970821. |
| [11] |
I. Lasiecka and R. Triggiani, Analyticity, and lack thereof, of thermo-elastic semigroups, in Control and Partial Differential Equations (Marseille-Luminy, 1997), vol. 4 of ESAIM Proc., Soc. Math. Appl. Indust., Paris, 1998,199–222(electronic).
doi: 10.1051/proc:1998029. |
| [12] |
I. Lasiecka and R. Triggiani, Analyticity of thermo-elastic semigroups with free boundary conditions, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 27 (1998), 457–482(1999), URL http://www.numdam.org/item?id=ASNSP_1998_4_27_3-4_457_0. |
| [13] |
I. Lasiecka and M. Wilke,
Maximal regularity and global existence of solutions to a quasilinear thermoelastic plate system, Discrete Contin. Dyn. Syst., 33 (2013), 5189-5202.
doi: 10.3934/dcds.2013.33.5189. |
| [14] |
K. Liu and Z. Liu,
Exponential stability and analyticity of abstract linear thermoelastic systems, Z. Angew. Math. Phys., 48 (1997), 885-904.
doi: 10.1007/s000330050071. |
| [15] |
Z.-Y. Liu and M. Renardy,
A note on the equations of a thermoelastic plate, Appl. Math. Lett., 8 (1995), 1-6.
doi: 10.1016/0893-9659(95)00020-Q. |
| [16] |
Z. Liu and J. Yong,
Qualitative properties of certain $ C_0 $ semigroups arising in elastic systems with various dampings, Adv. Differential Equations, 3 (1998), 643-686.
|
| [17] |
Z. Liu and S. Zheng,
Exponential stability of the Kirchhoff plate with thermal or viscoelastic damping, Quart. Appl. Math., 55 (1997), 551-564.
doi: 10.1090/qam/1466148. |
| [18] |
J. E. Muñoz Rivera and R. Racke,
Smoothing properties, decay, and global existence of solutions to nonlinear coupled systems of thermoelastic type, SIAM J. Math. Anal., 26 (1995), 1547-1563.
doi: 10.1137/S0036142993255058. |
| [19] |
Y. Naito,
On the $ L_p $-$ L_q $ maximal regularity for the linear thermoelastic plate equation in a bounded domain, Math. Methods Appl. Sci., 32 (2009), 1609-1637.
doi: 10.1002/mma.1100. |
| [20] |
Y. Naito and Y. Shibata, On the $ L_p $ analytic semigroup associated with the linear thermoelastic plate equations in the half-space, J. Math. Soc. Japan, 61 (2009), 971–1011, URL http://projecteuclid.org/euclid.jmsj/1257520498.
doi: 10.2969/jmsj/06140971. |
| [21] |
K. Schade and Y. Shibata,
On strong dynamics of compressible nematic liquid crystals, SIAM J. Math. Anal., 47 (2015), 3963-3992.
doi: 10.1137/140970628. |
| [22] |
Y. Shibata,
On the exponential decay of the energy of a linear thermoelastic plate, Mat. Apl. Comput., 13 (1994), 81-102.
|
| [23] |
L. Weis,
Operator-valued Fourier multiplier theorems and maximal $ L_p $-regularity, Math. Ann., 319 (2001), 735-758.
doi: 10.1007/PL00004457. |
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