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June 2019, 8(2): 301-313. doi: 10.3934/eect.2019016

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

Received  June 2018 Revised  October 2018 Published  March 2019

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.

Citation: Robert Denk, Yoshihiro Shibata. Generation of semigroups for the thermoelastic plate equation with free boundary conditions. Evolution Equations & Control Theory, 2019, 8 (2) : 301-313. doi: 10.3934/eect.2019016
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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