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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

Robert Denk 1, and  Yoshihiro Shibata 2,

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  June 2019 Early access  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.

Keywords: Thermoelastic plate equations, operator-valued Fourier multipliers, generation of analytic semigroups.
Mathematics Subject Classification: Primary: 35K35, 35J40; Secondary: 42B15.
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.  Google Scholar

[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.  Google Scholar

[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).  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

Y. Shibata, On the exponential decay of the energy of a linear thermoelastic plate, Mat. Apl. Comput., 13 (1994), 81-102.   Google Scholar

[23]

L. Weis, Operator-valued Fourier multiplier theorems and maximal $ L_p $-regularity, Math. Ann., 319 (2001), 735-758.  doi: 10.1007/PL00004457.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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).  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

Y. Shibata, On the exponential decay of the energy of a linear thermoelastic plate, Mat. Apl. Comput., 13 (1994), 81-102.   Google Scholar

[23]

L. Weis, Operator-valued Fourier multiplier theorems and maximal $ L_p $-regularity, Math. Ann., 319 (2001), 735-758.  doi: 10.1007/PL00004457.  Google Scholar

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