Generation of semigroups for the thermoelastic plate equation with free boundary conditions

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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 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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