American Institute of Mathematical Sciences

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December  2019, 8(4): 755-784. doi: 10.3934/eect.2019037

On some nonlinear problem for the thermoplate equations

Suma'inna 1,2,, , Hirokazu Saito 3, and  Yoshihiro Shibata 4,5,

1. 

Department of Pure and Applied Mathematics, Graduate School of Waseda Univeristy, 3-4-1 Ohkubo, Shinjuku-ku, Tokyo 169-8555, Japan
2. 

Department of Mathematics, Faculty of Science and Technology, Syarif Hiayatullah State Islamic University, Jl. Ir. H. Juanda No. 95, Ciputat Tangerang 15412, Indonesia
3. 

Faculty of Industrial Science and Technology, Tokyo University of Science, 102-1 Tomino, Oshamambe-cho, Yamakoshi-gun, Hokkaido 049-3514, Japan
4. 

Department of Pure and Applied Mathematics and Research Institute of Science and Engineering, Waseda Univeristy, Ohkubo 3-4-1, Shinjuku-ku, Tokyo 169-8555, Japan
5. 

Department of Mechanical Engineering and Material Science, University of Pittsburgh, USA

* Corresponding author: Suma'inna

Received  July 2018 Revised  January 2019 Published  December 2019 Early access  June 2019

In this paper, we prove the local and global well-posedness of some nonlinear thermoelastic plate equations with Dirichlet boundary conditions. The main tool for proving the local well-posedness is the maximal $ L_p $-$ L_q $ regularity theorem for the linearized equations, and the main tool for proving the global well-posedness is the exponential stability of $ C_0 $ analytic semigroup associated with linear thermoelastic plate equations with Dirichlet boundary conditions.

Keywords: Thermoplate equations, $ \mathcal{R} $-boundedness, maximal $ L_p $-$ L_q $ regularity, analytic semigroup, exponential stability.
Mathematics Subject Classification: Primary: 35K51, 74F05; Secondary: 93B05.
Citation: Suma'inna, Hirokazu Saito, Yoshihiro Shibata. On some nonlinear problem for the thermoplate equations. Evolution Equations and Control Theory, 2019, 8 (4) : 755-784. doi: 10.3934/eect.2019037
References:
[1]

J. Bourgain, Vector-valued singular integrals and the H1-BMO duality, Probability Theory and Harmonic Analysis, Monogr. Textbooks Pure Appl. Math., Dekker, New York, 98 (1986), 1–19.

[2]

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.

[3]

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+114 pp. doi: 10.1090/memo/0788.

[4]

R. Denk and R. Racke, Lp-resolvent estimates and time decay for generalized thermoelastic plate equations, Electron. J. Differential Equations (electronic), 48 (2006), 16pp.

[5]

R. Denk, R. Racke and Y. Shibata, Lp theory for the linear thermoelastic plate equations in bounded and exterior domains, Adv. Differential Equations, 14 (2009), 685-715. 

[6]

R. Denk, R. Racke and Y. Shibata, Local energy decay estimate of soluions to the thermoelastic plate equations in two- and three-dimensional exterior domains, Z. Anal. Anwend., 29 (2010), 21-62.  doi: 10.4171/ZAA/1396.

[7]

R. Denk and R. Schnaubelt, A structurally damped plate equations with Dirichlet-Neumann boundary conditions, J. Differential Equations, 259 (2015), 1323-1353.  doi: 10.1016/j.jde.2015.02.043.

[8]

R. Denk and Y. Shibata, Maximal regularity for the thermoelastic plate equations with free boundary conditions, J. Evolution Equations, 17 (2017), 215-261.  doi: 10.1007/s00028-016-0367-x.

[9]

R. Denk and Y. Shibata, Generation of semigroups for the thermoelastic plate equation with free boundary conditions, preprint.

[10]

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.

[11]

Su ma'inna, The existence of $ {\mathcal R}$-bounded solution operators of the thermoelastic plate equation with Dirichlet boundary conditions, Math. Methods Appl. Sci., 41 (2018), 1578-1599.  doi: 10.1002/mma.4687.

[12]

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.

[13]

P. C. Kunstmann and L. Weis, Maximal Lp-regularity for parabolic equations, Fourier multiplier theorems and $H^\infty$-functional calculus, Functional Analytic Methods for Evolution Equations, Lecture Notes in Math., Springer, Berlin, 1855 (2004), 65–311. doi: 10.1007/978-3-540-44653-8_2.

[14]

J. E. Lagnese, Boundary Stabilization of Thin Plates, 10. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1989. doi: 10.1137/1.9781611970821.

[15]

I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations, Volume 1, Abstract Parabolic Systems: Continuous and Approximation, Encyclopedia of Mathematics and its Applications, Cambridge University Press, Chapter 3, 2000.

[16]

I. Lasiecka and R. Triggiani, Analyticity, and lack thereof, of thermo-elastic semigroups, In Control and Partial Differential Equations (Marseille-Luminy, 1997), ESAIM Proc., 4, Soc. Math. Appl. Indust., Paris, 4 (1998), 199–222. doi: 10.1051/proc:1998029.

[17]

I. Lasiecka and R. Triggiani, Analyticity of thermo-elastic semigroups with free boundary conditions, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 27 (1998), 457–482.

[18]

I. Lasiecka and R. Triggiani, Analyticity of thermo-elastic semigroups with coupled hinged/Neumann B.C., Abstr. Appl. Anal., 3 (1998), 153-169.  doi: 10.1155/S1085337598000487.

[19]

I. Lasiecka and M. Wilke, Maximal regularity and global existence of soluiotns to a quasilinear thermoelastic plate system, Discrete Contin. Dyn. Syst., 33 (2013), 5189-5202.  doi: 10.3934/dcds.2013.33.5189.

[20]

I. Lasiecka, S. Maad and A. Sasane, Existence and exponential decay of solutions to a quasilinear thermoelastic plate system, Nonlinear Differential Equations and Applications NoDEA, 16 (2008), 689-715.  doi: 10.1007/s00030-008-0011-8.

[21]

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.

[22]

Z. Liu and J. Yong, Qualitative properties of certain C0 semigroups arising in elastic systems with varioius dampings, Adv. Differential Equations, 3 (1998), 643-686. 

[23]

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.

[24]

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.

[25]

J. E. Munoz 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.

[26]

Y. Naito, On the Lp-Lq 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.

[27]

Y. Naito and Y. Shibata, On the Lp analytic semigroup associated with the linear thermoelastic plate equations in the half-space, J. Math. Soc. Japan, 61 (2009), 971-1011.  doi: 10.2969/jmsj/06140971.

[28]

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

[29]

Y. Shibata, On the $\mathcal{R}$-boundedness of solution oeprators for the Stokes equations with free boundary condition, Differential Integral Equations, 27 (2014), 313-368. 

[30]

Y. Shibata and S. Shimizu, On the Lp-Lq maximal regularity of the Neumann problem for the Stokes equations in a bounded domain, J. Reine Angew. Math., 615 (2008), 157-209.  doi: 10.1515/CRELLE.2008.013.

[31]

Y. Shibata and S. Shimizu, On the maxial Lp-Lq regularity of the Stokes problem with first order boundary condition; Model problems, J. Math. Soc. Japan, 64 (2012), 561-626.  doi: 10.2969/jmsj/06420561.

[32]

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

show all references

References:
[1]

J. Bourgain, Vector-valued singular integrals and the H1-BMO duality, Probability Theory and Harmonic Analysis, Monogr. Textbooks Pure Appl. Math., Dekker, New York, 98 (1986), 1–19.

[2]

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.

[3]

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+114 pp. doi: 10.1090/memo/0788.

[4]

R. Denk and R. Racke, Lp-resolvent estimates and time decay for generalized thermoelastic plate equations, Electron. J. Differential Equations (electronic), 48 (2006), 16pp.

[5]

R. Denk, R. Racke and Y. Shibata, Lp theory for the linear thermoelastic plate equations in bounded and exterior domains, Adv. Differential Equations, 14 (2009), 685-715. 

[6]

R. Denk, R. Racke and Y. Shibata, Local energy decay estimate of soluions to the thermoelastic plate equations in two- and three-dimensional exterior domains, Z. Anal. Anwend., 29 (2010), 21-62.  doi: 10.4171/ZAA/1396.

[7]

R. Denk and R. Schnaubelt, A structurally damped plate equations with Dirichlet-Neumann boundary conditions, J. Differential Equations, 259 (2015), 1323-1353.  doi: 10.1016/j.jde.2015.02.043.

[8]

R. Denk and Y. Shibata, Maximal regularity for the thermoelastic plate equations with free boundary conditions, J. Evolution Equations, 17 (2017), 215-261.  doi: 10.1007/s00028-016-0367-x.

[9]

R. Denk and Y. Shibata, Generation of semigroups for the thermoelastic plate equation with free boundary conditions, preprint.

[10]

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.

[11]

Su ma'inna, The existence of $ {\mathcal R}$-bounded solution operators of the thermoelastic plate equation with Dirichlet boundary conditions, Math. Methods Appl. Sci., 41 (2018), 1578-1599.  doi: 10.1002/mma.4687.

[12]

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.

[13]

P. C. Kunstmann and L. Weis, Maximal Lp-regularity for parabolic equations, Fourier multiplier theorems and $H^\infty$-functional calculus, Functional Analytic Methods for Evolution Equations, Lecture Notes in Math., Springer, Berlin, 1855 (2004), 65–311. doi: 10.1007/978-3-540-44653-8_2.

[14]

J. E. Lagnese, Boundary Stabilization of Thin Plates, 10. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1989. doi: 10.1137/1.9781611970821.

[15]

I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations, Volume 1, Abstract Parabolic Systems: Continuous and Approximation, Encyclopedia of Mathematics and its Applications, Cambridge University Press, Chapter 3, 2000.

[16]

I. Lasiecka and R. Triggiani, Analyticity, and lack thereof, of thermo-elastic semigroups, In Control and Partial Differential Equations (Marseille-Luminy, 1997), ESAIM Proc., 4, Soc. Math. Appl. Indust., Paris, 4 (1998), 199–222. doi: 10.1051/proc:1998029.

[17]

I. Lasiecka and R. Triggiani, Analyticity of thermo-elastic semigroups with free boundary conditions, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 27 (1998), 457–482.

[18]

I. Lasiecka and R. Triggiani, Analyticity of thermo-elastic semigroups with coupled hinged/Neumann B.C., Abstr. Appl. Anal., 3 (1998), 153-169.  doi: 10.1155/S1085337598000487.

[19]

I. Lasiecka and M. Wilke, Maximal regularity and global existence of soluiotns to a quasilinear thermoelastic plate system, Discrete Contin. Dyn. Syst., 33 (2013), 5189-5202.  doi: 10.3934/dcds.2013.33.5189.

[20]

I. Lasiecka, S. Maad and A. Sasane, Existence and exponential decay of solutions to a quasilinear thermoelastic plate system, Nonlinear Differential Equations and Applications NoDEA, 16 (2008), 689-715.  doi: 10.1007/s00030-008-0011-8.

[21]

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.

[22]

Z. Liu and J. Yong, Qualitative properties of certain C0 semigroups arising in elastic systems with varioius dampings, Adv. Differential Equations, 3 (1998), 643-686. 

[23]

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.

[24]

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.

[25]

J. E. Munoz 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.

[26]

Y. Naito, On the Lp-Lq 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.

[27]

Y. Naito and Y. Shibata, On the Lp analytic semigroup associated with the linear thermoelastic plate equations in the half-space, J. Math. Soc. Japan, 61 (2009), 971-1011.  doi: 10.2969/jmsj/06140971.

[28]

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

[29]

Y. Shibata, On the $\mathcal{R}$-boundedness of solution oeprators for the Stokes equations with free boundary condition, Differential Integral Equations, 27 (2014), 313-368. 

[30]

Y. Shibata and S. Shimizu, On the Lp-Lq maximal regularity of the Neumann problem for the Stokes equations in a bounded domain, J. Reine Angew. Math., 615 (2008), 157-209.  doi: 10.1515/CRELLE.2008.013.

[31]

Y. Shibata and S. Shimizu, On the maxial Lp-Lq regularity of the Stokes problem with first order boundary condition; Model problems, J. Math. Soc. Japan, 64 (2012), 561-626.  doi: 10.2969/jmsj/06420561.

[32]

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

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