# American Institute of Mathematical Sciences

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

## On some nonlinear problem for the thermoplate equations

 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.

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.

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##### 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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