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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 |
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.
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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