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A Gevrey class semigroup for a thermoelastic plate model with a fractional Laplacian: Between the Euler-Bernoulli and Kirchhoff models
1. | University of Puerto Rico, Rio Piedras Campus, Department of Mathematics, Faculty of Natural Sciences, 17 University AVE. STE 1701 San Juan PR 00925-2537, USA |
2. | Department of Mathematics and Statistics, Florida International University, Miami, FL 33199, USA |
3. | Department of Mathematical Sciences, George Mason University, Fairfax, VA 22030, USA |
In a bounded domain, we consider a thermoelastic plate with rotational forces. The rotational forces involve the spectral fractional Laplacian, with power parameter $ 0\le\theta\le 1 $. The model includes both the Euler-Bernoulli ($ \theta = 0 $) and Kirchhoff ($ \theta = 1 $) models for thermoelastic plate as special cases. First, we show that the underlying semigroup is of Gevrey class $ \delta $ for every $ \delta>(2-\theta)/(2-4\theta) $ for both the clamped and hinged boundary conditions when the parameter $ \theta $ lies in the interval $ (0, 1/2) $. Then, we show that the semigroup is exponentially stable for hinged boundary conditions, for all values of $ \theta $ in $ [0, 1] $. Finally, we prove, by constructing a counterexample, that, under hinged boundary conditions, the semigroup is not analytic, for all $ \theta $ in the interval $ (0, 1] $. The main features of our Gevrey class proof are: the frequency domain method, appropriate decompositions of the components of the system and the use of Lions' interpolation inequalities.
References:
[1] |
H. Antil, J. Pfefferer and M. Warma,
A note on semilinear fractional elliptic equation: Analysis and discretization, ESAIM Math. Model. Numer. Anal., 51 (2017), 2049-2067.
doi: 10.1051/m2an/2017023. |
[2] |
G. Avalos and I. Lasiecka,
Exponential stability of a thermoelastic system without mechanical dissipation, Rend. Istit. Mat. Univ. Trieste, 28 (1997), 1-28.
|
[3] |
G. Avalos and I. Lasiecka,
Exponential stability of a thermoelastic system with free boundary conditions without mechanical dissipation, SIAM J. Math. Anal., 29 (1998), 155-182.
doi: 10.1137/S0036141096300823. |
[4] |
S. P. Chen and R. Triggiani,
Gevrey class semigroups arising from elastic systems with gentle dissipation: The case $0 < \alpha < 1/2$, Proc. Am. Math. Soc., 110 (1990), 401-415.
doi: 10.2307/2048084. |
[5] |
C. M. Dafermos,
On the existence and the asymptotic stability of solutions to the equations of linear thermoelasticity, Arch. Rational Mech. Anal., 29 (1968), 241-271.
doi: 10.1007/BF00276727. |
[6] |
F. Dell'Oro, J. E. Mun oz-Rivera and V. Pata,
Stability properties of an abstract system with applications to linear thermoelastic plates, J. Evol. Equations, 13 (2013), 777-794.
doi: 10.1007/s00028-013-0202-6. |
[7] |
F. L. Huang,
Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differential Equations, 1 (1985), 43-56.
|
[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. |
[9] |
V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, RAM: Research in Applied Mathematics, Masson, Paris, John Wiley & Sons, Ltd., Chichester, 1994. |
[10] |
J. E. Lagnese, Boundary Stabilization of Thin Plates, SIAM Stud. Appl. Math. 10, SIAM, Philadelphia, PA, 1989.
doi: 10.1137/1.9781611970821. |
[11] |
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.
|
[12] |
I. Lasiecka and R. Triggiani,
Two direct proofs on the analyticity of the s.c. semigroup arising in abstract thermo-elastic equations, Adv. Differential Equations, 3 (1998), 387-416.
|
[13] |
I. Lasiecka and R. Triggiani,
Analyticity and lack thereof, of thermo-elastic semigroups, Control and Partial Differential Equations, ESAIM Proc., Soc. Math. Appl. Indust., Paris, 4 (1998), 199-222.
doi: 10.1051/proc:1998029. |
[14] |
I. Lasiecka and R. Triggiani,
Analyticity of thermo-elastic semigroups with coupled hinged/Neumann B.C. Abstr, Abstr. Appl. Anal., 3 (1998), 153-169.
doi: 10.1155/S1085337598000487. |
[15] |
I. Lasiecka and R. Triggiani,
Structural decomposition of thermo-elastic semigroups with rotational forces, Semigroup Forum, 60 (2000), 16-66.
doi: 10.1007/s002330010003. |
[16] |
G. Lebeau and E. Zuazua,
Decay rates for the three-dimensional linear system of thermoelasticity, Arch. Ration. Mech. Anal., 148 (1999), 179-231.
doi: 10.1007/s002050050160. |
[17] |
J.-L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation des Systèmes Distribués, Research in Applied Mathematics, 8. Masson, Paris, 1988. |
[18] |
K. S. Liu and Z. Y. Liu,
Exponential stability and analyticity of abstract linear thermoelastic systems, Z. Angew. Math. Phys., 48 (1997), 885-904.
doi: 10.1007/s000330050071. |
[19] |
Z.-Y. Liu and M. Renardy,
A note on the equations of thermoelastic plate, Appl. Math. Lett., 8 (1995), 1-6.
doi: 10.1016/0893-9659(95)00020-Q. |
[20] |
Z. Y. Liu and S. M. Zheng,
Exponential stability of the Kirchhoff plate with thermal or viscoelastic damping, Quarterly Appl. Math., 55 (1997), 551-564.
doi: 10.1090/qam/1466148. |
[21] |
W.-J. Liu and E. Zuazua,
Uniform stabilization of the higher dimensional system of thermoelasticity with a nonlinear boundary feedback, Quarterly Appl. Math., 59 (2001), 269-314.
doi: 10.1090/qam/1828455. |
[22] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[23] |
G. Perla-Menzala and E. Zuazua,
The energy decay rate for the modified von Kármán system of thermoelastic plates: An improvement, Applied Mathematics Letters, 16 (2003), 531-534.
doi: 10.1016/S0893-9659(03)00032-6. |
[24] |
J. Prüss,
On the spectrum of C0-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857.
doi: 10.2307/1999112. |
[25] |
S. W. Taylor, Gevrey Regularity of Solutions of Evolution Equations and Boundary Controllability, Thesis (Ph.D.)–University of Minnesota. 1989, 182 pp. |
[26] |
L. Tebou,
Stabilization of some coupled hyperbolic/parabolic equations, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 1601-1620.
doi: 10.3934/dcdsb.2010.14.1601. |
[27] |
L. Tebou,
Uniform analyticity and exponential decay of the semigroup associated with a thermoelastic plate equation with perturbed boundary conditions, C. R. Math. Acad. Sci. Paris, 351 (2013), 539-544.
doi: 10.1016/j.crma.2013.07.014. |
show all references
References:
[1] |
H. Antil, J. Pfefferer and M. Warma,
A note on semilinear fractional elliptic equation: Analysis and discretization, ESAIM Math. Model. Numer. Anal., 51 (2017), 2049-2067.
doi: 10.1051/m2an/2017023. |
[2] |
G. Avalos and I. Lasiecka,
Exponential stability of a thermoelastic system without mechanical dissipation, Rend. Istit. Mat. Univ. Trieste, 28 (1997), 1-28.
|
[3] |
G. Avalos and I. Lasiecka,
Exponential stability of a thermoelastic system with free boundary conditions without mechanical dissipation, SIAM J. Math. Anal., 29 (1998), 155-182.
doi: 10.1137/S0036141096300823. |
[4] |
S. P. Chen and R. Triggiani,
Gevrey class semigroups arising from elastic systems with gentle dissipation: The case $0 < \alpha < 1/2$, Proc. Am. Math. Soc., 110 (1990), 401-415.
doi: 10.2307/2048084. |
[5] |
C. M. Dafermos,
On the existence and the asymptotic stability of solutions to the equations of linear thermoelasticity, Arch. Rational Mech. Anal., 29 (1968), 241-271.
doi: 10.1007/BF00276727. |
[6] |
F. Dell'Oro, J. E. Mun oz-Rivera and V. Pata,
Stability properties of an abstract system with applications to linear thermoelastic plates, J. Evol. Equations, 13 (2013), 777-794.
doi: 10.1007/s00028-013-0202-6. |
[7] |
F. L. Huang,
Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differential Equations, 1 (1985), 43-56.
|
[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. |
[9] |
V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, RAM: Research in Applied Mathematics, Masson, Paris, John Wiley & Sons, Ltd., Chichester, 1994. |
[10] |
J. E. Lagnese, Boundary Stabilization of Thin Plates, SIAM Stud. Appl. Math. 10, SIAM, Philadelphia, PA, 1989.
doi: 10.1137/1.9781611970821. |
[11] |
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.
|
[12] |
I. Lasiecka and R. Triggiani,
Two direct proofs on the analyticity of the s.c. semigroup arising in abstract thermo-elastic equations, Adv. Differential Equations, 3 (1998), 387-416.
|
[13] |
I. Lasiecka and R. Triggiani,
Analyticity and lack thereof, of thermo-elastic semigroups, Control and Partial Differential Equations, ESAIM Proc., Soc. Math. Appl. Indust., Paris, 4 (1998), 199-222.
doi: 10.1051/proc:1998029. |
[14] |
I. Lasiecka and R. Triggiani,
Analyticity of thermo-elastic semigroups with coupled hinged/Neumann B.C. Abstr, Abstr. Appl. Anal., 3 (1998), 153-169.
doi: 10.1155/S1085337598000487. |
[15] |
I. Lasiecka and R. Triggiani,
Structural decomposition of thermo-elastic semigroups with rotational forces, Semigroup Forum, 60 (2000), 16-66.
doi: 10.1007/s002330010003. |
[16] |
G. Lebeau and E. Zuazua,
Decay rates for the three-dimensional linear system of thermoelasticity, Arch. Ration. Mech. Anal., 148 (1999), 179-231.
doi: 10.1007/s002050050160. |
[17] |
J.-L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation des Systèmes Distribués, Research in Applied Mathematics, 8. Masson, Paris, 1988. |
[18] |
K. S. Liu and Z. Y. Liu,
Exponential stability and analyticity of abstract linear thermoelastic systems, Z. Angew. Math. Phys., 48 (1997), 885-904.
doi: 10.1007/s000330050071. |
[19] |
Z.-Y. Liu and M. Renardy,
A note on the equations of thermoelastic plate, Appl. Math. Lett., 8 (1995), 1-6.
doi: 10.1016/0893-9659(95)00020-Q. |
[20] |
Z. Y. Liu and S. M. Zheng,
Exponential stability of the Kirchhoff plate with thermal or viscoelastic damping, Quarterly Appl. Math., 55 (1997), 551-564.
doi: 10.1090/qam/1466148. |
[21] |
W.-J. Liu and E. Zuazua,
Uniform stabilization of the higher dimensional system of thermoelasticity with a nonlinear boundary feedback, Quarterly Appl. Math., 59 (2001), 269-314.
doi: 10.1090/qam/1828455. |
[22] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[23] |
G. Perla-Menzala and E. Zuazua,
The energy decay rate for the modified von Kármán system of thermoelastic plates: An improvement, Applied Mathematics Letters, 16 (2003), 531-534.
doi: 10.1016/S0893-9659(03)00032-6. |
[24] |
J. Prüss,
On the spectrum of C0-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857.
doi: 10.2307/1999112. |
[25] |
S. W. Taylor, Gevrey Regularity of Solutions of Evolution Equations and Boundary Controllability, Thesis (Ph.D.)–University of Minnesota. 1989, 182 pp. |
[26] |
L. Tebou,
Stabilization of some coupled hyperbolic/parabolic equations, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 1601-1620.
doi: 10.3934/dcdsb.2010.14.1601. |
[27] |
L. Tebou,
Uniform analyticity and exponential decay of the semigroup associated with a thermoelastic plate equation with perturbed boundary conditions, C. R. Math. Acad. Sci. Paris, 351 (2013), 539-544.
doi: 10.1016/j.crma.2013.07.014. |
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