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May  2020, 40(5): 2875-2889. doi: 10.3934/dcds.2020152

A Gevrey class semigroup for a thermoelastic plate model with a fractional Laplacian: Between the Euler-Bernoulli and Kirchhoff models

Valentin Keyantuo 1, , Louis Tebou 2,, and  Mahamadi Warma 3,

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

* Corresponding author: Louis Tebou

Received  July 2019 Revised  December 2019 Published  March 2020

Fund Project: The work of V. Keyantuo and M. Warma is partially supported by the Air Force Office of Scientific Research under Award NO [FA9550-18-1-0242]

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.

Keywords: Thermoelastic plate, fractional rotational forces, Gevrey class semigroup, exponential stability.
Mathematics Subject Classification: 35Q74, 35B65, 47D03, 74K20, 93D20.
Citation: Valentin Keyantuo, Louis Tebou, Mahamadi Warma. A Gevrey class semigroup for a thermoelastic plate model with a fractional Laplacian: Between the Euler-Bernoulli and Kirchhoff models. Discrete & Continuous Dynamical Systems, 2020, 40 (5) : 2875-2889. doi: 10.3934/dcds.2020152
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.  Google Scholar

[2]

G. Avalos and I. Lasiecka, Exponential stability of a thermoelastic system without mechanical dissipation, Rend. Istit. Mat. Univ. Trieste, 28 (1997), 1-28.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

F. L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differential Equations, 1 (1985), 43-56.   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]

V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, RAM: Research in Applied Mathematics, Masson, Paris, John Wiley & Sons, Ltd., Chichester, 1994.  Google Scholar

[10]

J. E. Lagnese, Boundary Stabilization of Thin Plates, SIAM Stud. Appl. Math. 10, SIAM, Philadelphia, PA, 1989. doi: 10.1137/1.9781611970821.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

J.-L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation des Systèmes Distribués, Research in Applied Mathematics, 8. Masson, Paris, 1988.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

J. Prüss, On the spectrum of C0-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857.  doi: 10.2307/1999112.  Google Scholar

[25]

S. W. Taylor, Gevrey Regularity of Solutions of Evolution Equations and Boundary Controllability, Thesis (Ph.D.)–University of Minnesota. 1989, 182 pp.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

G. Avalos and I. Lasiecka, Exponential stability of a thermoelastic system without mechanical dissipation, Rend. Istit. Mat. Univ. Trieste, 28 (1997), 1-28.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

F. L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differential Equations, 1 (1985), 43-56.   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]

V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, RAM: Research in Applied Mathematics, Masson, Paris, John Wiley & Sons, Ltd., Chichester, 1994.  Google Scholar

[10]

J. E. Lagnese, Boundary Stabilization of Thin Plates, SIAM Stud. Appl. Math. 10, SIAM, Philadelphia, PA, 1989. doi: 10.1137/1.9781611970821.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

J.-L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation des Systèmes Distribués, Research in Applied Mathematics, 8. Masson, Paris, 1988.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

J. Prüss, On the spectrum of C0-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857.  doi: 10.2307/1999112.  Google Scholar

[25]

S. W. Taylor, Gevrey Regularity of Solutions of Evolution Equations and Boundary Controllability, Thesis (Ph.D.)–University of Minnesota. 1989, 182 pp.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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