# American Institute of Mathematical Sciences

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

 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.

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 and 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. [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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##### 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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