Uniform stability in a vectorial full Von Kármán thermoelastic system with solenoidal dissipation and free boundary conditions

Catherine Lebiedzik

doi:10.3934/eect.2020092

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2021, Volume 10, Issue 4: 767-796. Doi: 10.3934/eect.2020092

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Uniform stability in a vectorial full Von Kármán thermoelastic system with solenoidal dissipation and free boundary conditions

Catherine Lebiedzik

Wayne State University, Department of Mathematics, Detroit, MI 48201 USA

Received: February 2020

Revised: August 2020

Early access: September 2020

Published: December 2021

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We will consider the full von Kármán thermoelastic system with free boundary conditions and dissipation imposed only on the in-plane displacement. It will be shown that the corresponding solutions are exponentially stable, though there is no mechanical dissipation on the vertical displacements. The main tools used are: (i) partial analyticity of the linearized semigroup and (ii) trace estimates which exploit the hidden regularity harvested from partial analyticity.

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Mathematics Subject Classification: Primary:35B35, 35M30, 74B20, 74F05, 74K20;Secondary:35A01, 35A02.

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References

[1]	G. Avalos and I. Lasiecka, Exponential stability of a thermoelastic system without mechanical dissipation, Rend. Istit. Mat. Univ. Trieste, 28 (1997), 1-28.
[2]	M. Belishev and I. Lasiecka, The dynamical Lamé system: Regularity of solutions, boundary controllability and boundary data continuation, ESAIM, Control Optim. Calc. Var., 8 (2002), 143-167. doi: 10.1051/cocv:2002058.
[3]	A. Benabdallah, Modeling of Von Kármán system with thermal effects, Prepublications de L'equipe de Mathematiques de Besancon, 99 (1999).
[4]	A. Benabdallah and I. Lasiecka, Exponential decay rates for a full Von Kármán system of dynamic thermoelasticity, J. Differential Equations, 160 (2000), 51-93. doi: 10.1006/jdeq.1999.3656.
[5]	A. Benabdallah and I. Lasiecka, Exponential decay rates for a full Von Kármán thermoelastic system with nonlinear thermal coupling, ESIAM: Proceedings. Contrôle des systèmes gouvernés par des équations aux dérivées partielles, 8 (2000), 13-38. doi: 10.1051/proc:2000002.
[6]	A. Benabdallah and D. Teniou, Exponential stability of a Von Kármán model with thermal effects, Electronic Journal of Differential Equations, (1998), 1–13.
[7]	A. Bensoussan, G. Da Prato, M. Delfour and S. Mitter and K. Sanjoy, Representation and Control of Infinite Dimensional Systems, 2$^nd$ edition, Birkhäuser Boston, Inc., Boston, MA, 2007. doi: 10.1007/978-0-8176-4581-6.
[8]	P. G. Ciarlet, Mathematical Elasticity. Vol. Ⅲ. Theory of Shells, in Studies in Mathematics and its Applications, Vol. 29, North-Holland Publishing Co., Amsterdam, 2000.
[9]	P. G. Ciarlet and P.Rabier, Les Equations de Von Karman, Springer, Berlin, 1980.
[10]	C. M. Dafermos, On the existence and asymptotic stability of solutions to the equations of linear thermoelasticity, Archive for Rational Mechanics and Analysis, 29 (1968), 241-271. doi: 10.1007/BF00276727.
[11]	E. H. Dowell, A Modern Course in Aeroelasticity, 5th edition, Springer, Cham, 2015. doi: 10.1007/978-3-319-09453-3.
[12]	A. Favini, M. Horn, I. Lasiecka and D. Tataru, Global existence, uniqueness and regularity of solutions to a von Kármán system with nonlinear boundary dissipation, Differential Integral Equation, 9 (1996), 267-294.
[13]	I. Chueshov and I. Lasiecka, Von Kármán Evolution Equations, Springer, New York, 2010. doi: 10.1007/978-0-387-87712-9.
[14]	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.
[15]	H. Koch, Slow decay in linear thermoelasticity, Quart. Appl. Math, 58 (2000), 601-612. doi: 10.1090/qam/1788420.
[16]	H. Koch and I. Lasiecka, Hadamard well-posedness of weak solutions in nonlinear dynamic elasticity – full von Kármán systems, in Progress in Nonlinear Differential Equations and their Appl., Vol. 50, Birkhäuser, Basel, 2002.
[17]	W. T. Koiter, On the nonlinear theory of thin elastic shells, Nederl. Akad. Wetensch. Proc. Ser. B, 69 (1966), 1-54.
[18]	J. Lagnese, Boundary Stabilization of Thin Plates, SIAM, Philadelphia, PA, 1989. doi: 10.1137/1.9781611970821.
[19]	J. Lagnese and G. Leugering, Uniform stabilization of a nonlinear beam by nonlinear boundary feedback, J. Differential Equations, 91 (1991), 355-388. doi: 10.1016/0022-0396(91)90145-Y.
[20]	J. Lagnese and J.-L. Lions, Modelling analysis and control of thin plates, in Research in Applied Mathematics, Vol. 6, Masson, Paris, 1988.
[21]	I. Lasiecka, Uniform stabilizability of a full von Kármán system with nonlinear boundary feedback, SIAM J. Control Optim., 36 (1998), 1376-1422. doi: 10.1137/S0363012996301907.
[22]	I. Lasiecka, Uniform decay rates for full von Kármán system of dynamic thermoelasticity with free boundary conditions and partial boundary dissipation, Comm. Partial Differential Equations, 24 (1999), 1801-1847. doi: 10.1080/03605309908821483.
[23]	I. Lasiecka, Mathematical Control Theory of Coupled PDEs, SIAM, Philadelphia, PA, 2002. doi: 10.1137/1.9780898717099.
[24]	I. Lasiecka, J. L. Lions and R. Triggiani, Nonhomogeneous boundary value problems for second-order hyperbolic operators, J. Math. Pures Appl., 65 (1986), 149-192.
[25]	I. Lasiecka, T. Ma and R. Montiero, Global smooth attractors for dynamics of thermal shallow shells without vertical dissipation, Trans. Amer. Math. Soc., 371 (2019), 8051-8096. doi: 10.1090/tran/7756.
[26]	I. Lasiecka, T. Ma and R. Montiero, Long-time dynamics of vectorial von Kármán system with nonlinear thermal effects and free boundary condition, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 1037-1072. doi: 10.3934/dcdsb.2018141.
[27]	I. Lasiecka and R. Triggiani, Sharp regularity theory for second order hyperbolic equations of Neumann type. Part Ⅰ. $L_2$ nonhomogenous data, Ann. Mat. Pura Appl., 157 (1990), 285-367. doi: 10.1007/BF01765322.
[28]	I. Lasiecka and R. Triggiani, Analyticity of thermoelastic semigroups with free boundary conditions, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 27 (1999), 457-487.
[29]	I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories, in Encyclopedia of Mathematics and its Applications, Volume Ⅰ and Ⅱ, Cambridge University Press, Cambridge, 2000.
[30]	J. L. Lions, Quelques Methodes de Résolution des Problèmes aux Limits Nonlinéaires, Dunod, Gauthier-Villars, Paris, 1969.
[31]	Z. 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.
[32]	S. Miyatake, Mixed problem for hyperbolic equation of second order, J. Math. Kyoto Univ., 13 (1973), 435-487. doi: 10.1215/kjm/1250523319.
[33]	V. I. Sedenko, On uniqueness of the generalized solutions of initial boundary value problem for Marguerre-Vlasov nonlinear oscillations of the shallow shells, Russian Izvestiya, North-Caucasus Region, Ser. Natural Sciences, 1-2 (1994).
[34]	D. Tataru, On the regularity of boundary traces for the wave equation, Annali di Scuola Normale Superiore, 26 (1998), 185-206.
[35]	T. von Kármán, Festigkeitprobleme in Maschinenbau, Encyklopedie der Mathematischen Wissenschaften, 4 (1910), 314-385.