Long-time dynamics of vectorial von Karman system with nonlinear thermal effects and free boundary conditions

Irena Lasiecka; To Fu Ma; Rodrigo Nunes Monteiro

doi:10.3934/dcdsb.2018141

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May 2018, 23(3): 1037-1072. doi: 10.3934/dcdsb.2018141

Long-time dynamics of vectorial von Karman system with nonlinear thermal effects and free boundary conditions

Irena Lasiecka ^1,, , To Fu Ma ^2, and Rodrigo Nunes Monteiro ^2,

1.	Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, USA and IBS-Polish Academy of Sciences, Warsaw, Poland
2.	Institute of Mathematical and Computer Sciences, University of São Paulo, 13566-590 São Carlos, SP, Brazil

* Corresponding author: Irena Lasiecka

Received January 2017 Revised June 2017 Published February 2018

This paper is concerned with long-time dynamics of a full von Karman system subject to nonlinear thermal coupling and free boundary conditions. In contrast with scalar von Karman system, vectorial full von Karman system accounts for both vertical and in plane displacements. The corresponding PDE is of critical interest in flow structure interactions where nonlinear plate/shell dynamics interacts with perturbed flows [vicid or invicid] [8,9,15]. In this paper it is shown that the system admits a global attractor which is also smooth and of finite fractal dimension. The above result is shown to hold for plates without regularizing effects of rotational inertia and without any mechanical dissipation imposed on vertical displacements. This is in contrast with the literature on this topic [15] and references therein. In order to handle highly supercritical nature of the von Karman nonlinearities, new results on "hidden" trace regularity generated by thermal effects are exploited. These lead to asymptotic compensated compactness of trajectories which then allows to use newly developed tools pertaining to quasi stable dynamical systems [8].

Keywords: Full vectorial von Karman system, thermal effects, free boundary conditions without mechanical dissipation, hidden trace regularity, smooth and finite dimensional attractors.

Mathematics Subject Classification: Primary: 35B41; Secondary: 74K20.

Citation: Irena Lasiecka, To Fu Ma, Rodrigo Nunes Monteiro. Long-time dynamics of vectorial von Karman system with nonlinear thermal effects and free boundary conditions. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1037-1072. doi: 10.3934/dcdsb.2018141

References:

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[2]	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
[3]	G. Avalos and I. Lasiecka, Exponential stability of a thermoelastic system without mechanical dissipation, Dedicated to the memory of Pierre Grisvard, Rend. Istit. Mat. Univ. Trieste, 28 (1996), suppl., 1-28 (1997). Google Scholar
[4]	A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, Studies in Mathematics and its Application 25, North-Holland, Amsterdam, 1992. Google Scholar
[5]	A. Benabdallah and I. Lasiecka, Exponential decay rates for a full von Karman system of dynamic thermoelasticity, J. Differential Equations, 160 (2000), 51-93. doi: 10.1006/jdeq.1999.3656. Google Scholar
[6]	A. Bensoussan, G. Da Prato, M. C. Delfour and S. K. Mitter, Representation and Control of Infinite Dimensional Systems, Vol. Ⅱ. Systems & Control: Foundations & Applications, Birkhäuser Boston, Boston, 1993. Google Scholar
[7]	F. Bucci and I. Chueshov, Long time dynamics of a coupled system of nonlinear wave and thermoelastic plate equations, Discr. Cont. Dyn. Systems, 22 (2008), 557-586. doi: 10.3934/dcds.2008.22.557. Google Scholar
[8]	I. Chueshov, Dynamics of Quasi-Stable Dissipative Systems, Universitext. Springer, Cham, 2015. Google Scholar
[9]	I. Chueshov, E. Dowell, I. Lasiecka and J. Webster, Nonlinear elastic plate in a flow of gas: Recent results and conjectures, Appl. Math. Optim., 73 (2016), 475-500. doi: 10.1007/s00245-016-9349-1. Google Scholar
[10]	I. Chueshov, M. Eller and I. Lasiecka, On the attractor for a semilinear wave equation with critical exponent and nonlinear boundary dissipation, Comm. Partial Differential Equations, 27 (2002), 1901-1951. doi: 10.1081/PDE-120016132. Google Scholar
[11]	I. Chueshov, I. Lasiecka and J. T. Webster, Attractors for delayed, nonrotational von Karman plates with applications to flow-structure interactions without any damping, Comm. Partial Differential Equations, 39 (2014), 1965-1997. doi: 10.1080/03605302.2014.930484. Google Scholar
[12]	I. Chueshov and I. Lasiecka, Long-time behavior of second order evolutions with nonlinear damping, Mem. Amer. Math. Soc., 195 (2008), ⅷ+183 pp. Google Scholar
[13]	I. Chueshov and I. Lasiecka, Von Karman Evolution Equations. Well-posedness and Long Time Dynamics, Springer Monographs in Mathematics. Springer, New York, 2010. Google Scholar
[14]	I. Chueshov and I. Lasiecka, Attractors and long time behavior of von Karman thermoelastic plates, Appl. Math. Optim., 58 (2008), 195-241. doi: 10.1007/s00245-007-9031-8. Google Scholar
[15]	I. Chueshov and I. Ryzhkova, Unsteady interaction of a viscous fluid with an elastic shell modeled by full von Karman equations, J. Differential Equations, 254 (2013), 1833-1862. doi: 10.1016/j.jde.2012.11.006. Google Scholar
[16]	I. Chueshov and I. Ryzhkova, A global attractor for a fluid-plate interaction model, Commun. Pure Appl. Anal., 12 (2013), 1635-1656. Google Scholar
[17]	P. G. Ciarlet and P. Rabier, Les Équations de von Kármán, Springer Verlag, 1980. Google Scholar
[18]	P. G. Ciarlet, Mathematical Elasticity, Vol. II, Theory of Plates. Studies in Mathematics and its Applications, 27. North-Holland, Amsterdam, 1997. Google Scholar
[19]	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
[20]	M. Eller, V. Isakov, G. Nakamura and D. Tataru, On the uniqueness and stability in the Cauchy problem for Maxwell and elasticity systems, in: Nonlinear Partial Differential Equations and Their Applications. Collège de France Seminar, Vol. XIV (Paris, 1997/1998), Stud. Math. Appl., North-Holland, Amsterdam, 31 (2002), 329-349. Google Scholar
[21]	P. G. Geredeli, I. Lasiecka and J. T. Webster, Smooth attractors of finite dimension for von Karman evolutions with nonlinear frictional damping localized in a boundary layer, J. Differential Equations, 254 (2013), 1193-1229. doi: 10.1016/j.jde.2012.10.016. Google Scholar
[22]	J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs 25, AMS, Providence, 1988. Google Scholar
[23]	S. Hansen, Boundary control of a one dimensional linear thermoelastic rod, SIAM J. Control Optim., 32 (1994), 1052-1074. doi: 10.1137/S0363012991222607. Google Scholar
[24]	M. A. Horn, Sharp trace regularity for the solutions of the equations of dynamic elasticity, J. Math. Systems Estim. Control, 8 (1998), 11pp. Google Scholar
[25]	V. Isakov, A nonhyperbolic Cauchy problem for $\Box_a \Box_b $ and its applications to elasticity theory, Comm. Pure Appl. Math., 39 (1986), 747-767. doi: 10.1002/cpa.3160390603. Google Scholar
[26]	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
[27]	H. Koch, Slow decay in linear thermoelasticity, Quart. Appl. Math., 58 (2000), 601-612. doi: 10.1090/qam/1788420. Google Scholar
[28]	H. Koch and I. Lasiecka, Hadamard well-posedness of weak solutions in nonlinear dynamic elasticity-full von Karman systems, Evolution Equations, Semigroups and Functional Analysis (Milano, 2000), Progr. Nonlinear Differential Equations Appl., Birkhäuser, Basel, 50 (2002), 197-216. Google Scholar
[29]	H. Koch and A. Stahel, Global existence of classical solutions to the dynamical von Kármán equations, Math. Methods Appl. Sci., 16 (1993), 581-586. doi: 10.1002/mma.1670160806. Google Scholar
[30]	V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, RAM: Research in Applied Mathematics. Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994. ⅷ+156 pp. Google Scholar
[31]	O. Ladyzhenskaya, Attractors for Semi-groups and Evolution Equations, Cambridge University Press, 1991. Google Scholar
[32]	J. Lagnese, The reachability problem for thermoelastic plates, Arch. Rational Mech. Anal., 112 (1990), 223-267. doi: 10.1007/BF00381235. Google Scholar
[33]	J. E. Lagnese, Uniform boundary stabilization of thermoelastic plates, Control of boundaries and stabilization, (Clermont-Ferrand, 1988), Lecture Notes in Control and Inform. Sci., 125 (1989), 154-167. Google Scholar
[34]	J. E Lagnese, Boundary Stabilization of Thin Plates, SIAM Studies in Applied Mathematics, 10. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1989. SIAM, Philadelphia, PA, 1989. Google Scholar
[35]	J. E. Lagnese and J. L. Lions, Modeling Analysis and Control of Thin Plates, Research in Applied Mathematics, 6. Masson, Paris, 1988. Google Scholar
[36]	I. Lasiecka, Weak, classical and intermediate solutions to full von Karman system of dynamic nonlinear elasticity, Appl. Anal., 68 (1998), 121-145. doi: 10.1080/00036819808840625. Google Scholar
[37]	I. Lasiecka, Uniform stabilizability of a full von Karman system with nonlinear boundary feedback, SIAM J. Control Optim., 36 (1998), 1376-1422. doi: 10.1137/S0363012996301907. Google Scholar
[38]	I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equation with nonlinear boundary damping, Differential and Integral Equations, 6 (1993), 507-553. Google Scholar
[39]	I. Lasiecka, Uniform decay rates for full von Karman system of dynamic thermoelasticity with free boundary conditions and partial boundary dissipation, Comm. Partial Differential Equations, 24 (1999), 1801-1847. doi: 10.1080/03605309908821483. Google Scholar
[40]	I. Lasiecka, J.-L. Lions and R. Triggiani, Nonhomogenous boundary value problems for second order hyperbolic operators, J. Math. Pures Appl.(9), 65 (1986), 149-192. Google Scholar
[41]	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
[42]	I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories. I. Abstract Parabolic Systems, Encyclopedia of Mathematics and its Applications, 74. Cambridge University Press, Cambridge, 2000. Google Scholar
[43]	J. L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Vol. 1, Springer-Verlag, New York-Heidelberg, 1972. Google Scholar
[44]	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. Google Scholar
[45]	J. E. Muñoz 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. Google Scholar
[46]	G. Perla Menzala and F. Travessini De Cezaro, Global existence and uniqueness of weak and regular solutions of shallow shells with thermal effects, Appl. Math. Optim., 74 (2016), 229-271. doi: 10.1007/s00245-015-9313-5. Google Scholar
[47]	A. Ruiz, Unique continuation for weak solutions of the wave equation plus a potential, J. Math. Pures Appl.(9), 71 (1992), 455-467. Google Scholar
[48]	I. Ryzhkova, Dynamics of a thermoelastic von Kármán plate in a subsonic gas flow, Z. Angew. Math. Phys., 58 (2007), 246-261. doi: 10.1007/s00033-006-0080-7. Google Scholar
[49]	I. Ryzhkova, Stabilization of von Kármán plate in the presence of thermal effects in a subsonic potential flow of gas, J. Math. Anal. Appl., 294 (2004), 462-481. doi: 10.1016/j.jmaa.2004.02.021. Google Scholar
[50]	R. Sakamoto, Hyperbolic Boundary Value Problem, Cambridge University Press, 1982. Google Scholar
[51]	R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences 68, Springer-Verlag, New York, 1988. Google Scholar
[52]	P.-F. Yao, Observability inequalities for shallow shells, SIAM J. Control Optim., 38 (2000), 1729-1756. doi: 10.1137/S0363012999338692. Google Scholar

show all references

References:

[1]	G. Amendola, M. Fabrizio and J. M. Golden, Thermodynamics of Materials With Memory. Theory and Applications, Springer, New York, 2012. Google Scholar
[2]	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
[3]	G. Avalos and I. Lasiecka, Exponential stability of a thermoelastic system without mechanical dissipation, Dedicated to the memory of Pierre Grisvard, Rend. Istit. Mat. Univ. Trieste, 28 (1996), suppl., 1-28 (1997). Google Scholar
[4]	A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, Studies in Mathematics and its Application 25, North-Holland, Amsterdam, 1992. Google Scholar
[5]	A. Benabdallah and I. Lasiecka, Exponential decay rates for a full von Karman system of dynamic thermoelasticity, J. Differential Equations, 160 (2000), 51-93. doi: 10.1006/jdeq.1999.3656. Google Scholar
[6]	A. Bensoussan, G. Da Prato, M. C. Delfour and S. K. Mitter, Representation and Control of Infinite Dimensional Systems, Vol. Ⅱ. Systems & Control: Foundations & Applications, Birkhäuser Boston, Boston, 1993. Google Scholar
[7]	F. Bucci and I. Chueshov, Long time dynamics of a coupled system of nonlinear wave and thermoelastic plate equations, Discr. Cont. Dyn. Systems, 22 (2008), 557-586. doi: 10.3934/dcds.2008.22.557. Google Scholar
[8]	I. Chueshov, Dynamics of Quasi-Stable Dissipative Systems, Universitext. Springer, Cham, 2015. Google Scholar
[9]	I. Chueshov, E. Dowell, I. Lasiecka and J. Webster, Nonlinear elastic plate in a flow of gas: Recent results and conjectures, Appl. Math. Optim., 73 (2016), 475-500. doi: 10.1007/s00245-016-9349-1. Google Scholar
[10]	I. Chueshov, M. Eller and I. Lasiecka, On the attractor for a semilinear wave equation with critical exponent and nonlinear boundary dissipation, Comm. Partial Differential Equations, 27 (2002), 1901-1951. doi: 10.1081/PDE-120016132. Google Scholar
[11]	I. Chueshov, I. Lasiecka and J. T. Webster, Attractors for delayed, nonrotational von Karman plates with applications to flow-structure interactions without any damping, Comm. Partial Differential Equations, 39 (2014), 1965-1997. doi: 10.1080/03605302.2014.930484. Google Scholar
[12]	I. Chueshov and I. Lasiecka, Long-time behavior of second order evolutions with nonlinear damping, Mem. Amer. Math. Soc., 195 (2008), ⅷ+183 pp. Google Scholar
[13]	I. Chueshov and I. Lasiecka, Von Karman Evolution Equations. Well-posedness and Long Time Dynamics, Springer Monographs in Mathematics. Springer, New York, 2010. Google Scholar
[14]	I. Chueshov and I. Lasiecka, Attractors and long time behavior of von Karman thermoelastic plates, Appl. Math. Optim., 58 (2008), 195-241. doi: 10.1007/s00245-007-9031-8. Google Scholar
[15]	I. Chueshov and I. Ryzhkova, Unsteady interaction of a viscous fluid with an elastic shell modeled by full von Karman equations, J. Differential Equations, 254 (2013), 1833-1862. doi: 10.1016/j.jde.2012.11.006. Google Scholar
[16]	I. Chueshov and I. Ryzhkova, A global attractor for a fluid-plate interaction model, Commun. Pure Appl. Anal., 12 (2013), 1635-1656. Google Scholar
[17]	P. G. Ciarlet and P. Rabier, Les Équations de von Kármán, Springer Verlag, 1980. Google Scholar
[18]	P. G. Ciarlet, Mathematical Elasticity, Vol. II, Theory of Plates. Studies in Mathematics and its Applications, 27. North-Holland, Amsterdam, 1997. Google Scholar
[19]	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
[20]	M. Eller, V. Isakov, G. Nakamura and D. Tataru, On the uniqueness and stability in the Cauchy problem for Maxwell and elasticity systems, in: Nonlinear Partial Differential Equations and Their Applications. Collège de France Seminar, Vol. XIV (Paris, 1997/1998), Stud. Math. Appl., North-Holland, Amsterdam, 31 (2002), 329-349. Google Scholar
[21]	P. G. Geredeli, I. Lasiecka and J. T. Webster, Smooth attractors of finite dimension for von Karman evolutions with nonlinear frictional damping localized in a boundary layer, J. Differential Equations, 254 (2013), 1193-1229. doi: 10.1016/j.jde.2012.10.016. Google Scholar
[22]	J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs 25, AMS, Providence, 1988. Google Scholar
[23]	S. Hansen, Boundary control of a one dimensional linear thermoelastic rod, SIAM J. Control Optim., 32 (1994), 1052-1074. doi: 10.1137/S0363012991222607. Google Scholar
[24]	M. A. Horn, Sharp trace regularity for the solutions of the equations of dynamic elasticity, J. Math. Systems Estim. Control, 8 (1998), 11pp. Google Scholar
[25]	V. Isakov, A nonhyperbolic Cauchy problem for $\Box_a \Box_b $ and its applications to elasticity theory, Comm. Pure Appl. Math., 39 (1986), 747-767. doi: 10.1002/cpa.3160390603. Google Scholar
[26]	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
[27]	H. Koch, Slow decay in linear thermoelasticity, Quart. Appl. Math., 58 (2000), 601-612. doi: 10.1090/qam/1788420. Google Scholar
[28]	H. Koch and I. Lasiecka, Hadamard well-posedness of weak solutions in nonlinear dynamic elasticity-full von Karman systems, Evolution Equations, Semigroups and Functional Analysis (Milano, 2000), Progr. Nonlinear Differential Equations Appl., Birkhäuser, Basel, 50 (2002), 197-216. Google Scholar
[29]	H. Koch and A. Stahel, Global existence of classical solutions to the dynamical von Kármán equations, Math. Methods Appl. Sci., 16 (1993), 581-586. doi: 10.1002/mma.1670160806. Google Scholar
[30]	V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, RAM: Research in Applied Mathematics. Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994. ⅷ+156 pp. Google Scholar
[31]	O. Ladyzhenskaya, Attractors for Semi-groups and Evolution Equations, Cambridge University Press, 1991. Google Scholar
[32]	J. Lagnese, The reachability problem for thermoelastic plates, Arch. Rational Mech. Anal., 112 (1990), 223-267. doi: 10.1007/BF00381235. Google Scholar
[33]	J. E. Lagnese, Uniform boundary stabilization of thermoelastic plates, Control of boundaries and stabilization, (Clermont-Ferrand, 1988), Lecture Notes in Control and Inform. Sci., 125 (1989), 154-167. Google Scholar
[34]	J. E Lagnese, Boundary Stabilization of Thin Plates, SIAM Studies in Applied Mathematics, 10. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1989. SIAM, Philadelphia, PA, 1989. Google Scholar
[35]	J. E. Lagnese and J. L. Lions, Modeling Analysis and Control of Thin Plates, Research in Applied Mathematics, 6. Masson, Paris, 1988. Google Scholar
[36]	I. Lasiecka, Weak, classical and intermediate solutions to full von Karman system of dynamic nonlinear elasticity, Appl. Anal., 68 (1998), 121-145. doi: 10.1080/00036819808840625. Google Scholar
[37]	I. Lasiecka, Uniform stabilizability of a full von Karman system with nonlinear boundary feedback, SIAM J. Control Optim., 36 (1998), 1376-1422. doi: 10.1137/S0363012996301907. Google Scholar
[38]	I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equation with nonlinear boundary damping, Differential and Integral Equations, 6 (1993), 507-553. Google Scholar
[39]	I. Lasiecka, Uniform decay rates for full von Karman system of dynamic thermoelasticity with free boundary conditions and partial boundary dissipation, Comm. Partial Differential Equations, 24 (1999), 1801-1847. doi: 10.1080/03605309908821483. Google Scholar
[40]	I. Lasiecka, J.-L. Lions and R. Triggiani, Nonhomogenous boundary value problems for second order hyperbolic operators, J. Math. Pures Appl.(9), 65 (1986), 149-192. Google Scholar
[41]	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
[42]	I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories. I. Abstract Parabolic Systems, Encyclopedia of Mathematics and its Applications, 74. Cambridge University Press, Cambridge, 2000. Google Scholar
[43]	J. L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Vol. 1, Springer-Verlag, New York-Heidelberg, 1972. Google Scholar
[44]	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. Google Scholar
[45]	J. E. Muñoz 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. Google Scholar
[46]	G. Perla Menzala and F. Travessini De Cezaro, Global existence and uniqueness of weak and regular solutions of shallow shells with thermal effects, Appl. Math. Optim., 74 (2016), 229-271. doi: 10.1007/s00245-015-9313-5. Google Scholar
[47]	A. Ruiz, Unique continuation for weak solutions of the wave equation plus a potential, J. Math. Pures Appl.(9), 71 (1992), 455-467. Google Scholar
[48]	I. Ryzhkova, Dynamics of a thermoelastic von Kármán plate in a subsonic gas flow, Z. Angew. Math. Phys., 58 (2007), 246-261. doi: 10.1007/s00033-006-0080-7. Google Scholar
[49]	I. Ryzhkova, Stabilization of von Kármán plate in the presence of thermal effects in a subsonic potential flow of gas, J. Math. Anal. Appl., 294 (2004), 462-481. doi: 10.1016/j.jmaa.2004.02.021. Google Scholar
[50]	R. Sakamoto, Hyperbolic Boundary Value Problem, Cambridge University Press, 1982. Google Scholar
[51]	R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences 68, Springer-Verlag, New York, 1988. Google Scholar
[52]	P.-F. Yao, Observability inequalities for shallow shells, SIAM J. Control Optim., 38 (2000), 1729-1756. doi: 10.1137/S0363012999338692. Google Scholar

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[19]	Juan-Ming Yuan, Jiahong Wu. The complex KdV equation with or without dissipation. Discrete & Continuous Dynamical Systems - B, 2005, 5 (2) : 489-512. doi: 10.3934/dcdsb.2005.5.489
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American Institute of Mathematical Sciences

Long-time dynamics of vectorial von Karman system with nonlinear thermal effects and free boundary conditions

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American Institute of Mathematical Sciences

Long-time dynamics of vectorial von Karman system with nonlinear thermal effects and free boundary conditions

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