Maximal regularity and global existence of solutions to a quasilinear thermoelastic plate system

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November 2013, 33(11&12): 5189-5202. doi: 10.3934/dcds.2013.33.5189

Maximal regularity and global existence of solutions to a quasilinear thermoelastic plate system

Irena Lasiecka ^1, and Mathias Wilke ^2,

1.	Department of Mathematics, University of Virginia, Charlottesville, VA 22903
2.	Institut für Mathematik, Martin-Luther Universität Halle-Wittenberg, 06099 Halle

Received November 2011 Revised October 2012 Published May 2013

Abstract
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We consider a quasilinear PDE system which models nonlinear vibrations of a thermoelastic plate defined on a bounded domain in $\mathbb{R}^n$. Global Well-posedness of solutions is shown by applying the theory of maximal parabolic regularity of type $L_p$. In addition, we prove exponential decay rates for strong solutions and their derivatives.

Keywords: existence and uniqueness of strong solutions, maximal regularity, Quasilinear thermoelastic plates, exponential decay..

Mathematics Subject Classification: Primary: 74F05; Secondary: 35B30, 35B40, 74H4.

Citation: Irena Lasiecka, Mathias Wilke. Maximal regularity and global existence of solutions to a quasilinear thermoelastic plate system. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5189-5202. doi: 10.3934/dcds.2013.33.5189

References:

[1]	S. A. Ambartsumian, M. V. Belubekyan and M. M. Minasyan, On the problem of vibrations of nonlinear elastic electroconductive plates in transverse and longitudinal magnetic fields,, International Journal of Nonlinear Mechanics, 19 (1983), 141.
[2]	P. Acquistapace and B. Terreni, Some existence and regularity results for abstract non-autonomous parabolic equations,, Journal of Mathematical Analysis and Applications, 99 (1984), 9. doi: 10.1016/0022-247X(84)90234-8.
[3]	S. Angenent, Nonlinear analytic semiflows,, Proc. Roy. Soc. Edinburgh Sect. A, 115 (1990), 91. doi: 10.1017/S0308210500024598.
[4]	G. Avalos and I. Lasiecka, Exponential stability of a thermoelastic system with free boundary conditions without mechanical dissipation,, SIAM Journal of Mathematical Analysis, 29 (1998), 155. doi: 10.1137/S0036141096300823.
[5]	G. Avalos and I. Lasiecka, Exponential stability of a thermoelastic system without dissipation,, Rend. Istit. Mat. Univ. Trieste, XXVIII (1997), 1.
[6]	G. Avalos and I. Lasiecka, On the null-controllability of thermoelastic plates and singularity of the associated minimal energy function,, Journal of Mathematical Analysis and its Applications, 10 (2004), 34. doi: 10.1016/j.jmaa.2004.01.035.
[7]	G. Avalos and I. Lasiecka, Uniform decays in nonlinear thermoelasticity,, in, 15 (1998), 1.
[8]	A. Benabdallah and M. G. Naso, Nullcontrolability of thermoelastic plates,, Abstract and Applied Analysis, 7 (2002), 585. doi: 10.1155/S108533750220408X.
[9]	G. Y. Bagdasaryan, "Vibrations and Stability of Magnetoelastic Systems,", Yerevan, (1999).
[10]	C. Dafermos, On the existence and asymptotic stability of solutions to the equations of nonlinear thermoelasticity,, Arch. Rat. Mechanics. Anal., 29 (1968), 241. doi: 10.1007/BF00276727.
[11]	C. Dafermos and L. Hsiao, Development of singularities in solutions of the equations of nonlinear thermoelasticity,, Quart. Appl. Math., 44 (1986), 463.
[12]	G. Da Prato and P. Grisvard, Maximal regularity for evolution equations by interpolation and extrapolation,, Journal of Functional Analysis, 58 (1984), 107. doi: 10.1016/0022-1236(84)90034-X.
[13]	K. Deimling, "Nonlinear Functional Analysis,", Springer, (1985).
[14]	R. Denk, M. Hieber and J. Prüss, $\mathcalR$-boundedness, Fourier Multipliers and Problems of elliptic and parabolic type,, Memoirs of the AMS, (2003).
[15]	R. Denk and R. Racke, $L^p$-resolvent estimates and time decay for generalized thermoelastic plate equations,, Electronic Journal of Differential Equations, (2006).
[16]	R. Denk, Y. Shibata and R. Racke, $L^p$ theory for the linear thermoelastic plate equations in bounded and exterior domains,, Konstanzer Schriften in Mathematik und Informatik, 240 (2008).
[17]	M. Eller, I. Lasiecka and R. Triggiani, Simultaneous exact-approximate boundary controllability of thermo-elastic plates with variable thermal coefficients and moment control,, Journal of Mathematical Analysis and its Applications, 251 (2000), 452. doi: 10.1006/jmaa.2000.7015.
[18]	M. Eller, I. Lasiecka and R. Triggiani, Unique continuation result for thermoelastic plates,, Inverse and Ill-Posed Problems, 9 (2001), 109.
[19]	D. Hasanyan, N. Hovakimyan, A. J. Sasane and V. Stepanyan, Analysis of nonlinear thermoelastic plate equations,, in, 2 (2004), 1514. doi: 10.1109/CDC.2004.1430258.
[20]	S. Hansen and B. Zhang, Boundary control of a linear thermoelastic beam,, Journal of Mathematical Analysis and its Applications, 210 (1997), 182. doi: 10.1006/jmaa.1997.5437.
[21]	S. Hansen, Exponential decay in a linear thermoelastic rod,, J. Math. Anal. Appl., 187 (1992), 428. doi: 10.1016/0022-247X(92)90217-2.
[22]	A. A. Ilyushin, "Plasticity. Part One. Elasticity-Plastic Deformations,", OGIZ, (1948).
[23]	S. Jiang and R. Racke, "Evolution Equations in Thermoelasticity,", Chapman and Hall, (2000).
[24]	J. U. Kim, On the energy decay of a linear thermoelastic bar and plate,, SIAM Journal of Mathematical Analysis, 23 (1992), 889. doi: 10.1137/0523047.
[25]	H. Koch and I.Lasiecka, Backward uniqueness in linear thermo-elasticity with variable coefficients,, Functional Analysis and Evolution Equations, (2007).
[26]	M. Köhne, J. Prüss and M. Wilke, On quasilinear parabolic evolution equations in wheighted $L_p$-spaces,, J. Evol. Equ., 10 (2010), 443. doi: 10.1007/s00028-010-0056-0.
[27]	J.Lagnese, The reachability problem for thermoelastic plates,, Archive for Rational Mechanics and Analysis, 112 (1990), 223. doi: 10.1007/BF00381235.
[28]	J. Lagnese, "Boundary Stabilization of Thin Plates,", SIAM, (1989). doi: 10.1137/1.9781611970821.
[29]	I. Lasiecka, Uniform decay rates for full von Karman system of dynamic thermoelasticity with free boundary conditions and partial boundary dissipation,, Communications in Partial Differential Equations, 24 (1999), 1801. doi: 10.1080/03605309908821483.
[30]	I. Lasiecka and C. Lebiedzik, Asymptotic behavior of nonlinear structural acoustic interactions with thermal effects on the interface,, Nonlinear Analysis, 49 (2002), 703. doi: 10.1016/S0362-546X(01)00135-3.
[31]	I. Lasiecka and C. Lebiedzik, Boundary stabilizability of nonlinear structural acoustic models with thermal effects on the interface,, C.R. Acad. Sci. Paris, 328 (2000), 187. doi: 10.1016/S1287-4620(00)00111-3.
[32]	I. Lasiecka, S. Maad and A. Sasane, Existence and exponential decay of solutions to a quasilinear thermoelastic plate system,, NODEA, 15 (2008), 689. doi: 10.1007/s00030-008-0011-8.
[33]	I. Lasiecka and T. Seidman, Blowup estimates for observability of a thermoelastic system,, Asymptotic Analysis, 50 (2006), 93.
[34]	I. Lasiecka and R. Triggiani, Structural decomposition of thermoelastic semigroups with rotational forces,, Semigroup Forum, 60 (2000), 16. doi: 10.1007/s002330010003.
[35]	I. Lasiecka and R. Triggiani, "Control Theory for PDEs,", Cambridge University Press, 1 (2000).
[36]	I. Lasiecka, M. Renardy and R. Triggiani, Backward uniqueness of thermoelastic plates with rotational forces,, Semigroup Forum, 62 (2001), 217. doi: 10.1007/s002330010035.
[37]	I. Lasiecka and R. Triggiani, Exact null-controllability of structurally damped and thermoelastic parabolic models,, Rend. Mat. Acta Lincei, 9 (1998), 43.
[38]	I. Lasiecka and R. Triggiani, Analyticity, and lack thereof, of thermoelastic semigroups,, European Series in Applied and Industrial Mathematics, 4 (1998), 199. doi: 10.1051/proc:1998029.
[39]	L. Librescu, "Elastostatics and Kinetics of Anisotropic and Heterogeneous Shell-type Structures,", Noordhoff, (1975). doi: 10.1115/1.3423921.
[40]	L. Librescu, D. Hasanyan, Z. Qin and D. Ambur, Nonlinear magnetothermoelasticity of anisotropic plates in a magnetic field,, Journal of Thermal Stresses, 26 (2003), 1277. doi: 10.1080/714050886.
[41]	Z. Liu and M. Renardy, A note on the equation of a thermoelastic plate,, Appl. Math. Letters, 8 (1995), 1. doi: 10.1016/0893-9659(95)00020-Q.
[42]	A. Lunardi, "Analytic Semigroups and Optimal Regularity in Parabolic Problems,", Birkhauser, (1995). doi: 10.1007/978-3-0348-9234-6.
[43]	A. Lunardi, Abstract quasilinear parabolic equations,, Math. Ann., 267 (1984), 395. doi: 10.1007/BF01456097.
[44]	A. Lunardi, Global solutions of abstract quasilinear parabolic equations,, Journal Differential Equations, 58 (1985), 228. doi: 10.1016/0022-0396(85)90014-2.
[45]	A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Applied Mathematical Sciences, 44 (1983). doi: 10.1007/978-1-4612-5561-1.
[46]	J. Prüss, Maximal regularity for evolution equations in $L_p$-spaces,, Conf. Semin. Mat. Univ. Bari, (2002), 1.
[47]	J. Prüss and G. Simonett, Maximal regularity for evolution equations in weighted $L_p$-spaces,, Arch. Math., 82 (2004), 415. doi: 10.1007/s00013-004-0585-2.
[48]	J. E. Muñoz Rivera and R. Racke, Smoothing properties, decay, and global existence of solutions to nonlinear coupled systems of thermoelastic type,, SIAM Journal on Mathematical Analysis, 26 (1995), 1547. doi: 10.1137/S0036142993255058.
[49]	J. E. Muñoz Rivera and R. Racke, Large solutions and smoothing properties for nonlinear thermoelastic systems,, Journal of Differential Equations, 127 (1996), 454. doi: 10.1006/jdeq.1996.0078.
[50]	H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators,", North-Holland, (1978).

show all references

References:

[1]	S. A. Ambartsumian, M. V. Belubekyan and M. M. Minasyan, On the problem of vibrations of nonlinear elastic electroconductive plates in transverse and longitudinal magnetic fields,, International Journal of Nonlinear Mechanics, 19 (1983), 141.
[2]	P. Acquistapace and B. Terreni, Some existence and regularity results for abstract non-autonomous parabolic equations,, Journal of Mathematical Analysis and Applications, 99 (1984), 9. doi: 10.1016/0022-247X(84)90234-8.
[3]	S. Angenent, Nonlinear analytic semiflows,, Proc. Roy. Soc. Edinburgh Sect. A, 115 (1990), 91. doi: 10.1017/S0308210500024598.
[4]	G. Avalos and I. Lasiecka, Exponential stability of a thermoelastic system with free boundary conditions without mechanical dissipation,, SIAM Journal of Mathematical Analysis, 29 (1998), 155. doi: 10.1137/S0036141096300823.
[5]	G. Avalos and I. Lasiecka, Exponential stability of a thermoelastic system without dissipation,, Rend. Istit. Mat. Univ. Trieste, XXVIII (1997), 1.
[6]	G. Avalos and I. Lasiecka, On the null-controllability of thermoelastic plates and singularity of the associated minimal energy function,, Journal of Mathematical Analysis and its Applications, 10 (2004), 34. doi: 10.1016/j.jmaa.2004.01.035.
[7]	G. Avalos and I. Lasiecka, Uniform decays in nonlinear thermoelasticity,, in, 15 (1998), 1.
[8]	A. Benabdallah and M. G. Naso, Nullcontrolability of thermoelastic plates,, Abstract and Applied Analysis, 7 (2002), 585. doi: 10.1155/S108533750220408X.
[9]	G. Y. Bagdasaryan, "Vibrations and Stability of Magnetoelastic Systems,", Yerevan, (1999).
[10]	C. Dafermos, On the existence and asymptotic stability of solutions to the equations of nonlinear thermoelasticity,, Arch. Rat. Mechanics. Anal., 29 (1968), 241. doi: 10.1007/BF00276727.
[11]	C. Dafermos and L. Hsiao, Development of singularities in solutions of the equations of nonlinear thermoelasticity,, Quart. Appl. Math., 44 (1986), 463.
[12]	G. Da Prato and P. Grisvard, Maximal regularity for evolution equations by interpolation and extrapolation,, Journal of Functional Analysis, 58 (1984), 107. doi: 10.1016/0022-1236(84)90034-X.
[13]	K. Deimling, "Nonlinear Functional Analysis,", Springer, (1985).
[14]	R. Denk, M. Hieber and J. Prüss, $\mathcalR$-boundedness, Fourier Multipliers and Problems of elliptic and parabolic type,, Memoirs of the AMS, (2003).
[15]	R. Denk and R. Racke, $L^p$-resolvent estimates and time decay for generalized thermoelastic plate equations,, Electronic Journal of Differential Equations, (2006).
[16]	R. Denk, Y. Shibata and R. Racke, $L^p$ theory for the linear thermoelastic plate equations in bounded and exterior domains,, Konstanzer Schriften in Mathematik und Informatik, 240 (2008).
[17]	M. Eller, I. Lasiecka and R. Triggiani, Simultaneous exact-approximate boundary controllability of thermo-elastic plates with variable thermal coefficients and moment control,, Journal of Mathematical Analysis and its Applications, 251 (2000), 452. doi: 10.1006/jmaa.2000.7015.
[18]	M. Eller, I. Lasiecka and R. Triggiani, Unique continuation result for thermoelastic plates,, Inverse and Ill-Posed Problems, 9 (2001), 109.
[19]	D. Hasanyan, N. Hovakimyan, A. J. Sasane and V. Stepanyan, Analysis of nonlinear thermoelastic plate equations,, in, 2 (2004), 1514. doi: 10.1109/CDC.2004.1430258.
[20]	S. Hansen and B. Zhang, Boundary control of a linear thermoelastic beam,, Journal of Mathematical Analysis and its Applications, 210 (1997), 182. doi: 10.1006/jmaa.1997.5437.
[21]	S. Hansen, Exponential decay in a linear thermoelastic rod,, J. Math. Anal. Appl., 187 (1992), 428. doi: 10.1016/0022-247X(92)90217-2.
[22]	A. A. Ilyushin, "Plasticity. Part One. Elasticity-Plastic Deformations,", OGIZ, (1948).
[23]	S. Jiang and R. Racke, "Evolution Equations in Thermoelasticity,", Chapman and Hall, (2000).
[24]	J. U. Kim, On the energy decay of a linear thermoelastic bar and plate,, SIAM Journal of Mathematical Analysis, 23 (1992), 889. doi: 10.1137/0523047.
[25]	H. Koch and I.Lasiecka, Backward uniqueness in linear thermo-elasticity with variable coefficients,, Functional Analysis and Evolution Equations, (2007).
[26]	M. Köhne, J. Prüss and M. Wilke, On quasilinear parabolic evolution equations in wheighted $L_p$-spaces,, J. Evol. Equ., 10 (2010), 443. doi: 10.1007/s00028-010-0056-0.
[27]	J.Lagnese, The reachability problem for thermoelastic plates,, Archive for Rational Mechanics and Analysis, 112 (1990), 223. doi: 10.1007/BF00381235.
[28]	J. Lagnese, "Boundary Stabilization of Thin Plates,", SIAM, (1989). doi: 10.1137/1.9781611970821.
[29]	I. Lasiecka, Uniform decay rates for full von Karman system of dynamic thermoelasticity with free boundary conditions and partial boundary dissipation,, Communications in Partial Differential Equations, 24 (1999), 1801. doi: 10.1080/03605309908821483.
[30]	I. Lasiecka and C. Lebiedzik, Asymptotic behavior of nonlinear structural acoustic interactions with thermal effects on the interface,, Nonlinear Analysis, 49 (2002), 703. doi: 10.1016/S0362-546X(01)00135-3.
[31]	I. Lasiecka and C. Lebiedzik, Boundary stabilizability of nonlinear structural acoustic models with thermal effects on the interface,, C.R. Acad. Sci. Paris, 328 (2000), 187. doi: 10.1016/S1287-4620(00)00111-3.
[32]	I. Lasiecka, S. Maad and A. Sasane, Existence and exponential decay of solutions to a quasilinear thermoelastic plate system,, NODEA, 15 (2008), 689. doi: 10.1007/s00030-008-0011-8.
[33]	I. Lasiecka and T. Seidman, Blowup estimates for observability of a thermoelastic system,, Asymptotic Analysis, 50 (2006), 93.
[34]	I. Lasiecka and R. Triggiani, Structural decomposition of thermoelastic semigroups with rotational forces,, Semigroup Forum, 60 (2000), 16. doi: 10.1007/s002330010003.
[35]	I. Lasiecka and R. Triggiani, "Control Theory for PDEs,", Cambridge University Press, 1 (2000).
[36]	I. Lasiecka, M. Renardy and R. Triggiani, Backward uniqueness of thermoelastic plates with rotational forces,, Semigroup Forum, 62 (2001), 217. doi: 10.1007/s002330010035.
[37]	I. Lasiecka and R. Triggiani, Exact null-controllability of structurally damped and thermoelastic parabolic models,, Rend. Mat. Acta Lincei, 9 (1998), 43.
[38]	I. Lasiecka and R. Triggiani, Analyticity, and lack thereof, of thermoelastic semigroups,, European Series in Applied and Industrial Mathematics, 4 (1998), 199. doi: 10.1051/proc:1998029.
[39]	L. Librescu, "Elastostatics and Kinetics of Anisotropic and Heterogeneous Shell-type Structures,", Noordhoff, (1975). doi: 10.1115/1.3423921.
[40]	L. Librescu, D. Hasanyan, Z. Qin and D. Ambur, Nonlinear magnetothermoelasticity of anisotropic plates in a magnetic field,, Journal of Thermal Stresses, 26 (2003), 1277. doi: 10.1080/714050886.
[41]	Z. Liu and M. Renardy, A note on the equation of a thermoelastic plate,, Appl. Math. Letters, 8 (1995), 1. doi: 10.1016/0893-9659(95)00020-Q.
[42]	A. Lunardi, "Analytic Semigroups and Optimal Regularity in Parabolic Problems,", Birkhauser, (1995). doi: 10.1007/978-3-0348-9234-6.
[43]	A. Lunardi, Abstract quasilinear parabolic equations,, Math. Ann., 267 (1984), 395. doi: 10.1007/BF01456097.
[44]	A. Lunardi, Global solutions of abstract quasilinear parabolic equations,, Journal Differential Equations, 58 (1985), 228. doi: 10.1016/0022-0396(85)90014-2.
[45]	A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Applied Mathematical Sciences, 44 (1983). doi: 10.1007/978-1-4612-5561-1.
[46]	J. Prüss, Maximal regularity for evolution equations in $L_p$-spaces,, Conf. Semin. Mat. Univ. Bari, (2002), 1.
[47]	J. Prüss and G. Simonett, Maximal regularity for evolution equations in weighted $L_p$-spaces,, Arch. Math., 82 (2004), 415. doi: 10.1007/s00013-004-0585-2.
[48]	J. E. Muñoz Rivera and R. Racke, Smoothing properties, decay, and global existence of solutions to nonlinear coupled systems of thermoelastic type,, SIAM Journal on Mathematical Analysis, 26 (1995), 1547. doi: 10.1137/S0036142993255058.
[49]	J. E. Muñoz Rivera and R. Racke, Large solutions and smoothing properties for nonlinear thermoelastic systems,, Journal of Differential Equations, 127 (1996), 454. doi: 10.1006/jdeq.1996.0078.
[50]	H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators,", North-Holland, (1978).

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