# American Institute of Mathematical Sciences

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

 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

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.
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. Google Scholar [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. Google Scholar [3] S. Angenent, Nonlinear analytic semiflows,, Proc. Roy. Soc. Edinburgh Sect. A, 115 (1990), 91. doi: 10.1017/S0308210500024598. Google Scholar [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. Google Scholar [5] G. Avalos and I. Lasiecka, Exponential stability of a thermoelastic system without dissipation,, Rend. Istit. Mat. Univ. Trieste, XXVIII (1997), 1. Google Scholar [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. Google Scholar [7] G. Avalos and I. Lasiecka, Uniform decays in nonlinear thermoelasticity,, in, 15 (1998), 1. Google Scholar [8] A. Benabdallah and M. G. Naso, Nullcontrolability of thermoelastic plates,, Abstract and Applied Analysis, 7 (2002), 585. doi: 10.1155/S108533750220408X. Google Scholar [9] G. Y. Bagdasaryan, "Vibrations and Stability of Magnetoelastic Systems,", Yerevan, (1999). Google Scholar [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. Google Scholar [11] C. Dafermos and L. Hsiao, Development of singularities in solutions of the equations of nonlinear thermoelasticity,, Quart. Appl. Math., 44 (1986), 463. Google Scholar [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. Google Scholar [13] K. Deimling, "Nonlinear Functional Analysis,", Springer, (1985). Google Scholar [14] R. Denk, M. Hieber and J. Prüss, $\mathcalR$-boundedness, Fourier Multipliers and Problems of elliptic and parabolic type,, Memoirs of the AMS, (2003). Google Scholar [15] R. Denk and R. Racke, $L^p$-resolvent estimates and time decay for generalized thermoelastic plate equations,, Electronic Journal of Differential Equations, (2006). Google Scholar [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). Google Scholar [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. Google Scholar [18] M. Eller, I. Lasiecka and R. Triggiani, Unique continuation result for thermoelastic plates,, Inverse and Ill-Posed Problems, 9 (2001), 109. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [22] A. A. Ilyushin, "Plasticity. Part One. Elasticity-Plastic Deformations,", OGIZ, (1948). Google Scholar [23] S. Jiang and R. Racke, "Evolution Equations in Thermoelasticity,", Chapman and Hall, (2000). Google Scholar [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. Google Scholar [25] H. Koch and I.Lasiecka, Backward uniqueness in linear thermo-elasticity with variable coefficients,, Functional Analysis and Evolution Equations, (2007). Google Scholar [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. Google Scholar [27] J.Lagnese, The reachability problem for thermoelastic plates,, Archive for Rational Mechanics and Analysis, 112 (1990), 223. doi: 10.1007/BF00381235. Google Scholar [28] J. Lagnese, "Boundary Stabilization of Thin Plates,", SIAM, (1989). doi: 10.1137/1.9781611970821. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [33] I. Lasiecka and T. Seidman, Blowup estimates for observability of a thermoelastic system,, Asymptotic Analysis, 50 (2006), 93. Google Scholar [34] I. Lasiecka and R. Triggiani, Structural decomposition of thermoelastic semigroups with rotational forces,, Semigroup Forum, 60 (2000), 16. doi: 10.1007/s002330010003. Google Scholar [35] I. Lasiecka and R. Triggiani, "Control Theory for PDEs,", Cambridge University Press, 1 (2000). Google Scholar [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. Google Scholar [37] I. Lasiecka and R. Triggiani, Exact null-controllability of structurally damped and thermoelastic parabolic models,, Rend. Mat. Acta Lincei, 9 (1998), 43. Google Scholar [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. Google Scholar [39] L. Librescu, "Elastostatics and Kinetics of Anisotropic and Heterogeneous Shell-type Structures,", Noordhoff, (1975). doi: 10.1115/1.3423921. Google Scholar [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. Google Scholar [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. Google Scholar [42] A. Lunardi, "Analytic Semigroups and Optimal Regularity in Parabolic Problems,", Birkhauser, (1995). doi: 10.1007/978-3-0348-9234-6. Google Scholar [43] A. Lunardi, Abstract quasilinear parabolic equations,, Math. Ann., 267 (1984), 395. doi: 10.1007/BF01456097. Google Scholar [44] A. Lunardi, Global solutions of abstract quasilinear parabolic equations,, Journal Differential Equations, 58 (1985), 228. doi: 10.1016/0022-0396(85)90014-2. Google Scholar [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. Google Scholar [46] J. Prüss, Maximal regularity for evolution equations in $L_p$-spaces,, Conf. Semin. Mat. Univ. Bari, (2002), 1. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [50] H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators,", North-Holland, (1978). Google Scholar

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##### 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. Google Scholar [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. Google Scholar [3] S. Angenent, Nonlinear analytic semiflows,, Proc. Roy. Soc. Edinburgh Sect. A, 115 (1990), 91. doi: 10.1017/S0308210500024598. Google Scholar [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. Google Scholar [5] G. Avalos and I. Lasiecka, Exponential stability of a thermoelastic system without dissipation,, Rend. Istit. Mat. Univ. Trieste, XXVIII (1997), 1. Google Scholar [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. Google Scholar [7] G. Avalos and I. Lasiecka, Uniform decays in nonlinear thermoelasticity,, in, 15 (1998), 1. Google Scholar [8] A. Benabdallah and M. G. Naso, Nullcontrolability of thermoelastic plates,, Abstract and Applied Analysis, 7 (2002), 585. doi: 10.1155/S108533750220408X. Google Scholar [9] G. Y. Bagdasaryan, "Vibrations and Stability of Magnetoelastic Systems,", Yerevan, (1999). Google Scholar [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. Google Scholar [11] C. Dafermos and L. Hsiao, Development of singularities in solutions of the equations of nonlinear thermoelasticity,, Quart. Appl. Math., 44 (1986), 463. Google Scholar [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. Google Scholar [13] K. Deimling, "Nonlinear Functional Analysis,", Springer, (1985). Google Scholar [14] R. Denk, M. Hieber and J. Prüss, $\mathcalR$-boundedness, Fourier Multipliers and Problems of elliptic and parabolic type,, Memoirs of the AMS, (2003). Google Scholar [15] R. Denk and R. Racke, $L^p$-resolvent estimates and time decay for generalized thermoelastic plate equations,, Electronic Journal of Differential Equations, (2006). Google Scholar [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). Google Scholar [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. Google Scholar [18] M. Eller, I. Lasiecka and R. Triggiani, Unique continuation result for thermoelastic plates,, Inverse and Ill-Posed Problems, 9 (2001), 109. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [22] A. A. Ilyushin, "Plasticity. Part One. Elasticity-Plastic Deformations,", OGIZ, (1948). Google Scholar [23] S. Jiang and R. Racke, "Evolution Equations in Thermoelasticity,", Chapman and Hall, (2000). Google Scholar [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. Google Scholar [25] H. Koch and I.Lasiecka, Backward uniqueness in linear thermo-elasticity with variable coefficients,, Functional Analysis and Evolution Equations, (2007). Google Scholar [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. Google Scholar [27] J.Lagnese, The reachability problem for thermoelastic plates,, Archive for Rational Mechanics and Analysis, 112 (1990), 223. doi: 10.1007/BF00381235. Google Scholar [28] J. Lagnese, "Boundary Stabilization of Thin Plates,", SIAM, (1989). doi: 10.1137/1.9781611970821. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [33] I. Lasiecka and T. Seidman, Blowup estimates for observability of a thermoelastic system,, Asymptotic Analysis, 50 (2006), 93. Google Scholar [34] I. Lasiecka and R. Triggiani, Structural decomposition of thermoelastic semigroups with rotational forces,, Semigroup Forum, 60 (2000), 16. doi: 10.1007/s002330010003. Google Scholar [35] I. Lasiecka and R. Triggiani, "Control Theory for PDEs,", Cambridge University Press, 1 (2000). Google Scholar [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. Google Scholar [37] I. Lasiecka and R. Triggiani, Exact null-controllability of structurally damped and thermoelastic parabolic models,, Rend. Mat. Acta Lincei, 9 (1998), 43. Google Scholar [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. Google Scholar [39] L. Librescu, "Elastostatics and Kinetics of Anisotropic and Heterogeneous Shell-type Structures,", Noordhoff, (1975). doi: 10.1115/1.3423921. Google Scholar [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. Google Scholar [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. Google Scholar [42] A. Lunardi, "Analytic Semigroups and Optimal Regularity in Parabolic Problems,", Birkhauser, (1995). doi: 10.1007/978-3-0348-9234-6. Google Scholar [43] A. Lunardi, Abstract quasilinear parabolic equations,, Math. Ann., 267 (1984), 395. doi: 10.1007/BF01456097. Google Scholar [44] A. Lunardi, Global solutions of abstract quasilinear parabolic equations,, Journal Differential Equations, 58 (1985), 228. doi: 10.1016/0022-0396(85)90014-2. Google Scholar [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. Google Scholar [46] J. Prüss, Maximal regularity for evolution equations in $L_p$-spaces,, Conf. Semin. Mat. Univ. Bari, (2002), 1. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [50] H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators,", North-Holland, (1978). Google Scholar
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