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.

[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).

[1]

P. Gidoni, G. B. Maggiani, R. Scala. Existence and regularity of solutions for an evolution model of perfectly plastic plates. Communications on Pure & Applied Analysis, 2019, 18 (4) : 1783-1826. doi: 10.3934/cpaa.2019084

[2]

Robert Jensen, Andrzej Świech. Uniqueness and existence of maximal and minimal solutions of fully nonlinear elliptic PDE. Communications on Pure & Applied Analysis, 2005, 4 (1) : 199-207. doi: 10.3934/cpaa.2005.4.187

[3]

T. Tachim Medjo. Existence and uniqueness of strong periodic solutions of the primitive equations of the ocean. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1491-1508. doi: 10.3934/dcds.2010.26.1491

[4]

Salim A. Messaoudi, Abdelfeteh Fareh. Exponential decay for linear damped porous thermoelastic systems with second sound. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 599-612. doi: 10.3934/dcdsb.2015.20.599

[5]

Maria Grazia Naso. Controllability to trajectories for semilinear thermoelastic plates. Conference Publications, 2005, 2005 (Special) : 672-681. doi: 10.3934/proc.2005.2005.672

[6]

Ramón Quintanilla, Reinhard Racke. Stability for thermoelastic plates with two temperatures. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6333-6352. doi: 10.3934/dcds.2017274

[7]

Yuming Qin, Xinguang Yang, Zhiyong Ma. Global existence of solutions for the thermoelastic Bresse system. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1395-1406. doi: 10.3934/cpaa.2014.13.1395

[8]

Zijuan Wen, Meng Fan, Asim M. Asiri, Ebraheem O. Alzahrani, Mohamed M. El-Dessoky, Yang Kuang. Global existence and uniqueness of classical solutions for a generalized quasilinear parabolic equation with application to a glioblastoma growth model. Mathematical Biosciences & Engineering, 2017, 14 (2) : 407-420. doi: 10.3934/mbe.2017025

[9]

Vo Anh Khoa, Le Thi Phuong Ngoc, Nguyen Thanh Long. Existence, blow-up and exponential decay of solutions for a porous-elastic system with damping and source terms. Evolution Equations & Control Theory, 2019, 8 (2) : 359-395. doi: 10.3934/eect.2019019

[10]

Barbara Kaltenbacher, Irena Lasiecka. Global existence and exponential decay rates for the Westervelt equation. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 503-523. doi: 10.3934/dcdss.2009.2.503

[11]

Jun Xie, Jinlong Yuan, Dongxia Wang, Weili Liu, Chongyang Liu. Uniqueness of solutions to fuzzy relational equations regarding Max-av composition and strong regularity of the matrices in Max-av algebra. Journal of Industrial & Management Optimization, 2018, 14 (3) : 1007-1022. doi: 10.3934/jimo.2017087

[12]

Daoyi Xu, Weisong Zhou. Existence-uniqueness and exponential estimate of pathwise solutions of retarded stochastic evolution systems with time smooth diffusion coefficients. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 2161-2180. doi: 10.3934/dcds.2017093

[13]

Pablo Ochoa, Julio Alejo Ruiz. A study of comparison, existence and regularity of viscosity and weak solutions for quasilinear equations in the Heisenberg group. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1091-1115. doi: 10.3934/cpaa.2019053

[14]

Stéphane Gerbi, Belkacem Said-Houari. Exponential decay for solutions to semilinear damped wave equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 559-566. doi: 10.3934/dcdss.2012.5.559

[15]

Sachiko Ishida, Tomomi Yokota. Boundedness in a quasilinear fully parabolic Keller-Segel system via maximal Sobolev regularity. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 211-232. doi: 10.3934/dcdss.2020012

[16]

Le Thi Phuong Ngoc, Nguyen Thanh Long. Existence and exponential decay for a nonlinear wave equation with nonlocal boundary conditions. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2001-2029. doi: 10.3934/cpaa.2013.12.2001

[17]

Evgeny Yu. Panov. On a condition of strong precompactness and the decay of periodic entropy solutions to scalar conservation laws. Networks & Heterogeneous Media, 2016, 11 (2) : 349-367. doi: 10.3934/nhm.2016.11.349

[18]

Filippo Dell'Oro, Vittorino Pata. Memory relaxation of type III thermoelastic extensible beams and Berger plates. Evolution Equations & Control Theory, 2012, 1 (2) : 251-270. doi: 10.3934/eect.2012.1.251

[19]

M. Eller, Roberto Triggiani. Exact/approximate controllability of thermoelastic plates with variable thermal coefficients. Discrete & Continuous Dynamical Systems - A, 2001, 7 (2) : 283-302. doi: 10.3934/dcds.2001.7.283

[20]

Huajun Gong, Jinkai Li. Global existence of strong solutions to incompressible MHD. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1553-1561. doi: 10.3934/cpaa.2014.13.1553

2017 Impact Factor: 1.179

Metrics

  • PDF downloads (20)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]