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Partial regularity for parabolic systems with VMO-coefficients
A note on the energy transfer in coupled differential systems
Politecnico di Milano - Dipartimento di Matematica, Via Bonardi 9, 20133 Milano, Italy |
$ \begin{cases} \ddot u+u+\dot u = b\dot v\\ \ddot v+v-\epsilon \dot v = -b\dot u \end{cases} $ |
$ b $ |
References:
[1] |
F. Alabau-Boussouira, Z. Wang and L. Yu,
A one-step optimal energy decay formula for indirectly nonlinearly damped hyperbolic systems coupled by velocities, ESAIM Control Optim. Calc. Var., 23 (2017), 721-749.
doi: 10.1051/cocv/2016011. |
[2] |
M. S. Alves, C. Buriol, M. V. Ferreira, J. E. Muñoz Rivera, M. Sepúlveda and O. Vera,
Asymptotic behaviour for the vibrations modeled by the standard linear solid model with a thermal effect, J. Math. Anal. Appl., 399 (2013), 472-479.
doi: 10.1016/j.jmaa.2012.10.019. |
[3] |
K. Ammari and S. Nicaise,
Stabilization of a transmission wave/plate equation, J. Differ. Equ., 249 (2010), 707-727.
doi: 10.1016/j.jde.2010.03.007. |
[4] |
G. Avalos and I. Lasiecka,
Exponential stability of a thermoelastic system without mechanical dissipation, Rend. Istit. Mat. Univ. Trieste, 28 (1997), 1-28.
|
[5] |
G. Avalos and I. Lasiecka,
The strong stability of a semigroup arising from a coupled hyperbolic/parabolic system, Semigroup Forum, 57 (1998), 278-292.
doi: 10.1007/PL00005977. |
[6] |
M. Conti, V. Pata and R. Quintanilla,
Thermoelasticity of Moore-Gibson-Thompson type with history dependence in the temperature, Asymptot. Anal., 120 (2020), 1-21.
doi: 10.3233/asy-191576. |
[7] |
F. Dell'Oro and V. Pata,
On the stability of Timoshenko systems with Gurtin-Pipkin thermal law, J. Differ. Equ., 257 (2014), 523-548.
doi: 10.1016/j.jde.2014.04.009. |
[8] |
R. Denk and F. Kammerlander,
Exponential stability for a coupled system of damped-undamped plate equations, IMA J. Appl. Math., 83 (2018), 302-322.
doi: 10.1093/imamat/hxy002. |
[9] |
H. D. Fernández Sare and R. Racke,
On the stability of damped Timoshenko systems: Cattaneo versus Fourier law, Arch. Rational Mech. Anal., 194 (2009), 221-251.
doi: 10.1007/s00205-009-0220-2. |
[10] |
G. R. Goldstein, J. A. Goldstein and G. Perla Menzala,
On the overdamping phenomenon: a general result and applications, Quart. Appl. Math., 71 (2013), 183-199.
doi: 10.1090/S0033-569X-2012-01282-3. |
[11] |
J. Hao and Z. Liu,
Stability of an abstract system of coupled hyperbolic and parabolic equations, Z. Angew. Math. Phys., 64 (2013), 1145-1159.
doi: 10.1007/s00033-012-0274-0. |
[12] |
M. W. Hirsch and S. Smale, Differential Equations, Dynamical Systems and Linear Algebra, Academic Press, New York, 1974.
![]() ![]() |
[13] |
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. |
[14] |
Z. Liu and S. Zheng, Semigroups associated with dissipative systems, Chapman & Hall/CRC, Boca Raton, 1999.
doi: 0-8493-0615-9. |
[15] |
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. |
[16] |
A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[17] |
L. Perko, Differential equations and dynamical systems, Springer-Verlag, New York, 1991.
doi: 10.1007/978-1-4684-0392-3. |
[18] |
M. L. Santos, D. S. Almeida Júnior and J. E. Muñoz Rivera,
The stability number of the Timoshenko system with second sound, J. Differ. Equ., 253 (2012), 2715-2733.
doi: 10.1016/j.jde.2012.07.012. |
[19] |
R. Triggiani,
Heat-viscoelastic plate interaction: analyticity, spectral analysis, exponential decay, Evol. Equ. Control Theory, 7 (2018), 153-182.
doi: 10.3934/eect.2018008. |
show all references
References:
[1] |
F. Alabau-Boussouira, Z. Wang and L. Yu,
A one-step optimal energy decay formula for indirectly nonlinearly damped hyperbolic systems coupled by velocities, ESAIM Control Optim. Calc. Var., 23 (2017), 721-749.
doi: 10.1051/cocv/2016011. |
[2] |
M. S. Alves, C. Buriol, M. V. Ferreira, J. E. Muñoz Rivera, M. Sepúlveda and O. Vera,
Asymptotic behaviour for the vibrations modeled by the standard linear solid model with a thermal effect, J. Math. Anal. Appl., 399 (2013), 472-479.
doi: 10.1016/j.jmaa.2012.10.019. |
[3] |
K. Ammari and S. Nicaise,
Stabilization of a transmission wave/plate equation, J. Differ. Equ., 249 (2010), 707-727.
doi: 10.1016/j.jde.2010.03.007. |
[4] |
G. Avalos and I. Lasiecka,
Exponential stability of a thermoelastic system without mechanical dissipation, Rend. Istit. Mat. Univ. Trieste, 28 (1997), 1-28.
|
[5] |
G. Avalos and I. Lasiecka,
The strong stability of a semigroup arising from a coupled hyperbolic/parabolic system, Semigroup Forum, 57 (1998), 278-292.
doi: 10.1007/PL00005977. |
[6] |
M. Conti, V. Pata and R. Quintanilla,
Thermoelasticity of Moore-Gibson-Thompson type with history dependence in the temperature, Asymptot. Anal., 120 (2020), 1-21.
doi: 10.3233/asy-191576. |
[7] |
F. Dell'Oro and V. Pata,
On the stability of Timoshenko systems with Gurtin-Pipkin thermal law, J. Differ. Equ., 257 (2014), 523-548.
doi: 10.1016/j.jde.2014.04.009. |
[8] |
R. Denk and F. Kammerlander,
Exponential stability for a coupled system of damped-undamped plate equations, IMA J. Appl. Math., 83 (2018), 302-322.
doi: 10.1093/imamat/hxy002. |
[9] |
H. D. Fernández Sare and R. Racke,
On the stability of damped Timoshenko systems: Cattaneo versus Fourier law, Arch. Rational Mech. Anal., 194 (2009), 221-251.
doi: 10.1007/s00205-009-0220-2. |
[10] |
G. R. Goldstein, J. A. Goldstein and G. Perla Menzala,
On the overdamping phenomenon: a general result and applications, Quart. Appl. Math., 71 (2013), 183-199.
doi: 10.1090/S0033-569X-2012-01282-3. |
[11] |
J. Hao and Z. Liu,
Stability of an abstract system of coupled hyperbolic and parabolic equations, Z. Angew. Math. Phys., 64 (2013), 1145-1159.
doi: 10.1007/s00033-012-0274-0. |
[12] |
M. W. Hirsch and S. Smale, Differential Equations, Dynamical Systems and Linear Algebra, Academic Press, New York, 1974.
![]() ![]() |
[13] |
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. |
[14] |
Z. Liu and S. Zheng, Semigroups associated with dissipative systems, Chapman & Hall/CRC, Boca Raton, 1999.
doi: 0-8493-0615-9. |
[15] |
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. |
[16] |
A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[17] |
L. Perko, Differential equations and dynamical systems, Springer-Verlag, New York, 1991.
doi: 10.1007/978-1-4684-0392-3. |
[18] |
M. L. Santos, D. S. Almeida Júnior and J. E. Muñoz Rivera,
The stability number of the Timoshenko system with second sound, J. Differ. Equ., 253 (2012), 2715-2733.
doi: 10.1016/j.jde.2012.07.012. |
[19] |
R. Triggiani,
Heat-viscoelastic plate interaction: analyticity, spectral analysis, exponential decay, Evol. Equ. Control Theory, 7 (2018), 153-182.
doi: 10.3934/eect.2018008. |









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