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Variable lorentz estimate for stationary stokes system with partially BMO coefficients
On the decay rates for a one-dimensional porous elasticity system with past history
Department of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu 611130, China |
$\left\{\begin{array}{l}{\rho u_{t t}-\mu u_{x x}-b \phi_{x} = 0} \\ {J \phi_{t t}-\delta \phi_{x x}+b u_{x}+\xi \phi+\int_{0}^{\infty} g(s) \phi_{x x}(t-s) d s = 0}\end{array}\right.$ |
References:
[1] |
F. Alabau-Boussouira and P. Cannarsa,
A general method for proving sharp energy decay rates for memory-dissipative evolution equations, C. R. Acad. Sci. Paris Ser. I, 347 (2009), 867-872.
doi: 10.1016/j.crma.2009.05.011. |
[2] |
T. A. Apalara,
General decay of solutions in one-dimensional porous-elastic system with memory, J. Math. Anal. Appl., 469 (2019), 457-471.
doi: 10.1016/j.jmaa.2017.08.007. |
[3] |
T. A. Apalara,
Exponential decay in one-dimensional porous dissipation elasticity, Quart. J. Mech. Appl. Math., 70 (2017), 363-372.
doi: 10.1093/qjmam/hbx012. |
[4] |
T. A. Apalara, A general decay for a weakly nonlinearly damped porous system, J. Dyn. Contr. Sys., (2018).
doi: 10.1007/s10883-018-9407-x. |
[5] |
P. S. Casas and R. Quintanilla,
Exponential decay in one-dimensional porous-thermo-elasticity, Mech. Res. Comm., 32 (2005), 652-658.
doi: 10.1016/j.mechrescom.2005.02.015. |
[6] |
P. S. Casas and R. Quintanilla,
Exponential stability in thermoelasticity with microtemperatures, Int. J. Engrg. Sci., 43 (2005), 33-47.
doi: 10.1016/j.ijengsci.2004.09.004. |
[7] |
S. C. Cowin,
The viscoelsstic behavior of linear elastic materials with voids, J. Elasticity, 15 (1985), 185-191.
|
[8] |
S. C. Cowin and J. W. Nunsiato,
Linear elastic materials with voids, J. Elasticity, 13 (1983), 125-147.
|
[9] |
C. M. Dafermos,
Asymptotic stability in viscoelasticity, Arch. Ration. Mech. Anal., 37 (1970), 297-308.
doi: 10.1007/BF00251609. |
[10] |
B. Feng,
Uniform decay of energy for a porous thermoelasticity system with past history, Appl. Anal., 97 (2018), 210-229.
doi: 10.1080/00036811.2016.1258116. |
[11] |
B. Feng and M. Yin, Decay of solutions for a one-dimensional porous elasticity system with memory: the case of non-equal wave speeds, Math. Mech. Solids, (2018).
doi: 10.1177/1081286518757299. |
[12] |
M. M. Freitas, M. L. Santos and J. A. Langa,
Porous elastic system with nonlinear damping and sources terms, J. Differ. Equ., 264 (2018), 2970-3051.
doi: 10.1016/j.jde.2017.11.006. |
[13] |
A. Guesmia,
Asymototic stability of abstract dissipative systems with infinite memory, J. Math. Anal. Appl., 382 (2011), 748-760.
doi: 10.1016/j.jmaa.2011.04.079. |
[14] |
A. Guesmia, S. A. Messaoudi and A. Soufyane,
Stabilization of a linear Timoshenko system with infinite history and applications to the Timoshenko-heat systems, Electron. J. Differ. Equ., 2012 (2012), 1-45.
|
[15] |
D. Ieşan,
A theories of thermoelastic materials with voids, Acta Mechanica, 60 (1986), 67-89.
|
[16] |
D. Ieşan, Thermoelastic Models of Continua, Kluwer Academic Publishers, Dordrecht, 2004. |
[17] |
I. Lasiecka and D. Tataru,
Uniform boundary stabilization of semilinear wave equation with nonlinear boundary damping, Differ. Integral Equ., 6 (1993), 507-533.
|
[18] |
Z. Liu and S. Zheng, Semigroups Associated with Dissipative Systems, Chapman and Hall/CRC, Boca Raton, 1999. |
[19] |
A. Magaña and R. Quintanilla,
On the time decay of solutions in one-dimensional theories of porous materials, Int. Solids Structures, 43 (2006), 3414-3427.
doi: 10.1016/j.ijsolstr.2005.06.077. |
[20] |
S. A. Messaoudi and A. Fareh,
General decay for a porous thermoelastic system with memory: the case of equal speeds, Nonlinear Anal., 74 (2011), 6895-6906.
doi: 10.1016/j.na.2011.07.012. |
[21] |
S. A. Messaoudi and A. Fareh,
General decay for a porous thermoelastic system with memory: the case of nonequal speeds, Acta Math. Sci., 33B (2013), 1-19.
doi: 10.1016/S0252-9602(12)60192-1. |
[22] |
J. E. Muñoz Rivera and H. D. Fernández Sare,
Stability of Timoshenko systems with past history, J. Math. Anal. Appl., 339 (2008), 482-502.
doi: 10.1016/j.jmaa.2007.07.012. |
[23] |
J. E. Muñoz Rivera and R. Quintanilla,
On the time polynomial decay in elastic solids with viods, J. Math. Anal. Appl., 338 (2008), 1296-1309.
doi: 10.1016/j.jmaa.2007.06.005. |
[24] |
W. Nunziato and S. C. Cowin,
A nonlinear theory of elastic materials with voids, Arch. Rational Mech. Anal., 72 (1979), 175-201.
doi: 10.1007/BF00249363. |
[25] |
P. X. Pamplona, J. E. Muñoz Rivera and R. Quintanilla,
Stabilization in elastic solids with voids, J. Math. Anal. Appl., 1 (2009), 37-49.
doi: 10.1016/j.jmaa.2008.09.026. |
[26] |
P. X. Pamplona, J. E. Muñoz Rivera and R. Quintanilla,
On the decay of solutions for porous-elastic system with history, J. Math. Anal. Appl., 379 (2011), 682-705.
doi: 10.1016/j.jmaa.2011.01.045. |
[27] |
V. Pata and A. Zucchi,
Attractors for a damped hyperbolic equation with linear memory, Adv. Math. Sci. Appl., 11 (2001), 505-529.
|
[28] |
R. Quintanilla,
Slow decay for one-dimensional porous dissipation elasticity, Appl. Math. Lett., 16 (2003), 487-491.
doi: 10.1016/S0893-9659(03)00025-9. |
[29] |
M. L Santos and D. S. Almeida Júnior,
On porous-elastic system with localized damping, Z. Angew. Math. Phys., 67 (2016), 1-18.
doi: 10.1007/s00033-016-0622-6. |
[30] |
M. L Santos and D. S. Almeida Júnior,
On the porous-elastic system with Kelvin-Voigt damping, J. Math. Anal. Appl., 445 (2017), 498-512.
doi: 10.1016/j.jmaa.2016.08.005. |
[31] |
M. L. Santos, A. D. S. Campelo and D. S. Almeida Júnior,
On the decay rates of porous elastic systems, J. Elasticity, 127 (2017), 79-101.
doi: 10.1007/s10659-016-9597-y. |
[32] |
M. L. Santos, A. D. S. Campelo and D. S. Almeida Júnior,
Rates of decay for porous elastic system weakly dissipative, Acta Appl. Math., 151 (2017), 1-26.
doi: 10.1007/s10440-017-0100-y. |
[33] |
M. L. Santos, A. D. S. Campelo and M. L. S. Oliveira,
On porous-elastic systems with Fourier law, Appl. Anal., (2018), 1-17.
doi: 10.1080/00036811.2017.1419197. |
[34] |
A. Soufyane,
Energy decay for porous-thermo-elasticity systems of memory type, Appl. Anal., 87 (2008), 451-464.
doi: 10.1080/00036810802035634. |
[35] |
A. Soufyane, M. Afilal and M. Chacha, Boundary stabilization of memory type for the porous-thermoelasticity system, Abstr. Appl. Anal., 2009 (2009), ID 280790.
doi: 10.1155/2009/280790. |
[36] |
A. Soufyane, M. Afilal, T. Aouam and M. Chacha,
General decay of solutions of a linear one-dimensional porous-thermoelasticity system with a boundary control of memory type, Nonlinear Anal., 72 (2010), 3903-3910.
doi: 10.1016/j.na.2010.01.004. |
show all references
References:
[1] |
F. Alabau-Boussouira and P. Cannarsa,
A general method for proving sharp energy decay rates for memory-dissipative evolution equations, C. R. Acad. Sci. Paris Ser. I, 347 (2009), 867-872.
doi: 10.1016/j.crma.2009.05.011. |
[2] |
T. A. Apalara,
General decay of solutions in one-dimensional porous-elastic system with memory, J. Math. Anal. Appl., 469 (2019), 457-471.
doi: 10.1016/j.jmaa.2017.08.007. |
[3] |
T. A. Apalara,
Exponential decay in one-dimensional porous dissipation elasticity, Quart. J. Mech. Appl. Math., 70 (2017), 363-372.
doi: 10.1093/qjmam/hbx012. |
[4] |
T. A. Apalara, A general decay for a weakly nonlinearly damped porous system, J. Dyn. Contr. Sys., (2018).
doi: 10.1007/s10883-018-9407-x. |
[5] |
P. S. Casas and R. Quintanilla,
Exponential decay in one-dimensional porous-thermo-elasticity, Mech. Res. Comm., 32 (2005), 652-658.
doi: 10.1016/j.mechrescom.2005.02.015. |
[6] |
P. S. Casas and R. Quintanilla,
Exponential stability in thermoelasticity with microtemperatures, Int. J. Engrg. Sci., 43 (2005), 33-47.
doi: 10.1016/j.ijengsci.2004.09.004. |
[7] |
S. C. Cowin,
The viscoelsstic behavior of linear elastic materials with voids, J. Elasticity, 15 (1985), 185-191.
|
[8] |
S. C. Cowin and J. W. Nunsiato,
Linear elastic materials with voids, J. Elasticity, 13 (1983), 125-147.
|
[9] |
C. M. Dafermos,
Asymptotic stability in viscoelasticity, Arch. Ration. Mech. Anal., 37 (1970), 297-308.
doi: 10.1007/BF00251609. |
[10] |
B. Feng,
Uniform decay of energy for a porous thermoelasticity system with past history, Appl. Anal., 97 (2018), 210-229.
doi: 10.1080/00036811.2016.1258116. |
[11] |
B. Feng and M. Yin, Decay of solutions for a one-dimensional porous elasticity system with memory: the case of non-equal wave speeds, Math. Mech. Solids, (2018).
doi: 10.1177/1081286518757299. |
[12] |
M. M. Freitas, M. L. Santos and J. A. Langa,
Porous elastic system with nonlinear damping and sources terms, J. Differ. Equ., 264 (2018), 2970-3051.
doi: 10.1016/j.jde.2017.11.006. |
[13] |
A. Guesmia,
Asymototic stability of abstract dissipative systems with infinite memory, J. Math. Anal. Appl., 382 (2011), 748-760.
doi: 10.1016/j.jmaa.2011.04.079. |
[14] |
A. Guesmia, S. A. Messaoudi and A. Soufyane,
Stabilization of a linear Timoshenko system with infinite history and applications to the Timoshenko-heat systems, Electron. J. Differ. Equ., 2012 (2012), 1-45.
|
[15] |
D. Ieşan,
A theories of thermoelastic materials with voids, Acta Mechanica, 60 (1986), 67-89.
|
[16] |
D. Ieşan, Thermoelastic Models of Continua, Kluwer Academic Publishers, Dordrecht, 2004. |
[17] |
I. Lasiecka and D. Tataru,
Uniform boundary stabilization of semilinear wave equation with nonlinear boundary damping, Differ. Integral Equ., 6 (1993), 507-533.
|
[18] |
Z. Liu and S. Zheng, Semigroups Associated with Dissipative Systems, Chapman and Hall/CRC, Boca Raton, 1999. |
[19] |
A. Magaña and R. Quintanilla,
On the time decay of solutions in one-dimensional theories of porous materials, Int. Solids Structures, 43 (2006), 3414-3427.
doi: 10.1016/j.ijsolstr.2005.06.077. |
[20] |
S. A. Messaoudi and A. Fareh,
General decay for a porous thermoelastic system with memory: the case of equal speeds, Nonlinear Anal., 74 (2011), 6895-6906.
doi: 10.1016/j.na.2011.07.012. |
[21] |
S. A. Messaoudi and A. Fareh,
General decay for a porous thermoelastic system with memory: the case of nonequal speeds, Acta Math. Sci., 33B (2013), 1-19.
doi: 10.1016/S0252-9602(12)60192-1. |
[22] |
J. E. Muñoz Rivera and H. D. Fernández Sare,
Stability of Timoshenko systems with past history, J. Math. Anal. Appl., 339 (2008), 482-502.
doi: 10.1016/j.jmaa.2007.07.012. |
[23] |
J. E. Muñoz Rivera and R. Quintanilla,
On the time polynomial decay in elastic solids with viods, J. Math. Anal. Appl., 338 (2008), 1296-1309.
doi: 10.1016/j.jmaa.2007.06.005. |
[24] |
W. Nunziato and S. C. Cowin,
A nonlinear theory of elastic materials with voids, Arch. Rational Mech. Anal., 72 (1979), 175-201.
doi: 10.1007/BF00249363. |
[25] |
P. X. Pamplona, J. E. Muñoz Rivera and R. Quintanilla,
Stabilization in elastic solids with voids, J. Math. Anal. Appl., 1 (2009), 37-49.
doi: 10.1016/j.jmaa.2008.09.026. |
[26] |
P. X. Pamplona, J. E. Muñoz Rivera and R. Quintanilla,
On the decay of solutions for porous-elastic system with history, J. Math. Anal. Appl., 379 (2011), 682-705.
doi: 10.1016/j.jmaa.2011.01.045. |
[27] |
V. Pata and A. Zucchi,
Attractors for a damped hyperbolic equation with linear memory, Adv. Math. Sci. Appl., 11 (2001), 505-529.
|
[28] |
R. Quintanilla,
Slow decay for one-dimensional porous dissipation elasticity, Appl. Math. Lett., 16 (2003), 487-491.
doi: 10.1016/S0893-9659(03)00025-9. |
[29] |
M. L Santos and D. S. Almeida Júnior,
On porous-elastic system with localized damping, Z. Angew. Math. Phys., 67 (2016), 1-18.
doi: 10.1007/s00033-016-0622-6. |
[30] |
M. L Santos and D. S. Almeida Júnior,
On the porous-elastic system with Kelvin-Voigt damping, J. Math. Anal. Appl., 445 (2017), 498-512.
doi: 10.1016/j.jmaa.2016.08.005. |
[31] |
M. L. Santos, A. D. S. Campelo and D. S. Almeida Júnior,
On the decay rates of porous elastic systems, J. Elasticity, 127 (2017), 79-101.
doi: 10.1007/s10659-016-9597-y. |
[32] |
M. L. Santos, A. D. S. Campelo and D. S. Almeida Júnior,
Rates of decay for porous elastic system weakly dissipative, Acta Appl. Math., 151 (2017), 1-26.
doi: 10.1007/s10440-017-0100-y. |
[33] |
M. L. Santos, A. D. S. Campelo and M. L. S. Oliveira,
On porous-elastic systems with Fourier law, Appl. Anal., (2018), 1-17.
doi: 10.1080/00036811.2017.1419197. |
[34] |
A. Soufyane,
Energy decay for porous-thermo-elasticity systems of memory type, Appl. Anal., 87 (2008), 451-464.
doi: 10.1080/00036810802035634. |
[35] |
A. Soufyane, M. Afilal and M. Chacha, Boundary stabilization of memory type for the porous-thermoelasticity system, Abstr. Appl. Anal., 2009 (2009), ID 280790.
doi: 10.1155/2009/280790. |
[36] |
A. Soufyane, M. Afilal, T. Aouam and M. Chacha,
General decay of solutions of a linear one-dimensional porous-thermoelasticity system with a boundary control of memory type, Nonlinear Anal., 72 (2010), 3903-3910.
doi: 10.1016/j.na.2010.01.004. |
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