On the decay rates for a one-dimensional porous elasticity system with past history

Baowei Feng

doi:10.3934/cpaa.2019130

Article Contents

2019, Volume 18, Issue 6: 2905-2921. Doi: 10.3934/cpaa.2019130

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On the decay rates for a one-dimensional porous elasticity system with past history

Baowei Feng

Department of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu 611130, China

Received: January 31, 2018

Revised: June 30, 2018

Published: May 2019

The author is supported by the National Natural Science Foundation of China (No. 11701465) and by the Fundamental Research Funds for the Central Universities (No. JBK1902026)

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Abstract

This paper studies a porous elasticity system with past history

$\left\{\begin{array}{l}{\rho u_{t t}-\mu u_{x x}-b \phi_{x} = 0} \\ {J \phi_{t t}-\delta \phi_{x x}+b u_{x}+\xi \phi+\int_{0}^{\infty} g(s) \phi_{x x}(t-s) d s = 0}\end{array}\right.$

By introducing a new variable, we establish an explicit and a general decay of energy for the case of equal-speed wave propagation as well as for the nonequalspeed case. To establish our results, we mainly adopt the method developed by Guesmia, Messaoudi and Soufyane [Electron. J. Differ. Equa. 2012(2012), 1-45] and some properties of convex functions developed by Alabau-Boussouira and Cannarsa [C. R. Acad. Sci. Paris Ser. I, 347(2009), 867-872], Lasiecka and Tataru [Differ. Inte. Equa., 6(1993), 507-533]. In addition we remove the assumption that b is positive constant in [J. Math. Anal. Appl., 469(2019), 457-471] and hence improve the result.

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Mathematics Subject Classification: Primary: 35B35, 35B40; Secondary: 74Dxx, 93D20.

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[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.