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November  2019, 18(6): 2905-2921. doi: 10.3934/cpaa.2019130

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  February 2018 Revised  July 2018 Published  May 2019

Fund Project: 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).

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.
Keywords: Porous system, general decay, memory, convexity, stability.
Mathematics Subject Classification: Primary: 35B35, 35B40; Secondary: 74Dxx, 93D20.
Citation: Baowei Feng. On the decay rates for a one-dimensional porous elasticity system with past history. Communications on Pure and Applied Analysis, 2019, 18 (6) : 2905-2921. doi: 10.3934/cpaa.2019130
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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