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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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

S. C. Cowin, The viscoelsstic behavior of linear elastic materials with voids, J. Elasticity, 15 (1985), 185-191.   Google Scholar

[8]

S. C. Cowin and J. W. Nunsiato, Linear elastic materials with voids, J. Elasticity, 13 (1983), 125-147.   Google Scholar

[9]

C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Ration. Mech. Anal., 37 (1970), 297-308.  doi: 10.1007/BF00251609.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[15]

D. Ieşan, A theories of thermoelastic materials with voids, Acta Mechanica, 60 (1986), 67-89.   Google Scholar

[16]

D. Ieşan, Thermoelastic Models of Continua, Kluwer Academic Publishers, Dordrecht, 2004. Google Scholar

[17]

I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equation with nonlinear boundary damping, Differ. Integral Equ., 6 (1993), 507-533.   Google Scholar

[18]

Z. Liu and S. Zheng, Semigroups Associated with Dissipative Systems, Chapman and Hall/CRC, Boca Raton, 1999.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

V. Pata and A. Zucchi, Attractors for a damped hyperbolic equation with linear memory, Adv. Math. Sci. Appl., 11 (2001), 505-529.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

A. Soufyane, Energy decay for porous-thermo-elasticity systems of memory type, Appl. Anal., 87 (2008), 451-464.  doi: 10.1080/00036810802035634.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

S. C. Cowin, The viscoelsstic behavior of linear elastic materials with voids, J. Elasticity, 15 (1985), 185-191.   Google Scholar

[8]

S. C. Cowin and J. W. Nunsiato, Linear elastic materials with voids, J. Elasticity, 13 (1983), 125-147.   Google Scholar

[9]

C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Ration. Mech. Anal., 37 (1970), 297-308.  doi: 10.1007/BF00251609.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[15]

D. Ieşan, A theories of thermoelastic materials with voids, Acta Mechanica, 60 (1986), 67-89.   Google Scholar

[16]

D. Ieşan, Thermoelastic Models of Continua, Kluwer Academic Publishers, Dordrecht, 2004. Google Scholar

[17]

I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equation with nonlinear boundary damping, Differ. Integral Equ., 6 (1993), 507-533.   Google Scholar

[18]

Z. Liu and S. Zheng, Semigroups Associated with Dissipative Systems, Chapman and Hall/CRC, Boca Raton, 1999.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

V. Pata and A. Zucchi, Attractors for a damped hyperbolic equation with linear memory, Adv. Math. Sci. Appl., 11 (2001), 505-529.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

A. Soufyane, Energy decay for porous-thermo-elasticity systems of memory type, Appl. Anal., 87 (2008), 451-464.  doi: 10.1080/00036810802035634.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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