The lack of polynomial stability to mixtures with memory
DOI:
https://doi.org/10.5269/bspm.47878Abstract
We consider the system modeling a mixture of n materials with memory. We show that the corresponding semigroup is exponentially stable if and only if the imaginary axis is contained in the resolvent set of the innitesimal generator. In particular this implies the lack of polynomial stability to the corresponding semigroup.
References
1. M. S. Alves, J. E. MuËœnoz Rivera, M. Sepulveda , and O. V. Villagran, Analyticity of Semigroups Associated with Thermoviscoelastic Mixtures of Solids. Journal of Thermal Stresses 32(10) (2009), 986-1004. https://doi.org/10.1080/01495730903103028
2. M. S. Alves, J. E. MuËœnoz Rivera, and R. Quintanilla, Exponential decay in a thermoelastic mixture of solids. International Journal Solids Structures 46(7,8) (2009), 1659-1666. https://doi.org/10.1016/j.ijsolstr.2008.12.005
3. M. S. Alves, J. E. MuËœnoz Rivera, M. Sepulveda , and O. V. Villagran, Exponential stability in thermoviscoelastic mixtures of solids. International Journal Solids Structures 46(24) (2009), 4151-4162. https://doi.org/10.1016/j.ijsolstr.2009.07.026
4. M. S. Alves, J. E. MuËœnoz Rivera, M. Sepulveda , and O. V. Villagran, Stabilization in three-dimensional porous thermoviscoelastic mixtures. The Quarterly Journal of Mathematics 64(1)(2013), 37-57. https://doi.org/10.1093/qmath/har029
5. R. J. Atkin and R. E. Craine, Continuum theories of mixtures: basic theory and historical development. The Quarterly Journal Mech. Appl. Math. 29 (1976), 209-243. https://doi.org/10.1093/qjmam/29.2.209
6. A. Bedford and D. S. Drumheller, Theories of immiscible and structured mixtures. International Journal of Engineering Science 21 (1983), 863-960. https://doi.org/10.1016/0020-7225(83)90071-X
7. D. S. Bernstein, Matrix Mathematics: theory, facts and formulas. Princenton, New Jersey, United States, 2009. https://doi.org/10.1515/9781400833344
8. R. M. Bowen, Continuum Physics III: Theory of mixtures. A. C. Erigen, ed., Academic Press, New York, 1976, 689-722. https://doi.org/10.1016/B978-0-12-240803-8.50017-7
9. R. M. Bowen and J. C. Wiese, Diffusion in mixtures of elastic materials. International Journal of Engineering Science 7, 1969, 689-722. https://doi.org/10.1016/0020-7225(69)90048-2
10. F. F. Cordova Puma and J. E. MuËœnoz Rivera, The lack of Polynomial stability to mixtures with frictional dissipation. Journal of Mathematical Analysis and Applications (2016).
11. C. M. Dafermos, On abstract Volterra equation with applications to linear viscoelasticity. Differential and Integral Equations Vol. 7(1), (1970), 554-569. https://doi.org/10.1016/0022-0396(70)90101-4
12. M. Fabrizio and A. Morro, Mathematical problems in linear viscoelasticity. SIAM - Studies in Applied Mathematics Vol. 12, Philadelphia, (1992). https://doi.org/10.1137/1.9781611970807
13. H. D. Fernandez Sare, J. E. MuËœnoz Rivera and R. Quintanilla, On the rate of decay in interacting continua with memory. Journal of Differential Equations 251(12) (2011), 3583-3605. https://doi.org/10.1016/j.jde.2011.08.015
14. A. E. Green and P. M. Naghdi, A dynamical theory of interacting continua. International Journal of Engineering Science 3 (1965), 231-241. https://doi.org/10.1016/0020-7225(65)90046-7
15. A. E. Green and P. M. Naghdi, A note on mixtures. International Journal of Engineering Science 6 (1968), 631-635. https://doi.org/10.1016/0020-7225(68)90064-5
16. R. A. Horn and C. R. Johnson, Matrix Analysis. Cambridge University Press, United States of America, (1999).
17. F. L. Huang, Strong asymptotic stability theory for linear dynamical systems in Banach spaces. Journal of Differential Equations, Vol. 104 (1993), 307-324. https://doi.org/10.1006/jdeq.1993.1074
18. D. Iesan and R. Quintanilla, Existence and continuous dependence results in the theory of interacting continua. J. Elasticity 36(1) (1994), 85-98. https://doi.org/10.1007/BF00042493
19. Z. Liu and S. Zheng, Semigroups associated with dissipative systems. Research Notes in Mathematics, vol. 398, Chapman and Hall/CRC, Boca Raton, FL, (1999).
20. F. Martinez and R. Quintanilla, Some qualitative results for the linear theory of binary mixtures of thermoelastic solids. Collect. Math. 46(3) (1995), 263-277.
21. J. E. MuËœnoz Rivera, M. G. Naso and R. Quintanilla, Decay of solutions for a mixture of thermoelastic one dimensional solids. Computers and Mathematics with Applications 66(1) (2013), 41-55. https://doi.org/10.1016/j.camwa.2013.03.022
22. J. E. MuËœnoz Rivera, M. G. Naso and R. Quintanilla, Decay of solutions for a mixture of thermoelastic solids with different temperatures. Computers and Mathematics with Applications 71 (2016), 991-1009. https://doi.org/10.1016/j.camwa.2016.01.010
23. J. Pruss, On the spectrum of C0− semigroups. Trans. Amer. Math. Soc. 284 (1984), 847-857. https://doi.org/10.1090/S0002-9947-1984-0743749-9
24. R. Quintanilla, Existence and exponential decay in the linear theory of viscoelastic mixtures. Eur. J. Mech. A Solids 24(2) (2005), 311-324. https://doi.org/10.1016/j.euromechsol.2004.11.008
25. R. Quintanilla, Exponential decay in mixtures with localized dissipative term. Appl. math. Lett. 18(12) (2005), 1381-1388. https://doi.org/10.1016/j.aml.2005.02.023
26. K. R. Rajagopal and L. Tao, Mechanics of mixtures. World Scientific, Singapore, 1995. https://doi.org/10.1142/2197
27. C. Truesdell and R. A. Toupin, The classical field theories, Vol. III/1. Springer-Verlag, Berlin, 1960. https://doi.org/10.1007/978-3-642-45943-6_2
2. M. S. Alves, J. E. MuËœnoz Rivera, and R. Quintanilla, Exponential decay in a thermoelastic mixture of solids. International Journal Solids Structures 46(7,8) (2009), 1659-1666. https://doi.org/10.1016/j.ijsolstr.2008.12.005
3. M. S. Alves, J. E. MuËœnoz Rivera, M. Sepulveda , and O. V. Villagran, Exponential stability in thermoviscoelastic mixtures of solids. International Journal Solids Structures 46(24) (2009), 4151-4162. https://doi.org/10.1016/j.ijsolstr.2009.07.026
4. M. S. Alves, J. E. MuËœnoz Rivera, M. Sepulveda , and O. V. Villagran, Stabilization in three-dimensional porous thermoviscoelastic mixtures. The Quarterly Journal of Mathematics 64(1)(2013), 37-57. https://doi.org/10.1093/qmath/har029
5. R. J. Atkin and R. E. Craine, Continuum theories of mixtures: basic theory and historical development. The Quarterly Journal Mech. Appl. Math. 29 (1976), 209-243. https://doi.org/10.1093/qjmam/29.2.209
6. A. Bedford and D. S. Drumheller, Theories of immiscible and structured mixtures. International Journal of Engineering Science 21 (1983), 863-960. https://doi.org/10.1016/0020-7225(83)90071-X
7. D. S. Bernstein, Matrix Mathematics: theory, facts and formulas. Princenton, New Jersey, United States, 2009. https://doi.org/10.1515/9781400833344
8. R. M. Bowen, Continuum Physics III: Theory of mixtures. A. C. Erigen, ed., Academic Press, New York, 1976, 689-722. https://doi.org/10.1016/B978-0-12-240803-8.50017-7
9. R. M. Bowen and J. C. Wiese, Diffusion in mixtures of elastic materials. International Journal of Engineering Science 7, 1969, 689-722. https://doi.org/10.1016/0020-7225(69)90048-2
10. F. F. Cordova Puma and J. E. MuËœnoz Rivera, The lack of Polynomial stability to mixtures with frictional dissipation. Journal of Mathematical Analysis and Applications (2016).
11. C. M. Dafermos, On abstract Volterra equation with applications to linear viscoelasticity. Differential and Integral Equations Vol. 7(1), (1970), 554-569. https://doi.org/10.1016/0022-0396(70)90101-4
12. M. Fabrizio and A. Morro, Mathematical problems in linear viscoelasticity. SIAM - Studies in Applied Mathematics Vol. 12, Philadelphia, (1992). https://doi.org/10.1137/1.9781611970807
13. H. D. Fernandez Sare, J. E. MuËœnoz Rivera and R. Quintanilla, On the rate of decay in interacting continua with memory. Journal of Differential Equations 251(12) (2011), 3583-3605. https://doi.org/10.1016/j.jde.2011.08.015
14. A. E. Green and P. M. Naghdi, A dynamical theory of interacting continua. International Journal of Engineering Science 3 (1965), 231-241. https://doi.org/10.1016/0020-7225(65)90046-7
15. A. E. Green and P. M. Naghdi, A note on mixtures. International Journal of Engineering Science 6 (1968), 631-635. https://doi.org/10.1016/0020-7225(68)90064-5
16. R. A. Horn and C. R. Johnson, Matrix Analysis. Cambridge University Press, United States of America, (1999).
17. F. L. Huang, Strong asymptotic stability theory for linear dynamical systems in Banach spaces. Journal of Differential Equations, Vol. 104 (1993), 307-324. https://doi.org/10.1006/jdeq.1993.1074
18. D. Iesan and R. Quintanilla, Existence and continuous dependence results in the theory of interacting continua. J. Elasticity 36(1) (1994), 85-98. https://doi.org/10.1007/BF00042493
19. Z. Liu and S. Zheng, Semigroups associated with dissipative systems. Research Notes in Mathematics, vol. 398, Chapman and Hall/CRC, Boca Raton, FL, (1999).
20. F. Martinez and R. Quintanilla, Some qualitative results for the linear theory of binary mixtures of thermoelastic solids. Collect. Math. 46(3) (1995), 263-277.
21. J. E. MuËœnoz Rivera, M. G. Naso and R. Quintanilla, Decay of solutions for a mixture of thermoelastic one dimensional solids. Computers and Mathematics with Applications 66(1) (2013), 41-55. https://doi.org/10.1016/j.camwa.2013.03.022
22. J. E. MuËœnoz Rivera, M. G. Naso and R. Quintanilla, Decay of solutions for a mixture of thermoelastic solids with different temperatures. Computers and Mathematics with Applications 71 (2016), 991-1009. https://doi.org/10.1016/j.camwa.2016.01.010
23. J. Pruss, On the spectrum of C0− semigroups. Trans. Amer. Math. Soc. 284 (1984), 847-857. https://doi.org/10.1090/S0002-9947-1984-0743749-9
24. R. Quintanilla, Existence and exponential decay in the linear theory of viscoelastic mixtures. Eur. J. Mech. A Solids 24(2) (2005), 311-324. https://doi.org/10.1016/j.euromechsol.2004.11.008
25. R. Quintanilla, Exponential decay in mixtures with localized dissipative term. Appl. math. Lett. 18(12) (2005), 1381-1388. https://doi.org/10.1016/j.aml.2005.02.023
26. K. R. Rajagopal and L. Tao, Mechanics of mixtures. World Scientific, Singapore, 1995. https://doi.org/10.1142/2197
27. C. Truesdell and R. A. Toupin, The classical field theories, Vol. III/1. Springer-Verlag, Berlin, 1960. https://doi.org/10.1007/978-3-642-45943-6_2
Downloads
Published
2022-01-26
Issue
Section
Research Articles
License
When the manuscript is accepted for publication, the authors agree automatically to transfer the copyright to the (SPM).
The journal utilize the Creative Common Attribution (CC-BY 4.0).
