Pamplona, Paulo Xavier; Muñoz Rivera, Jaime E.; Quintanilla, Ramón Stabilization in elastic solids with voids. (English) Zbl 1153.74016 J. Math. Anal. Appl. 350, No. 1, 37-49 (2009). Summary: We study the asymptotic behavior and analyticity of the solutions of one-dimensional poro-elasticity problem with thermal effect. Our main result is to prove the lack of exponential stability in the case of poro-elasticity with thermal effect when viscoelasticity is present. We prove the analyticity of the problem when a porous viscosity is present. We conclude by showing the impossibility of localization in time of the solutions in the isothermal case. Cited in 62 Documents MSC: 74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) 74F05 Thermal effects in solid mechanics 74D99 Materials of strain-rate type and history type, other materials with memory (including elastic materials with viscous damping, various viscoelastic materials) 74H30 Regularity of solutions of dynamical problems in solid mechanics 74H40 Long-time behavior of solutions for dynamical problems in solid mechanics 35Q72 Other PDE from mechanics (MSC2000) Keywords:polynomial stability; exponential stability; analiticity; poro-elasticity PDFBibTeX XMLCite \textit{P. X. Pamplona} et al., J. Math. Anal. Appl. 350, No. 1, 37--49 (2009; Zbl 1153.74016) Full Text: DOI References: [1] Batkai, A.; Engel, K. J.; Prüss, J.; Schnaubelt, R., Polynomial stability of operators semigroup, Math. Nachr., 279, 1425-1440 (2006) · Zbl 1118.47034 [2] Brun, L., Méthodes énergétiques dans les systèmes évolutifs linéaires. Premier Partie: Séparation des énergies. Deuxième Partie: Thèorémes d’unicité, J. Mécanique, 8, 125-166 (1969) · Zbl 0179.54101 [3] Casas, P. S.; Quintanilla, R., Exponential stability in thermoelasticity with microtemperatures, Internat. J. Engrg. Sci., 43, 33-47 (2005) · Zbl 1211.74060 [4] Casas, P. S.; Quintanilla, R., Exponential decay in one-dimensional porous-themoelasticity, Mech. Res. Comm., 32, 652-658 (2005) · Zbl 1192.74156 [5] Cowin, S. C.; Nunziato, J. W., Linear elastic materials with voids, J. Elasticity, 13, 125-147 (1983) · Zbl 0523.73008 [6] Cowin, S. C., The viscoelastic behavior of linear elastic materials with voids, J. Elasticity, 15, 185-191 (1985) · Zbl 0564.73044 [7] Flavin, J. N.; Rionero, S., Qualitative Estimates for Partial Differential Equations, An Introduction (1996), CRC Press: CRC Press Boca Raton, FL · Zbl 0862.35001 [8] Ieşan, D., A theory of thermoelastic materials with voids, Acta Mech., 60, 67-89 (1986) · Zbl 0597.73007 [9] Ieşan, D., On a theory of micromorphic elastic solids with microtemperatures, J. Thermal Stresses, 24, 737-752 (2001) [10] Ieşan, D., Thermoelastic Models of Continua (2004), Springer · Zbl 1108.74004 [11] Ieşan, D.; Quintanilla, R., A theory of porous thermoviscoelastic mixtures, J. Thermal Stresses, 30, 693-714 (2007) [12] Liu, Z.; Rao, B., Characterization of polynomial decay for the solution of linear evolution equation, Z. Angew. Math. Phys., 56, 630-644 (2005) · Zbl 1100.47036 [13] Liu, Z.; Zheng, S., Semigroups Associated to Dissipative Systems, Res. Notes Math., vol. 398 (1999), Chapman & Hall/CRC: Chapman & Hall/CRC Boca Raton, FL [14] Magaña, A.; Quintanilla, R., On the exponential decay of solutions in one-dimensional generalized porous-thermo-elasticity, Asymptot. Anal., 49, 173-187 (2006) · Zbl 1115.35023 [15] Magaña, A.; Quintanilla, R., On the time decay of solutions in one-dimensional theories of porous materials, Internat. J. Solids Structures, 43, 3414-3427 (2006) · Zbl 1121.74361 [16] Magaña, A.; Quintanilla, R., On the time decay of solutions in porous elasticity with quasi-static microvoids, J. Math. Anal. Appl., 331, 617-630 (2007) · Zbl 1114.35024 [17] Muñoz Rivera, J. E.; Quintanilla, R., On the time polynomial decay in elastic solids with voids, J. Math. Anal. Appl., 338, 1296-1309 (2008) · Zbl 1131.74019 [18] Nunziato, J. W.; Cowin, S. C., A nonlinear theory of elastic materials with voids, Arch. Ration. Mech. Anal., 72, 175-201 (1979) · Zbl 0444.73018 [19] Prüss, J., On the spectrum of \(C_0\)-semigroups, Trans. Amer. Math. Soc., 284, 2, 847-857 (1984) · Zbl 0572.47030 [20] Quintanilla, R., Slow decay for one-dimensional porous dissipation elasticity, Appl. Math. Lett., 16, 487-491 (2003) · Zbl 1040.74023 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.