Exponential and polynomial decay for first order linear Volterra evolution equations
Authors:
Edoardo Mainini and Gianluca Mola
Journal:
Quart. Appl. Math. 67 (2009), 93-111
MSC (2000):
Primary 35B41, 37L30, 45J05, 80A22
DOI:
https://doi.org/10.1090/S0033-569X-09-01145-X
Published electronically:
January 7, 2009
MathSciNet review:
2495073
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: We consider, in an abstract setting, an instance of the Coleman-Gurtin model for heat conduction with memory, that is, the Volterra integro-differential equation \[ \partial _t u(t) - \beta \Delta u(t) - \int _{0}^{t}k(s)\Delta u(t-s)ds = 0. \] We establish new results for the exponential and polynomial decay of solutions, by means of conditions on the convolution kernel which are weaker than the classical differential inequalities.
References
- F. Ammar-Khodja, A. Benabdallah, J. E. Muñoz Rivera, and R. Racke, Energy decay for Timoshenko systems of memory type, J. Differential Equations 194 (2003), no. 1, 82–115. MR 2001030, DOI https://doi.org/10.1016/S0022-0396%2803%2900185-2
- Carlo Cattaneo, Sulla conduzione del calore, Atti Sem. Mat. Fis. Univ. Modena 3 (1949), 83–101 (Italian). MR 0032898
- Vladimir V. Chepyzhov, Edoardo Mainini, and Vittorino Pata, Stability of abstract linear semigroups arising from heat conduction with memory, Asymptot. Anal. 50 (2006), no. 3-4, 269–291. MR 2294601
- V. V. Chepyzhov and V. Pata, Some remarks on stability of semigroups arising from linear viscoelasticity, Asymptot. Anal. 46 (2006), no. 3-4, 251–273. MR 2215885
- Bernard D. Coleman and Morton E. Gurtin, Equipresence and constitutive equations for rigid heat conductors, Z. Angew. Math. Phys. 18 (1967), 199–208 (English, with German summary). MR 214334, DOI https://doi.org/10.1007/BF01596912
- Monica Conti, Stefania Gatti, and Vittorino Pata, Uniform decay properties of linear Volterra integro-differential equations, Math. Models Methods Appl. Sci. 18 (2008), no. 1, 21–45. MR 2378082, DOI https://doi.org/10.1142/S0218202508002590
- Constantine M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal. 37 (1970), 297–308. MR 281400, DOI https://doi.org/10.1007/BF00251609
- M. Fabrizio, G. Gentili, and D. W. Reynolds, On rigid linear heat conductors with memory, Internat. J. Engrg. Sci. 36 (1998), no. 7-8, 765–782. MR 1629806, DOI https://doi.org/10.1016/S0020-7225%2897%2900123-7
- Mauro Fabrizio and Sergio Polidoro, Asymptotic decay for some differential systems with fading memory, Appl. Anal. 81 (2002), no. 6, 1245–1264. MR 1956060, DOI https://doi.org/10.1080/0003681021000035588
- C. Giorgi and G. Gentili, Thermodynamic properties and stability for the heat flux equation with linear memory, Quart. Appl. Math. 51 (1993), no. 2, 343–362. MR 1218373, DOI https://doi.org/10.1090/qam/1218373
- Claudio Giorgi, Maria Grazia Naso, and Vittorino Pata, Exponential stability in linear heat conduction with memory: a semigroup approach, Commun. Appl. Anal. 5 (2001), no. 1, 121–133. MR 1844676
- Maurizio Grasselli and Vittorino Pata, Uniform attractors of nonautonomous dynamical systems with memory, Evolution equations, semigroups and functional analysis (Milano, 2000) Progr. Nonlinear Differential Equations Appl., vol. 50, Birkhäuser, Basel, 2002, pp. 155–178. MR 1944162
- Morton E. Gurtin and A. C. Pipkin, A general theory of heat conduction with finite wave speeds, Arch. Rational Mech. Anal. 31 (1968), no. 2, 113–126. MR 1553521, DOI https://doi.org/10.1007/BF00281373
- R. K. Miller, An integro-differential equation for rigid heat conductors with memory, J. Math. Anal. Appl. 66 (1978), no. 2, 313–332. MR 515894, DOI https://doi.org/10.1016/0022-247X%2878%2990234-2
- Jaime E. Muñoz Rivera and Eugenio Cabanillas Lapa, Decay rates of solutions of an anisotropic inhomogeneous $n$-dimensional viscoelastic equation with polynomially decaying kernels, Comm. Math. Phys. 177 (1996), no. 3, 583–602. MR 1385077
- Jaime E. Muñoz Rivera, Maria Grazia Naso, and Elena Vuk, Asymptotic behaviour of the energy for electromagnetic systems with memory, Math. Methods Appl. Sci. 27 (2004), no. 7, 819–841. MR 2055321, DOI https://doi.org/10.1002/mma.473
- Jaime E. Muñoz Rivera and Reinhard Racke, Magneto-thermo-elasticity—large-time behavior for linear systems, Adv. Differential Equations 6 (2001), no. 3, 359–384. MR 1799490
- Vittorino Pata, Exponential stability in linear viscoelasticity, Quart. Appl. Math. 64 (2006), no. 3, 499–513. MR 2259051, DOI https://doi.org/10.1090/S0033-569X-06-01010-4
- Vittorino Pata and Adele Zucchi, Attractors for a damped hyperbolic equation with linear memory, Adv. Math. Sci. Appl. 11 (2001), no. 2, 505–529. MR 1907454
- A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. MR 710486
References
- F. Ammar-Khodja, A. Benabdallah, J. E. Muñoz Rivera, R. Racke, Energy decay for Timoshenko systems of memory type, J. Differential Equations 194, 82-115 (2003). MR 2001030 (2004f:74032)
- C. Cattaneo, Sulla conduzione del calore, Atti Sem. Mat. Fis. Univ. Modena 3, 83–101 (1948). MR 0032898 (11:362d)
- V. V. Chepyzhov, E. Mainini, V. Pata, Stability of abstract linear semigroups arising from heat conduction with memory, Asymptot. Anal. 50, 269–291 (2006). MR 2294601 (2007k:35284)
- V. V. Chepyzhov, V. Pata, Some remarks on stability of semigroups arising from linear viscoelasticity, Asymptot. Anal. 46, 251–273 (2006). MR 2215885 (2007c:47053)
- B. D. Coleman, M. E. Gurtin, Equipresence and constitutive equations for rigid heat conductors, Z. Angew. Math. Phys. 18, 199–208 (1967). MR 0214334 (35:5185)
- M. Conti, S. Gatti, V. Pata, Uniform decay properties of linear Volterra integro-differential equations, Math. Models Methods Appl. Sci. 18, 21–45 (2008). MR 2378082
- C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal. 37, 297–308 (1970). MR 0281400 (43:7117)
- M. Fabrizio, G. Gentili, D.W. Reynolds, On rigid linear heat condution with memory, Int. J. Eng. Sci. 36, 765–782 (1998). MR 1629806 (99i:80006)
- M. Fabrizio, S. Polidoro, Asymptotic decay for some differential systems with fading memory, Appl. Anal. 81, 1245–1264 (2002). MR 1956060 (2004a:45015)
- C. Giorgi, G. Gentili, Thermodynamic properties and stability for the heat flux equation with linear memory, Quart. Appl. Math. 51, 343–362 (1993). MR 1218373 (94j:80004)
- C. Giorgi, M.G. Naso, V. Pata, Exponential stability in linear heat conduction with memory: A semigroup approach, Comm. Appl. Anal. 5, 2001 (121–134). MR 1844676 (2002e:35233)
- M. Grasselli, V. Pata, Uniform attractors of nonautonomous systems with memory, in “Evolution Equations, Semigroups and Functional Analysis” (A. Lorenzi and B. Ruf, Eds.), pp. 155–178, Progr. Nonlinear Differential Equations Appl. no. 50, Birkhäuser, Boston (2002). MR 1944162 (2003j:37135)
- M. E. Gurtin, A. C. Pipkin, A general theory of heat conduction with finite wave speeds, Arch. Rational Mech. Anal. 31, 113-126 (1968). MR 1553521
- R. K. Miller, An integro-differential equation for rigid heat conductors with memory, J. Math. Anal. Appl. 66, 313–332 (1978). MR 515894 (80g:45015)
- J. E. Muñoz Rivera, E. Cabanillas Lapa, Decay rates of solutions of an anisotropic inhomogeneous n-dimensional viscoelastic equation with polynomially decaying kernels, Commun. Math. Phys. 177, 583–602 (1996). MR 1385077 (97e:73034)
- J. E. Muñoz Rivera, M.G. Naso, E.Vuk, Asymptotic behaviour of the energy for electromagnetic systems with memory, Math. Methods Appl. Sci. 27, 819–841 (2004). MR 2055321 (2005a:35268)
- J. E. Muñoz Rivera, R. Racke, Magneto-thermo-elasticity — large-time behavior for linear systems, Adv. Differential Equations 6, 359–384 (2001). MR 1799490 (2001j:74037)
- V. Pata, Exponential stability in linear viscoelasticity, Quart. Appl. Math. 64, 499–513 (2006). MR 2259051 (2007h:35211)
- V. Pata, A. Zucchi, Attractors for a damped hyperbolic equation with linear memory, Adv. Math. Sci. Appl. 11, 505–529 (2001). MR 1907454 (2003f:35027)
- A. Pazy, Semigroup of linear operators and application to partial differential equations, Springer-Verlag, New York, 1983. MR 710486 (85g:47061)
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC (2000):
35B41,
37L30,
45J05,
80A22
Retrieve articles in all journals
with MSC (2000):
35B41,
37L30,
45J05,
80A22
Additional Information
Edoardo Mainini
Affiliation:
Classe di Scienze Scuola Normale Superiore Piazza dei Cavalieri 7, I-56126 Pisa, Italy
Email:
edoardo.mainini@sns.it
Gianluca Mola
Affiliation:
Dipartimento di Matematica “F.Brioschi” Politecnico di Milano Via Bonardi 9, I-20133 Milano, Italy & Department of Applied Physics, Graduate School of Engineering, Osaka University, Suita, Osaka 565-0871, Japan
Email:
gianluca.mola@polimi.it
Received by editor(s):
July 9, 2007
Published electronically:
January 7, 2009
Additional Notes:
The second author was supported by the Postdoctoral Fellowship of the Japan Society for the Promotion of Sciences (No. PE06067).
Article copyright:
© Copyright 2009
Brown University
The copyright for this article reverts to public domain 28 years after publication.