Quarterly of Applied Mathematics

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
		Online ISSN 1552-4485; Print ISSN 0033-569X

Exponential and polynomial decay for first order linear Volterra evolution equations

Author(s): Edoardo Mainini; Gianluca Mola
Journal: Quart. Appl. Math. 67 (2009), 93-111.
MSC (2000): Primary 35B41, 37L30, 45J05, 80A22
Posted: January 7, 2009
MathSciNet review: 2495073
Retrieve article in: PDF

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

$\displaystyle \partial_t u(t) - \beta \Delta u(t) - \int_{0}^{t}k(s)\Delta u(t-s)ds = 0.$

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:

1.: 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)
2.: C. Cattaneo, Sulla conduzione del calore, Atti Sem. Mat. Fis. Univ. Modena 3, 83-101 (1948). MR 0032898 (11:362d)
3.: 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)
4.: 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)
5.: 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)
6.: 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
7.: C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal. 37, 297-308 (1970). MR 0281400 (43:7117)
8.: 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)
9.: M. Fabrizio, S. Polidoro, Asymptotic decay for some differential systems with fading memory, Appl. Anal. 81, 1245-1264 (2002). MR 1956060 (2004a:45015)
10.: 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)
11.: 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)
12.: 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)
13.: 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
14.: 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)
15.: 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)
16.: 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)
17.: 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)
18.: V. Pata, Exponential stability in linear viscoelasticity, Quart. Appl. Math. 64, 499-513 (2006). MR 2259051 (2007h:35211)
19.: 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)
20.: A. Pazy, Semigroup of linear operators and application to partial differential equations, Springer-Verlag, New York, 1983. MR 710486 (85g:47061)

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|