Abstract
In this paper we are concerned with some new identification problems related to isotropic thermal bodies with a loss of memory. We assume that they are governed by the following state equation where A and B are given linear (differential) operators, while u and h 0 ,h 1 stand for the temperature and the memory functions. Moreover, function α —the loss of memory — satisfies 0 ≤ α(t) < t for any t ∈ (0,T).
Dedicated to the memory of Brunello Terreni and to his courageous wife Raimonda and daughters Ester, Maria Pia and Noemi.
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References
J. Baumeister: Boundary control of an integrodifferential equation, J. Math. Anal. Appl. 93 (1983), 550–570.
B.D. Coleman, M.E. Gurtin: Equipresence and constitutive equations for rigid heat conductors, Z. Angew. Math. Phys. 18 (1967), 199–208.
B.D. Coleman, V.J. Mizel: Norms and semigroups in the theory of fading memory, Arch. Rational Mech. Anal. 23 (1966), 87–123
B.D. Coleman, W. Noll: Foundations of linear viscoelasticity, Rev. of Modern Physics 33 (1961), 239–249
G. Fichera: Sui materiali elastici con memoria, (in Italian), Atti Acc. Lincei Rend, fis., 82 (1998), pp. 473–478
M. Gurtin, A. C. Pipkin: A general theory of heat conduction with finite wave speed, Arch. Rational Mech. Anal. 31 (1968), 113–126
H. Grambüller: On linear theory of heat conduction in materials with memory. Existence and uniqueness theorems for the final value problem, Proc. Royal Soc. Edinburgh 76A (1976), 119–137.
M. Grasselli: An identification problem for a linear integro-differential equation occurring in heat flow, Math. Meth. in Appl. Sci. 15 (1992), 167–186.
J. Janno, L. v. Wolfersdorf: On identification of memory kernels in linear theory of heat conduction, Math. Meth. Appl. Sci. 17 (1994), 919–932.
J. Janno, L. v. Wolfersdorf: Identification of weakly singular memory kernels in heat conduction, Z. Angew. Math. Mech. 77 (1997), 243–257.
J. Janno, L. v. Wolfersdorf: Identification of memory kernels in general linear flow, J. Inv. Ill-Posed Problems 6 (1998), 141–164.
J. Janno, L. von Wolfersdorf: An inverse problem for identification of a time-and space-dependent memory kernel of a special kind in heat conduction, Inverse Problems 15 (1999), 1455–1467.
J. Janno, L. von Wolfersdorf: Identification of memory kernels in heat conduction and viscoelasticity, Optimal control of partial differential equations (Chemnitz, 1998), 301–308, Internat. Ser. Numer. Math., 133, Birkhäuser, Basel, 1999.
J. Janno, L. von Wolfersdorf: Inverse problems for memory kernel by Laplace transform method, Z.A.A 19 (2000), 489–510.
A. Lorenzi, E. Sinestrari: Stability results for a partial integro-differential inverse problem, Pitman Research Notes in Math. vol. 190, (1989), 271–294 (Proceedings of the Meeting on “Volterra Integro-differential Equations in Banach Spaces and Applications”, Trento (1987))
A. Lorenzi, E. Sinestrari: An inverse problem in the theory of materials with memory I, Nonlinear Anal. 12 (1988), 1317–1335
A. Lunardi: Analytic semigroups and optimal regularity in parabolic problems, Birkhäuser, 1995
J. Meixner: On the linear theory of heat conduction, Arch. Rational Mech. Anal. 39 (1970), 108–130
J. W. Nunziato: On heat conduction in materials with memory, Quart. Appl. Math. 29 (1971), 187–204.
E. Sinestrari: On the abstract Cauchy problem of parabolic type in spaces of continuous functions, J. Math. Anal. Appl. 107 (1985), 16–66.
H. Triebel: Interpolation Theory, Function Spaces, Differential Operators, North Holland, Amsterdam (1978).
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Lorenzi, A. (2002). Some Identification Problems Related to Thermal Materials with Loss of Memory. In: Lorenzi, A., Ruf, B. (eds) Evolution Equations, Semigroups and Functional Analysis. Progress in Nonlinear Differential Equations and Their Applications, vol 50. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8221-7_13
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DOI: https://doi.org/10.1007/978-3-0348-8221-7_13
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