Semigroup-theoretic approach to identification of linear diffusion coefficients

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June 2016, 9(3): 777-790. doi: 10.3934/dcdss.2016028

Semigroup-theoretic approach to identification of linear diffusion coefficients

Gianluca Mola ^1, , Noboru Okazawa ^2, , Jan Prüss ^3, and Tomomi Yokota ^4,

1.	Dipartimento di Matematica F. Brioschi, Politecnico di Milano, Via Bonardi 9, I-20133 Milano, Italy
2.	Department of Mathematics, Science University of Tokyo, 1-3 Kagurazaka, Sinjuku-ku, Tokyo 162-8601
3.	Institut für Mathematik, Martin-Luther Univ. Halle -Wittenberg, Theodor-Lieser-Strasse 506120 Halle (Saale), Germany
4.	Department of Mathematics, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601

Received March 2015 Revised January 2016 Published April 2016

Abstract
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Let $X$ be a complex Banach space and $A:\,D(A) \to X$ a quasi-$m$-sectorial operator in $X$. This paper is concerned with the identification of diffusion coefficients $\nu > 0$ in the initial-value problem: \[ (d/dt)u(t) + {\nu}Au(t) = 0, \quad t \in (0,T), \quad u(0) = x \in X, \] with additional condition $\|u(T)\| = \rho$, where $\rho >0$ is known. Except for the additional condition, the solution to the initial-value problem is given by $u(t) := e^{-t\,{\nu}A} x \in C([0,T];X) \cap C^{1}((0,T];X)$. Therefore, the identification of $\nu$ is reduced to solving the equation $\|e^{-{\nu}TA}x\| = \rho$. It will be shown that the unique root $\nu = \nu(x,\rho)$ depends on $(x,\rho)$ locally Lipschitz continuously if the datum $(x,\rho)$ fulfills the restriction $\|x\|> \rho$. This extends those results in Mola [6](2011).

Keywords: linear evolution equations in Banach spaces, linear parabolic equations, Identification problems, unknown constants, well-posedness results..

Mathematics Subject Classification: Primary: 35R30, 35K90; Secondary: 35K20, 35K25, 65N3.

Citation: Gianluca Mola, Noboru Okazawa, Jan Prüss, Tomomi Yokota. Semigroup-theoretic approach to identification of linear diffusion coefficients. Discrete and Continuous Dynamical Systems - S, 2016, 9 (3) : 777-790. doi: 10.3934/dcdss.2016028

References:

[1]	H. Brezis, Analyse Fonctionnelle, Théorie et Applications, Masson, Paris, 1983; (English Translation) Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011.
[2]	P. Drábek and J. Milota, Methods of Nonlinear Analysis, Applications to Differential Equations, Advanced Texts, Birkhäuser Verlag, Basel, 2007.
[3]	J. Goldstein, Semigroups of Linear Operators and Applications, Oxford Math. Monograph, Clarendon Press, Oxford University Press, New York, 1985.
[4]	T. Kato, Perturbation Theory for Linear Operators, Grundlehren math. Wissenschften, 132, Springer-Verlag, Berlin and New York, 1966; 2nd ed., 1976.
[5]	I. Miyadera, Nonlinear Semigroups, Translations of Math. Monograph 109, Amer. Math. Soc., Providence, RI, 1992.
[6]	G. Mola, Identification of the diffusion coefficient in linear evolution equations in Hilbert spaces, J. Abstr. Differ. Equ. Appl., 2 (2011), 14-28.
[7]	N. Okazawa, Sectorialness of second order elliptic operators in divergence form, Proc. Amer. Math. Soc., 113 (1991), 701-706. doi: 10.1090/S0002-9939-1991-1072347-4.
[8]	E. M. Ouhabaz, Analysis of Heat Equations on Domains, London Mathematical Society Monographs Series, 31, Princeton Univ. Press, Princeton and Oxford, 2005.
[9]	A. L. Ruoff, Materials Science, Englewood Cliffs, N.J., Prentice-Hall, 1973.
[10]	J. Voigt, The sector of holomorphy for symmetric sub-Markovian semigroups, in Functional Analysis (Trier, 1994), de Gruyter, Berlin, 1996, 449-453.
[11]	E. Zeidler, Nonlinear Functional Analysis and Its Applications. I. Fixed-point Theorems, Springer-Verlag, New York, 1986. doi: 10.1007/978-1-4612-4838-5.
[12]	E. Zeidler, Nonlinear Functional Analysis and Its Applications. II/B. Nonlinear Monotone Operators, Springer-Verlag, New York, 1990. doi: 10.1007/978-1-4612-0985-0.

show all references

References:

[1]	H. Brezis, Analyse Fonctionnelle, Théorie et Applications, Masson, Paris, 1983; (English Translation) Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011.
[2]	P. Drábek and J. Milota, Methods of Nonlinear Analysis, Applications to Differential Equations, Advanced Texts, Birkhäuser Verlag, Basel, 2007.
[3]	J. Goldstein, Semigroups of Linear Operators and Applications, Oxford Math. Monograph, Clarendon Press, Oxford University Press, New York, 1985.
[4]	T. Kato, Perturbation Theory for Linear Operators, Grundlehren math. Wissenschften, 132, Springer-Verlag, Berlin and New York, 1966; 2nd ed., 1976.
[5]	I. Miyadera, Nonlinear Semigroups, Translations of Math. Monograph 109, Amer. Math. Soc., Providence, RI, 1992.
[6]	G. Mola, Identification of the diffusion coefficient in linear evolution equations in Hilbert spaces, J. Abstr. Differ. Equ. Appl., 2 (2011), 14-28.
[7]	N. Okazawa, Sectorialness of second order elliptic operators in divergence form, Proc. Amer. Math. Soc., 113 (1991), 701-706. doi: 10.1090/S0002-9939-1991-1072347-4.
[8]	E. M. Ouhabaz, Analysis of Heat Equations on Domains, London Mathematical Society Monographs Series, 31, Princeton Univ. Press, Princeton and Oxford, 2005.
[9]	A. L. Ruoff, Materials Science, Englewood Cliffs, N.J., Prentice-Hall, 1973.
[10]	J. Voigt, The sector of holomorphy for symmetric sub-Markovian semigroups, in Functional Analysis (Trier, 1994), de Gruyter, Berlin, 1996, 449-453.
[11]	E. Zeidler, Nonlinear Functional Analysis and Its Applications. I. Fixed-point Theorems, Springer-Verlag, New York, 1986. doi: 10.1007/978-1-4612-4838-5.
[12]	E. Zeidler, Nonlinear Functional Analysis and Its Applications. II/B. Nonlinear Monotone Operators, Springer-Verlag, New York, 1990. doi: 10.1007/978-1-4612-0985-0.

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