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

Semigroup-theoretic approach to identification of linear diffusion coefficients

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

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).
Citation: Gianluca Mola, Noboru Okazawa, Jan Prüss, Tomomi Yokota. Semigroup-theoretic approach to identification of linear diffusion coefficients. Discrete & 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, (2011).

[2]

P. Drábek and J. Milota, Methods of Nonlinear Analysis,, Applications to Differential Equations, (2007).

[3]

J. Goldstein, Semigroups of Linear Operators and Applications,, Oxford Math. Monograph, (1985).

[4]

T. Kato, Perturbation Theory for Linear Operators,, Grundlehren math. Wissenschften, 132 (1966).

[5]

I. Miyadera, Nonlinear Semigroups,, Translations of Math. Monograph 109, 109 (1992).

[6]

G. Mola, Identification of the diffusion coefficient in linear evolution equations in Hilbert spaces,, J. Abstr. Differ. Equ. Appl., 2 (2011), 14.

[7]

N. Okazawa, Sectorialness of second order elliptic operators in divergence form,, Proc. Amer. Math. Soc., 113 (1991), 701. doi: 10.1090/S0002-9939-1991-1072347-4.

[8]

E. M. Ouhabaz, Analysis of Heat Equations on Domains,, London Mathematical Society Monographs Series, 31 (2005).

[9]

A. L. Ruoff, Materials Science,, Englewood Cliffs, (1973).

[10]

J. Voigt, The sector of holomorphy for symmetric sub-Markovian semigroups,, in Functional Analysis (Trier, (1994), 449.

[11]

E. Zeidler, Nonlinear Functional Analysis and Its Applications. I. Fixed-point Theorems,, Springer-Verlag, (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, (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, (2011).

[2]

P. Drábek and J. Milota, Methods of Nonlinear Analysis,, Applications to Differential Equations, (2007).

[3]

J. Goldstein, Semigroups of Linear Operators and Applications,, Oxford Math. Monograph, (1985).

[4]

T. Kato, Perturbation Theory for Linear Operators,, Grundlehren math. Wissenschften, 132 (1966).

[5]

I. Miyadera, Nonlinear Semigroups,, Translations of Math. Monograph 109, 109 (1992).

[6]

G. Mola, Identification of the diffusion coefficient in linear evolution equations in Hilbert spaces,, J. Abstr. Differ. Equ. Appl., 2 (2011), 14.

[7]

N. Okazawa, Sectorialness of second order elliptic operators in divergence form,, Proc. Amer. Math. Soc., 113 (1991), 701. doi: 10.1090/S0002-9939-1991-1072347-4.

[8]

E. M. Ouhabaz, Analysis of Heat Equations on Domains,, London Mathematical Society Monographs Series, 31 (2005).

[9]

A. L. Ruoff, Materials Science,, Englewood Cliffs, (1973).

[10]

J. Voigt, The sector of holomorphy for symmetric sub-Markovian semigroups,, in Functional Analysis (Trier, (1994), 449.

[11]

E. Zeidler, Nonlinear Functional Analysis and Its Applications. I. Fixed-point Theorems,, Springer-Verlag, (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, (1990). doi: 10.1007/978-1-4612-0985-0.

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