# American Institute of Mathematical Sciences

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June  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
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