# American Institute of Mathematical Sciences

• Previous Article
Stability of the wave equation with localized Kelvin-Voigt damping and boundary delay feedback
• DCDS-S Home
• This Issue
• Next Article
Existence of solutions for anisotropic Cahn-Hilliard and Allen-Cahn systems in higher space dimensions
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
##### References:

show all references

##### References:
 [1] Jean-Daniel Djida, Arran Fernandez, Iván Area. Well-posedness results for fractional semi-linear wave equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (2) : 569-597. doi: 10.3934/dcdsb.2019255 [2] Fatihcan M. Atay, Lavinia Roncoroni. Lumpability of linear evolution Equations in Banach spaces. Evolution Equations & Control Theory, 2017, 6 (1) : 15-34. doi: 10.3934/eect.2017002 [3] Tôn Việt Tạ. Existence results for linear evolution equations of parabolic type. Communications on Pure & Applied Analysis, 2018, 17 (3) : 751-785. doi: 10.3934/cpaa.2018039 [4] Alfredo Lorenzi, Gianluca Mola. Identification of a real constant in linear evolution equations in Hilbert spaces. Inverse Problems & Imaging, 2011, 5 (3) : 695-714. doi: 10.3934/ipi.2011.5.695 [5] Viorel Barbu, Gabriela Marinoschi. An identification problem for a linear evolution equation in a banach space. Discrete & Continuous Dynamical Systems - S, 2020, 13 (5) : 1429-1440. doi: 10.3934/dcdss.2020081 [6] Mustapha Mokhtar-Kharroubi. On permanent regimes for non-autonomous linear evolution equations in Banach spaces with applications to transport theory. Kinetic & Related Models, 2010, 3 (3) : 473-499. doi: 10.3934/krm.2010.3.473 [7] Goro Akagi, Mitsuharu Ôtani. Evolution equations and subdifferentials in Banach spaces. Conference Publications, 2003, 2003 (Special) : 11-20. doi: 10.3934/proc.2003.2003.11 [8] Alfredo Lorenzi, Ioan I. Vrabie. An identification problem for a linear evolution equation in a Banach space and applications. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 671-691. doi: 10.3934/dcdss.2011.4.671 [9] Radhia Ghanmi, Tarek Saanouni. Well-posedness issues for some critical coupled non-linear Klein-Gordon equations. Communications on Pure & Applied Analysis, 2019, 18 (2) : 603-623. doi: 10.3934/cpaa.2019030 [10] Matthias Hieber, Sylvie Monniaux. Well-posedness results for the Navier-Stokes equations in the rotational framework. Discrete & Continuous Dynamical Systems, 2013, 33 (11&12) : 5143-5151. doi: 10.3934/dcds.2013.33.5143 [11] Qunyi Bie, Qiru Wang, Zheng-An Yao. On the well-posedness of the inviscid Boussinesq equations in the Besov-Morrey spaces. Kinetic & Related Models, 2015, 8 (3) : 395-411. doi: 10.3934/krm.2015.8.395 [12] Giuseppe Floridia. Well-posedness for a class of nonlinear degenerate parabolic equations. Conference Publications, 2015, 2015 (special) : 455-463. doi: 10.3934/proc.2015.0455 [13] Vo Van Au, Jagdev Singh, Anh Tuan Nguyen. Well-posedness results and blow-up for a semi-linear time fractional diffusion equation with variable coefficients. Electronic Research Archive, 2021, 29 (6) : 3581-3607. doi: 10.3934/era.2021052 [14] Iñigo U. Erneta. Well-posedness for boundary value problems for coagulation-fragmentation equations. Kinetic & Related Models, 2020, 13 (4) : 815-835. doi: 10.3934/krm.2020028 [15] Ugur G. Abdulla. On the optimal control of the free boundary problems for the second order parabolic equations. I. Well-posedness and convergence of the method of lines. Inverse Problems & Imaging, 2013, 7 (2) : 307-340. doi: 10.3934/ipi.2013.7.307 [16] Xavier Carvajal, Liliana Esquivel, Raphael Santos. On local well-posedness and ill-posedness results for a coupled system of mkdv type equations. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2699-2723. doi: 10.3934/dcds.2020382 [17] Vishal Vasan, Bernard Deconinck. Well-posedness of boundary-value problems for the linear Benjamin-Bona-Mahony equation. Discrete & Continuous Dynamical Systems, 2013, 33 (7) : 3171-3188. doi: 10.3934/dcds.2013.33.3171 [18] Tarek Saanouni. A note on global well-posedness and blow-up of some semilinear evolution equations. Evolution Equations & Control Theory, 2015, 4 (3) : 355-372. doi: 10.3934/eect.2015.4.355 [19] Aissa Guesmia, Nasser-eddine Tatar. Some well-posedness and stability results for abstract hyperbolic equations with infinite memory and distributed time delay. Communications on Pure & Applied Analysis, 2015, 14 (2) : 457-491. doi: 10.3934/cpaa.2015.14.457 [20] Noufel Frikha, Valentin Konakov, Stéphane Menozzi. Well-posedness of some non-linear stable driven SDEs. Discrete & Continuous Dynamical Systems, 2021, 41 (2) : 849-898. doi: 10.3934/dcds.2020302

2020 Impact Factor: 2.425