American Institute of Mathematical Sciences

Advanced
  • 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

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

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 & 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).   Google Scholar

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[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.  Google Scholar

[8]

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

[9]

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

[10]

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

[11]

E. Zeidler, Nonlinear Functional Analysis and Its Applications. I. Fixed-point Theorems,, Springer-Verlag, (1986).  doi: 10.1007/978-1-4612-4838-5.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

H. Brezis, Analyse Fonctionnelle, Théorie et Applications, Masson, Paris, 1983;, (English Translation) Functional Analysis, (2011).   Google Scholar

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[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.  Google Scholar

[8]

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

[9]

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

[10]

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

[11]

E. Zeidler, Nonlinear Functional Analysis and Its Applications. I. Fixed-point Theorems,, Springer-Verlag, (1986).  doi: 10.1007/978-1-4612-4838-5.  Google Scholar

[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.  Google Scholar

[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]

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
[9]

Matthias Hieber, Sylvie Monniaux. Well-posedness results for the Navier-Stokes equations in the rotational framework. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5143-5151. doi: 10.3934/dcds.2013.33.5143
[10]

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
[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]

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
[14]

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
[15]

Vishal Vasan, Bernard Deconinck. Well-posedness of boundary-value problems for the linear Benjamin-Bona-Mahony equation. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 3171-3188. doi: 10.3934/dcds.2013.33.3171
[16]

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
[17]

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
[18]

Thomas Y. Hou, Congming Li. Global well-posedness of the viscous Boussinesq equations. Discrete & Continuous Dynamical Systems - A, 2005, 12 (1) : 1-12. doi: 10.3934/dcds.2005.12.1
[19]

Massimo Cicognani, Michael Reissig. Well-posedness for degenerate Schrödinger equations. Evolution Equations & Control Theory, 2014, 3 (1) : 15-33. doi: 10.3934/eect.2014.3.15
[20]

Timur Akhunov. Local well-posedness of quasi-linear systems generalizing KdV. Communications on Pure & Applied Analysis, 2013, 12 (2) : 899-921. doi: 10.3934/cpaa.2013.12.899

2018 Impact Factor: 0.545

Tools

Metrics

  • PDF downloads (33)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences