Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Journal of Inverse and Ill-posed Problems

Editor-in-Chief: Kabanikhin, Sergey I.


IMPACT FACTOR 2018: 0.881
5-year IMPACT FACTOR: 1.170

CiteScore 2018: 0.91

SCImago Journal Rank (SJR) 2018: 0.430
Source Normalized Impact per Paper (SNIP) 2018: 0.969

Mathematical Citation Quotient (MCQ) 2018: 0.66

Online
ISSN
1569-3945
See all formats and pricing
More options …
Volume 27, Issue 4

Issues

Unique continuation for a reaction-diffusion system with cross diffusion

Bin Wu
  • Corresponding author
  • School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, P. R. China
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Ying Gao
  • School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, P. R. China
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Zewen Wang / Qun Chen
  • School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, P. R. China
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2019-01-30 | DOI: https://doi.org/10.1515/jiip-2017-0094

Abstract

This paper concerns unique continuation for a reaction-diffusion system with cross diffusion, which is a drug war reaction-diffusion system describing a simple dynamic model of a drug epidemic in an idealized community. We first establish a Carleman estimate for this strongly coupled reaction-diffusion system. Then we apply the Carleman estimate to prove the unique continuation, which means that the Cauchy data on any lateral boundary determine the solution uniquely in the whole domain.

Keywords: Unique continuation; cross diffusion; Carleman

MSC 2010: 35L05; 35L10; 35R09; 35R30

References

  • [1]

    M. Bellassoued and M. Yamamoto, Carleman estimate and inverse source problem for Biot’s equations describing wave propagation in porous media, Inverse Problems 29 (2013), no. 11, Article ID 115002. Web of ScienceGoogle Scholar

  • [2]

    M. Boulakia, A.-C. Egloffe and C. Grandmont, Stability estimates for the unique continuation property of the Stokes system and for an inverse boundary coefficient problem, Inverse Problems 29 (2013), no. 11, Article ID 115001. Web of ScienceGoogle Scholar

  • [3]

    A. L. Bukhgeĭm and M. V. Klibanov, Uniqueness in the large of a class of multidimensional inverse problems, Dokl. Akad. Nauk SSSR 260 (1981), no. 2, 269–272. Google Scholar

  • [4]

    T. Carleman, Sur un problème d’unicité pur les systèmes d’équations aux dérivées partielles à deux variables indépendantes, Ark. Mat. Astr. Fys. 26 (1939), no. 17, 9. Google Scholar

  • [5]

    O. Y. Èmanuilov, Controllability of parabolic equations, Mat. Sb. 186 (1995), no. 6, 109–132. Google Scholar

  • [6]

    J. M. Epstein, Nonlinear Dynamics, Mathematical Biology, and Social Science, Lecture Notes Santa Fe Inst. Stud. Sci. Compl. 4, Addison-Wesley, Reading, 1997. Google Scholar

  • [7]

    J. Fan, M. Di Cristo, Y. Jiang and G. Nakamura, Inverse viscosity problem for the Navier–Stokes equation, J. Math. Anal. Appl. 365 (2010), no. 2, 750–757. CrossrefWeb of ScienceGoogle Scholar

  • [8]

    X. Fu, A weighted identity for partial differential operators of second order and its applications, C. R. Math. Acad. Sci. Paris 342 (2006), no. 8, 579–584. CrossrefGoogle Scholar

  • [9]

    A. V. Fursikov and O. Y. Imanuvilov, Controllability of Evolution Equations, Lect. Notes Ser. 34, Seoul National University, Seoul, 1996. Google Scholar

  • [10]

    O. Y. Imanuvilov and M. Yamamoto, Lipschitz stability in inverse parabolic problems by the Carleman estimate, Inverse Problems 14 (1998), no. 5, 1229–1245. CrossrefGoogle Scholar

  • [11]

    O. Y. Imanuvilov and M. Yamamoto, Carleman inequalities for parabolic equations in Sobolev spaces of negative order and exact controllability for semilinear parabolic equations, Publ. Res. Inst. Math. Sci. 39 (2003), no. 2, 227–274. CrossrefGoogle Scholar

  • [12]

    O. Y. Imanuvilov and M. Yamamoto, Carleman estimates for the non-stationary Lamé system and the application to an inverse problem, ESAIM Control Optim. Calc. Var. 11 (2005), no. 1, 1–56. CrossrefGoogle Scholar

  • [13]

    V. Isakov, Inverse Problems for Partial Differential Equations, Appl. Math. Sci. 127, Springer, New York, 1998. Google Scholar

  • [14]

    V. Isakov, On the uniqueness of the continuation for a thermoelasticity system, SIAM J. Math. Anal. 33 (2001), no. 3, 509–522. CrossrefGoogle Scholar

  • [15]

    V. Isakov and N. Kim, Carleman estimates with two large parameters for second order operators and applications to elasticity with residual stress, Appl. Math. (Warsaw) 35 (2008), no. 4, 447–465. CrossrefGoogle Scholar

  • [16]

    M. V. Klibanov, Carleman estimates for global uniqueness, stability and numerical methods for coefficient inverse problems, J. Inverse Ill-Posed Probl. 21 (2013), no. 4, 477–560. Web of ScienceGoogle Scholar

  • [17]

    M. V. Klibanov and A. E. Kolesov, Convexification of a 3-D coefficient inverse scattering problem, Comput. Math. Appl. (2018), 10.1016/j.camwa.2018.03.016. Web of ScienceGoogle Scholar

  • [18]

    M. V. Klibanov, A. E. Kolesov, L. Nguyen and A. Sullivan, Globally strictly convex cost functional for a 1-D inverse medium scattering problem with experimental data, SIAM J. Appl. Math. 77 (2017), no. 5, 1733–1755. CrossrefGoogle Scholar

  • [19]

    M. V. Klibanov and A. Timonov, Carleman Estimates for Coefficient Inverse Problems and Numerical Applications, Inverse Ill-posed Probl. Ser., VSP, Utrecht, 2004. Google Scholar

  • [20]

    M. M. Lavrent’ev, V. G. Romanov and S. P. Šišatskiĭ, Ill-Posed Problems of Mathematical Physics and Analysis, American Mathematical Society, Providence, 1980. Google Scholar

  • [21]

    Q. Lü and Z. Yin, Unique continuation for stochastic heat equations, ESAIM Control Optim. Calc. Var. 21 (2015), no. 2, 378–398. Web of ScienceCrossrefGoogle Scholar

  • [22]

    J.-P. Puel and M. Yamamoto, On a global estimate in a linear inverse hyperbolic problem, Inverse Problems 12 (1996), no. 6, 995–1002. CrossrefGoogle Scholar

  • [23]

    V. G. Romanov and M. Yamamoto, Recovering a Lamé kernel in a viscoelastic equation by a single boundary measurement, Appl. Anal. 89 (2010), no. 3, 377–390. CrossrefGoogle Scholar

  • [24]

    L. Rosier and B.-Y. Zhang, Null controllability of the complex Ginzburg-Landau equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), no. 2, 649–673. CrossrefGoogle Scholar

  • [25]

    I. Seo, Global unique continuation from a half space for the Schrödinger equation, J. Funct. Anal. 266 (2014), no. 1, 85–98. CrossrefGoogle Scholar

  • [26]

    S. Tang and X. Zhang, Null controllability for forward and backward stochastic parabolic equations, SIAM J. Control Optim. 48 (2009), no. 4, 2191–2216. Web of ScienceCrossrefGoogle Scholar

  • [27]

    M. Uesaka and M. Yamamoto, Carleman estimate and unique continuation for a structured population model, Appl. Anal. 95 (2016), no. 3, 599–614. CrossrefWeb of ScienceGoogle Scholar

  • [28]

    V. K. Vanag and I. R. Epstein, Cross-diffusion and pattern formation in reaction-diffusion systems, Phys. Chem. Chem. Phys. 11 (2009), 897–912. PubMedCrossrefWeb of ScienceGoogle Scholar

  • [29]

    B. Wu and J. Liu, Determination of an unknown source for a thermoelastic system with a memory effect, Inverse Problems 28 (2012), no. 9, Article ID 095012. Web of ScienceGoogle Scholar

  • [30]

    B. Wu and J. Yu, Hölder stability of an inverse problem for a strongly coupled reaction-diffusion system, IMA J. Appl. Math. 82 (2017), no. 2, 424–444. Google Scholar

  • [31]

    M. Yamamoto, Carleman estimates for parabolic equations and applications, Inverse Problems 25 (2009), no. 12, Article ID 123013. Web of ScienceGoogle Scholar

  • [32]

    G. Yuan and M. Yamamoto, Lipschitz stability in the determination of the principal part of a parabolic equation, ESAIM Control Optim. Calc. Var. 15 (2009), no. 3, 525–554. Web of ScienceCrossrefGoogle Scholar

About the article

Received: 2017-09-24

Revised: 2018-12-25

Accepted: 2019-01-01

Published Online: 2019-01-30

Published in Print: 2019-08-01


Funding Source: National Natural Science Foundation of China

Award identifier / Grant number: 11601240

Award identifier / Grant number: 11561003

This work is supported by NSFC (No. 11601240, No. 11561003) and the Foundation of Academic and Technical Leaders Program for Major Subjects in Jiangxi Province (No. 20172BCB22019).


Citation Information: Journal of Inverse and Ill-posed Problems, Volume 27, Issue 4, Pages 511–525, ISSN (Online) 1569-3945, ISSN (Print) 0928-0219, DOI: https://doi.org/10.1515/jiip-2017-0094.

Export Citation

© 2019 Walter de Gruyter GmbH, Berlin/Boston.Get Permission

Comments (0)

Please log in or register to comment.
Log in