Unique continuation for a reaction-diffusion system with cross diffusion

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. 10.1088/0266-5611/29/11/115002Search in Google 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. 10.1088/0266-5611/29/11/115001Search in Google 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. Search in 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. Search in Google Scholar

[5] O. Y. Èmanuilov, Controllability of parabolic equations, Mat. Sb. 186 (1995), no. 6, 109–132. 10.1070/SM1995v186n06ABEH000047Search in 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. Search in 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. 10.1016/j.jmaa.2009.12.012Search in Google 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. 10.1016/j.crma.2006.02.023Search in Google Scholar

[9] A. V. Fursikov and O. Y. Imanuvilov, Controllability of Evolution Equations, Lect. Notes Ser. 34, Seoul National University, Seoul, 1996. Search in 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. 10.1088/0266-5611/14/5/009Search in Google 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. 10.2977/prims/1145476103Search in Google 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. 10.1051/cocv:2004030Search in Google Scholar

[13] V. Isakov, Inverse Problems for Partial Differential Equations, Appl. Math. Sci. 127, Springer, New York, 1998. 10.1007/978-1-4899-0030-2Search in 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. 10.1137/S0036141000366509Search in Google 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. 10.4064/am35-4-4Search in Google 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. 10.1515/jip-2012-0072Search in Google 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. 10.1016/j.camwa.2018.03.016Search in Google 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. 10.1137/17M1122487Search in Google 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. 10.1515/9783110915549Search in 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. Search in 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. 10.1051/cocv/2014027Search in Google 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. 10.1088/0266-5611/12/6/013Search in Google 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. 10.1080/00036810903518975Search in Google 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. 10.1016/j.anihpc.2008.03.003Search in Google Scholar

[25] I. Seo, Global unique continuation from a half space for the Schrödinger equation, J. Funct. Anal. 266 (2014), no. 1, 85–98. 10.1016/j.jfa.2013.09.025Search in Google 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. 10.1137/050641508Search in Google 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. 10.1080/00036811.2015.1022157Search in Google 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. 10.1039/B813825GSearch in Google Scholar PubMed

[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. 10.1088/0266-5611/28/9/095012Search in Google 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. 10.1093/imamat/hxw058Search in Google Scholar

[31] M. Yamamoto, Carleman estimates for parabolic equations and applications, Inverse Problems 25 (2009), no. 12, Article ID 123013. 10.1088/0266-5611/25/12/123013Search in Google 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. 10.1051/cocv:2008043Search in Google Scholar