Local logarithmic stability of an inverse coefficient problem for a singular heat equation with an inverse-square potential

References

• Bebernes J, Eberly D. Mathematical problems from combustion theory. Berlin: Springer; 1989. (Applied Mathematical Sciences; vol. 83).
   Google Scholar
• Brezis H, Vzquez JL. Blow-up solutions of some nonlinear elliptic equations. Rev Mat Complut. 1997;10:443–469.
   Google Scholar
• Dold JW, Galaktionov VA, Lacey AA, et al. Rate of approach to a singular steady state in quasilinear reaction-diffusion equations. Ann Scuola Norm Sup Pisa Cl Sci. 1998;26:663–687.
   Google Scholar
• Galaktionov V, Vzquez JL. Continuation of blow-up solutions of nonlinear heat equations in several space dimensions. Commun Pure Appl Math. 1997;1:1–67.
   Google Scholar
• Baras P, Goldstein J. Remarks on the inverse square potential in quantum mechanics. In: Knowles I, Lewis R, editors. Diff Equat+. Vol. 92, North-Holland mathematical studies. Amsterdam: North-Holland; 1984. p. 31–35.
   Google Scholar
• De Castro AS. Bound states of the dirac equation for a class of effective quadratic plus inversely quadratic potentials. Ann Phys. 2004;311:170–181.
   Web of Science ®Google Scholar
• Reed M, Simon B. Methods of modern mathematical physics. Vol. II. New York: Academic Press; 1979.
   Google Scholar
• Adams RA, Fournier JJ. Sobolev spaces. Amsterdam: Elsevier Academic Press; 2003.
   Google Scholar
• Baras P, Goldstein J. The heat equation with a singular potential. Trans Am Math Soc. 1984;284:121–139.
   Web of Science ®Google Scholar
• Vazquez J, Zuazua E. The Hardy inequality and the asymptotic behavior of the heat equation with an inverse-square potential. J Funct Anal. 2000;173(1):103–153.
   Web of Science ®Google Scholar
• Cabr X, Martel Y. Existence versus explosion instantane pour des equations de la chaleur linaires avec potentiel singulier [Existence versus instantaneous blow-up for linear heat equations with singular potential]. CR Acad Sci Paris. 1999;329:973–978.
   Google Scholar
• Goldstein JA, Zhang QS. On a degenerate heat equation with a singular potential. J Funct Anal. 2001;186:342–359.
   Web of Science ®Google Scholar
• Goldstein JA, Zhang QS. Linear parabolic equations with strong singular potentials. Trans Am Math Soc. 2003;355(1):197–211.
   Web of Science ®Google Scholar
• Biccari U, Zuazua E. Null controllability for a heat equation with a singular inverse- square potential involving the distance to the boundary function. J Differ Equ. 2016;261(5):2809–2853.
   Web of Science ®Google Scholar
• Cristian C. Controllability of the heat equation with an inverse-square potential localized on the boundary. SIAM J Control Optim. 2014;52(4):2055–2089.
   Web of Science ®Google Scholar
• Ervedoza S. Control and stabilization properties for a singular heat equation with an inverse square potential. Commun Partial Differ Equ. 2010;33:1996–2019.
   Web of Science ®Google Scholar
• Vancostenoble J, Zuazua E. Null controllability for the heat equation with singular inverse-square potentials. J Funct Anal. 2008;254:1864–1902.
   Web of Science ®Google Scholar
• Qin X, Li S. The null controllability for a singular heat equation with variable coefficients. Appl Anal. DOI:10.1080/00036811.2020.1769076
   Google Scholar
• Carleman T. Sur un problème d'unicité pur les systèmes d'équations aux dérivées partielles à deux variables indépendantes. Ark Mat. 1939;26(17):1–9.
   Google Scholar
• Bukhgeim AL, Klibanov MV. Uniqueness in the large of a class of multidiimensional inverse problems. Dokl Akad Nauk SSSR. 1981;17:244–247.
   Google Scholar
• Baudouin L, Puel J-P. Uniqueness and stability in an inverse problem for the Schrödinger equation. Inverse Probl. 2002;18:1537–1554.
   Web of Science ®Google Scholar
• Beilina L, Cristofol M, Li S. Determining the conductivity for a nonautonomous hyperbolic operator in a cylindrical domain. Math Methods Appl Sci. 2018;41(5):2012–2030.
   Web of Science ®Google Scholar
• Beilina L, Cristofol M, Li S, et al. Lipschitz stability for an inverse hyperbolic problem of determining two coefficients by a finite number of observations. Inverse Probl. 2018;34:015001.
   Google Scholar
• Bellassoued M, Yamamoto M. Logarithmic stability in determination of a coefficient in an acoustic equation by arbitrary boundary observation. J Math Pures Appl. 2006;85(2):193–224.
   Web of Science ®Google Scholar
• Cristofol M, Li S, Soccorsi E. Determining the waveguide conductivity in a hyperbolic equation from a single measurement on the lateral boundary. Math Control Relat Fields. 2016;6(3):407–427.
   Web of Science ®Google Scholar
• Huang X, Imanuvilov O, Yamamoto M. Stability for inverse source problems by Carleman estimates. Inverse Probl. 2020;36(12):125006.
   Google Scholar
• Imanuvilov OY, Isakov V, Yamamoto M. An inverse problem for the dynamical Lamé system with two sets of boundary data. Commun Pure Appl Math. 2003;56(9):1366–1382.
   Web of Science ®Google Scholar
• Imanuvilov OY, Yamamoto M. Carleman estimate for a parabolic equation in a Sobolev space of negative order and its applications. In: Control of nonlinear distributed parameter systems. New York: Dekker; 2001. p. 113–137.
   Google Scholar
• Imanuvilov OY, Yamamoto M. Global lipschitz stability in an inverse hyperbolic problem by interior observations. Inverse Probl. 2001;17(4):717–728.
   Web of Science ®Google Scholar
• Imanuvilov OY, Yamamoto M. Determination of a coefficient in an acoustic equation with a single measurement. Inverse Probl. 2003;19(1):157–171.
   Web of Science ®Google Scholar
• Isakov V. Inverse source problems. Providence, RI: AMS Ebooks Program; 1990. p. 191. (Mathematical Surveys and Monographs; 34.
   Google Scholar
• Isakov V. Inverse problems for partial differential equations. New York: Springer; 1998, 2006. (Applied Mathematical Sciences; 127).
   Google Scholar
• Isakov V, Yamamoto M. Carleman estimate with the Neumann boundary condition and its applications to the observability inequality and inverse hyperbolic problems. Contemp Math. 2000;268:191–225.
   Google Scholar
• Kaltenbacher B, Klibanov MV. An inverse problem for a nonlinear parabolic equation with applications in population dynamics and magnetics. SIAM J Math Anal. 2008;39(6):1863–1889.
   Web of Science ®Google Scholar
• Klibanov MV. Inverse problems in the large and carleman estimates. Differ Uravn. 1984;20(6):1035–1041. Russian.
   Web of Science ®Google Scholar
• Klibanov MV. Uniqueness of solutions pf two inverse problems for the Maxwellian system. USSR J Comput Math Math Phys. 1986;26:1063–1071.
   Web of Science ®Google Scholar
• Klibanov MV. A class of inverse problems for nonlinear parabolic equations. Siberian Math J. 1987;27:698–708.
   Web of Science ®Google Scholar
• Klibanov MV. Inverse problems and Carleman estimates. Inverse Probl. 1992;8:575–596.
   Web of Science ®Google Scholar
• Klibanov MV. Carleman estimates for global uniqueness, stability and numerical methods for coefficient inverse problems. J Inverse Ill-Posed Probl. 2013;21:477–560.
   Web of Science ®Google Scholar
• Klibanov MV, Li J. Inverse problems and carleman estimates: global uniqueness, global convergence and experimental data. Berlin, Boston: De Gruyter; 2021.
   Google Scholar
• Klibanov MV, Li J, Zhang W. Convexification for an inverse parabolic problem. Inverse Probl. 2020;36(8):085008.
   Google Scholar
• Klibanov MV, Timonov A. Carleman estimates for coefficient inverse problems and numerical applications. Utrecht: VSP; 2004.
   Google Scholar
• Klibanov MV, Yamamoto M. Lipschitz stability of an inverse problem for an acoustic equation. Appl Anal. 2006;85:515–538.
   Google Scholar
• Lavrentíev MM, Romanov VG, Shishatskii SP. Ill-posed problems of mathematical physics and analysis. Providence (RI): AMS; 1986.
   Google Scholar
• Li S. Estimation of coefficients in a hyperbolic equation with impulsive inputs. J Inverse Ill-Posed Probl. 2006;14:891–904.
   Google Scholar
• Li S. Carleman estimates for second-order hyperbolic systems in anisotropic cases and applications. Part II: an inverse source problem. Appl Anal. 2014;94(11):2287–2307.
   Web of Science ®Google Scholar
• Li S, Miara B, Yamamoto M. A Carleman estimate for the linear shallow shell equation and an inverse source problem. Discrete Contin Dyn Syst. 2009;23:367–380.
   Web of Science ®Google Scholar
• Li S, Yamamoto M. An inverse problem for Maxwell's equations in anisotropic media. Chin Ann Math Ser B. 2007;28(1):35–54.
   Web of Science ®Google Scholar
• Li S, Yamamoto M. An inverse source problem for Maxwell's equations in anisotropic media. Appl Anal. 2005;84(10):1051–1067.
   Google Scholar
• Puel J-P, Yamamoto M. On a global estimate in a linear inverse hyperbolic problem. Inverse Probl. 1996;12:995–1002.
   Web of Science ®Google Scholar
• Qin X, Li S. A stability estimate for an inverse problem of determining a coefficient in a hyperbolic equation with a point source. Commun Math Stat. 2016;4(3):403–421.
   Web of Science ®Google Scholar
• Shang Y, Zhao L, Li S, et al. An inverse problem for Maxwell's equations in a uniaxially anisotropic medium. J Math Anal Appl. 2017;456(2):1415–1441.
   Web of Science ®Google Scholar
• Yamamoto M. Carleman estimates for parabolic equations and applications. Inverse Probl. 2009;25(12):123013.
   Google Scholar
• Yuan G, Yamamoto M. Lipschitz stability in the determination of the principal part of a parabolic equation. ESAIM Control Optim Calc Var. 2009;15(3):525–554.
   Web of Science ®Google Scholar
• Liskevich VA, Semenov YA. Some problems on Markov semigroups. In: Schrödinger operators, markov semigroups, wavelet analysis, operator algebras. Berlin: Akademie; 1996. p. 163–217.
   Google Scholar
• Vancostenoble J. Lipschitz stability in inverse source problems for singular parabolic equations. Commun Partial Differ Equ. 2011;36(8):1287–1317.
   Web of Science ®Google Scholar
• Fursikov AV, Imanuvilov OY. Controllability of evolution equations. Seoul: Seoul National University Research Institute of Mathematics Global Analysis Research Center; 1996. (Lecture Notes Series; vol. 34).
   Google Scholar
• Maz'ja VG. Sobolev spaces. New York: Springer; 1985.
   Google Scholar