References
- Brezis H, Vzquez JL. Blow-up solutions of some nonlinear elliptic equations. Rev Mat Complut. 1997;10:443–469.
- Bebernes J, Eberly D. Mathematical problems from combustion theory. Berlin: Springer; 1989. (Applied mathematical sciences; vol. 83).
- 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.
- Galaktionov V, Vzquez JL. Continuation of blow-up solutions of nonlinear heat equations in several space dimensions. Comm Pure Appl Math. 1997;1:1–67. doi: https://doi.org/10.1002/(SICI)1097-0312(199701)50:1<1::AID-CPA1>3.0.CO;2-H
- Baras P, Goldstein J. Remarks on the inverse square potential in quantum mechanics. In: Knowles I, Lewis R, editors. Differential equations. Amsterdam: North-Holland; 1984. p. 31–35. (North-Holland mathematical studies; vol. 92).
- 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. doi: https://doi.org/10.1016/j.aop.2003.12.007
- Reed M, Simon B. Methods of modern mathematical physics. Vol. II. New York: Academic Press; 1979.
- Goldstein JA, Zhang QS. Linear parabolic equations with strong singular potentials. Trans Am Math Soc. 2003;355(1):197–211. doi: https://doi.org/10.1090/S0002-9947-02-03057-X
- Baras P, Goldstein J. The heat equation with a singular potential. Trans Amer Math Soc. 1984;284:121–139. doi: https://doi.org/10.1090/S0002-9947-1984-0742415-3
- 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]. C R Acad Sci Paris. 1999;329:973–978. doi: https://doi.org/10.1016/S0764-4442(00)88588-2
- Goldstein JA, Zhang QS. On a degenerate heat equation with a singular potential. J Funct Anal. 2001;186:342–359. doi: https://doi.org/10.1006/jfan.2001.3792
- 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).
- Imanuvilov OY. Controllability of parabolic equations. Sb Math. 1995;186(6):879–900. doi: https://doi.org/10.1070/SM1995v186n06ABEH000047
- Lions J-L. Contrôlabilité Exacte, Stabilisation et Perturbations de Systémes Distribués [Exact controllability, perturbation and stabilization of distributed systems. Volume 1: exact controllability]. Tome 1. Contrôlabilité exacte, Volume RMA 8. Paris: Masson; 1988.
- Vancostenoble J, Zuazua E. Null controllability for the heat equation with singular inverse-square potentials. J Funct Anal. 2008;254:1864–1902. doi: https://doi.org/10.1016/j.jfa.2007.12.015
- Ervedoza S. Control and stabilization properties for a singular heat equation with an inverse square potential. Commun Partial Differ Equ. 2008;33:1996–2019. doi: https://doi.org/10.1080/03605300802402633
- Vancostenoble J. Lipschitz stability in inverse source problems for singular parabolic equations. Commun Partial Differ Equ. 2011;36(8):1287–1317. doi: https://doi.org/10.1080/03605302.2011.587491
- 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. doi: https://doi.org/10.1137/120862557
- 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. doi: https://doi.org/10.1016/j.jde.2016.05.019
- Benabdallah A, Dermenjian Y, Le Rousseau J. Carleman estimates for the one-dimensional heat equation with a discontinuous coefficient and applications to controllability and an inverse problem. J Math Anal Appl. 2007;336:865–887. doi: https://doi.org/10.1016/j.jmaa.2007.03.024
- Fernndez-Cara E, Gonzlez-Burgos M, Guerrero S, et al. Null controllability of the heat equation with boundary Fourier conditions: the linear case. ESAIM Control Optim Calc Var. 2006;12:442–465. doi: https://doi.org/10.1051/cocv:2006010
- Fernndez-Cara E, Zuazua E. Null and approximate controllability for weakly blowing up semilinear heat equations. Ann Inst H Poincare Anal Non Linaire. 2000;17:583–616. doi: https://doi.org/10.1016/S0294-1449(00)00117-7
- Fernndez-Cara E, Zuazua E. The cost of approximate controllability for heat equations: the linear case. Adv Differ Equ. 2000;5:465–514.
- Martinez P, Vancostenoble J. Carleman estimates for one-dimensional degenerate heat equations. J Evol Equ. 2006;6(2):325–362. doi: https://doi.org/10.1007/s00028-006-0214-6
- Maz'ja VG. Sobolev spaces. Translated from the Russian by Shaposhnikova TO. Berlin: Springer; 1985. (Springer series in soviet mathematics).
- Evans LC. Partial differential. 2nd ed. Providence (RI): AMS; 2010.
- Fernndez-Cara E, Guerrero S, Imanuvilov OY, et al. Some controllability results for the N-dimensional Navier-Stokes and Boussinesq systems with N-1 scalar controls. SIAM J Control Optim. 2006;45:146–173. doi: https://doi.org/10.1137/04061965X
- Garcia AJ, Peral I. Hardy inequalities and some criticial elliptic and parabolic problems. J Differ Equ. 1998;144:441–476. MR 99f:35099. doi: https://doi.org/10.1006/jdeq.1997.3375
- Vancostenoble J, Zuazua E. Hardy inequalities. Observability and control for the wave and Schrödinger equations with singular potentials. SIAM J Math Anal. 2009;41:1508–1532. doi: https://doi.org/10.1137/080731396
- Vazquez JL, 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. doi: https://doi.org/10.1006/jfan.1999.3556