Inverse problem for a degenerate/singular parabolic system with Neumann boundary conditions. (English) Zbl 1481.35389
Summary: In this paper, we study an inverse source problem for a degenerate and singular parabolic system where the boundary conditions are of Neumann type. We consider a problem with degenerate diffusion coefficients and singular lower-order terms, both vanishing at an interior point of the space domain. In particular, we address the question of well-posedness of the problem, and then we prove a stability estimate of Lipschitz type in determining the source term by data of only one component. Our method is based on Carleman estimates, cut-off procedures and a reflection technique.
MSC:
| 35R30 | Inverse problems for PDEs |
| 35K65 | Degenerate parabolic equations |
| 35K67 | Singular parabolic equations |
| 93B30 | System identification |
Keywords:
inverse problem; degenerate parabolic equations; singular parabolic equations; Hardy-Poincaré inequalities; Carleman estimatesReferences:
| [1] | E. M. Ait Ben Hassi, F. Ammar Khodja, A. Hajjaj and L. Maniar, Null controllability of degenerate parabolic cascade systems, Port. Math. 68 (2011), no. 3, 345-367. · Zbl 1231.35103 |
| [2] | E. M. Ait Ben Hassi, F. Ammar Khodja, A. Hajjaj and L. Maniar, Carleman estimates and null controllability of coupled degenerate systems, Evol. Equ. Control Theory 2 (2013), no. 3, 441-459. · Zbl 1272.35124 |
| [3] | F. Alabau-Boussouira, P. Cannarsa and G. Fragnelli, Carleman estimates for degenerate parabolic operators with applications to null controllability, J. Evol. Equ. 6 (2006), no. 2, 161-204. · Zbl 1103.35052 |
| [4] | W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, Vector-valued Laplace Transforms and Cauchy Problems, Monogr. Math. 96, Birkhäuser, Basel, 2001. · Zbl 0978.34001 |
| [5] | P. Baras and J. A. Goldstein, Remarks on the inverse square potential in quantum mechanics, Differential Equations (Birmingham 1983), North-Holland Math. Stud. 92, North-Holland, Amsterdam (1984), 31-35. · Zbl 0566.35035 |
| [6] | P. Baras and J. A. Goldstein, The heat equation with a singular potential, Trans. Amer. Math. Soc. 284 (1984), no. 1, 121-139. · Zbl 0556.35063 |
| [7] | A. Benabdallah, M. Cristofol, P. Gaitan and M. Yamamoto, Inverse problem for a parabolic system with two components by measurements of one component, Appl. Anal. 88 (2009), no. 5, 683-709. · Zbl 1221.35444 |
| [8] | A. Bensoussan, G. Da Prato, M. C. Delfour and S. K. Mitter, Representation and Control of Infinite-Dimensional Systems. Vol. 1, Systems Control Found. Appl., Birkhäuser, Boston, 1992. · Zbl 0781.93002 |
| [9] | U. Biccari and E. Zuazua, Null controllability for a heat equation with a singular inverse-square potential involving the distance to the boundary function, J. Differential Equations 261 (2016), no. 5, 2809-2853. · Zbl 1341.35085 |
| [10] | I. Boutaayamou, G. Fragnelli and L. Maniar, Lipschitz stability for linear parabolic systems with interior degeneracy, Electron. J. Differential Equations 2014 (2014), Paper No. 167. · Zbl 1298.35248 |
| [11] | I. Boutaayamou, G. Fragnelli and L. Maniar, Inverse problems for parabolic equations with interior degeneracy and Neumann boundary conditions, J. Inverse Ill-Posed Probl. 24 (2016), no. 3, 275-292. · Zbl 1342.35447 |
| [12] | I. Boutaayamou, A. Hajjaj and L. Maniar, Lipschitz stability for degenerate parabolic systems, Electron. J. Differential Equations 2014 (2014), Paper No. 149. · Zbl 1297.35119 |
| [13] | A. L. Bukhgeĭm, Multidimensional inverse problems of spectral analysis, Dokl. Akad. Nauk SSSR 284 (1985), no. 1, 21-24. · Zbl 0604.35076 |
| [14] | X. Cabré and Y. Martel, Existence versus explosion instantanée pour des équations de la chaleur linéaires avec potentiel singulier, C. R. Math. Acad. Sci. Paris 329 (1999), no. 11, 973-978. · Zbl 0940.35105 |
| [15] | P. Cannarsa and L. de Teresa, Controllability of 1-D coupled degenerate parabolic equations, Electron. J. Differential Equations 2009 (2009), Paper No. 73. · Zbl 1178.35216 |
| [16] | P. Cannarsa, P. Martinez and J. Vancostenoble, Null controllability of degenerate heat equations, Adv. Differential Equations 10 (2005), no. 2, 153-190. · Zbl 1145.35408 |
| [17] | P. Cannarsa, P. Martinez and J. Vancostenoble, Carleman estimates for a class of degenerate parabolic operators, SIAM J. Control Optim. 47 (2008), no. 1, 1-19. · Zbl 1168.35025 |
| [18] | P. Cannarsa, P. Martinez and J. Vancostenoble, Global Carleman estimates for degenerate parabolic operators with applications, Mem. Amer. Math. Soc. 239 (2016), no. 1133, 1-209. · Zbl 1168.35025 |
| [19] | P. Cannarsa, J. Tort and M. Yamamoto, Determination of source terms in a degenerate parabolic equation, Inverse Problems 26 (2010), no. 10, Article ID 105003. · Zbl 1200.35319 |
| [20] | C. Cazacu, Controllability of the heat equation with an inverse-square potential localized on the boundary, SIAM J. Control Optim. 52 (2014), no. 4, 2055-2089. · Zbl 1303.35119 |
| [21] | J. B. Conway, A Course in Functional Analysis, 2nd ed., Grad. Texts in Math. 96, Springer, New York, 1990. · Zbl 0706.46003 |
| [22] | M. Cristofol, P. Gaitan and H. Ramoul, Inverse problems for a 2\times 2 reaction-diffusion system using a Carleman estimate with one observation, Inverse Problems 22 (2006), no. 5, 1561-1573. · Zbl 1105.35140 |
| [23] | M. Cristofol, P. Gaitan, H. Ramoul and M. Yamamoto, Identification of two coefficients with data of one component for a nonlinear parabolic system, Appl. Anal. 91 (2012), no. 11, 2073-2081. · Zbl 1254.35241 |
| [24] | S. Ervedoza, Control and stabilization properties for a singular heat equation with an inverse-square potential, Comm. Partial Differential Equations 33 (2008), no. 10-12, 1996-2019. · Zbl 1170.35331 |
| [25] | G. Fragnelli, G. R. Goldstein, J. A. Goldstein and S. Romanelli, Generators with interior degeneracy on spaces of L^2 type, Electron. J. Differential Equations 2012 (2012), Paper No. 189. · Zbl 1301.47065 |
| [26] | G. Fragnelli, G. Marinoschi, R. M. Mininni and S. Romanelli, A control approach for an identification problem associated to a strongly degenerate parabolic system with interior degeneracy, New Prospects in Direct, Inverse and Control Problems for Evolution Equations, Springer INdAM Ser. 10, Springer, Cham (2014), 121-139. · Zbl 1391.35416 |
| [27] | G. Fragnelli, G. Marinoschi, R. M. Mininni and S. Romanelli, Identification of a diffusion coefficient in strongly degenerate parabolic equations with interior degeneracy, J. Evol. Equ. 15 (2015), no. 1, 27-51. · Zbl 1325.35278 |
| [28] | G. Fragnelli and D. Mugnai, Carleman estimates and observability inequalities for parabolic equations with interior degeneracy, Adv. Nonlinear Anal. 2 (2013), no. 4, 339-378. · Zbl 1282.35101 |
| [29] | G. Fragnelli and D. Mugnai, Carleman estimates, observability inequalities and null controllability for interior degenerate nonsmooth parabolic equations, Mem. Amer. Math. Soc. 242 (2016), no. 1146, 1-84. · Zbl 1377.93043 |
| [30] | G. Fragnelli and D. Mugnai, Carleman estimates for singular parabolic equations with interior degeneracy and non-smooth coefficients, Adv. Nonlinear Anal. 6 (2017), no. 1, 61-84. · Zbl 1358.35219 |
| [31] | A. V. Fursikov and O. Y. Imanuvilov, Controllability of Evolution Equations, Lect. Notes Ser. 34, Seoul National University, Seoul, 1996. · Zbl 0862.49004 |
| [32] | A. Hajjaj, L. Maniar and J. Salhi, Carleman estimates and null controllability of degenerate/singular parabolic systems, Electron. J. Differential Equations 2016 (2016), Paper No. 292. · Zbl 1353.35186 |
| [33] | O. Y. Imanuvilov and M. Yamamoto, Lipschitz stability in inverse parabolic problems by the Carleman estimate, Inverse Problems 14 (1998), no. 5, 1229-1245. · Zbl 0992.35110 |
| [34] | J. Salhi, Null controllability for a singular coupled system of degenerate parabolic equations in nondivergence form, Electron. J. Qual. Theory Differ. Equ. 2018 (2018), Paper No. 31. · Zbl 1413.35269 |
| [35] | J. Vancostenoble, Improved Hardy-Poincaré inequalities and sharp Carleman estimates for degenerate/singular parabolic problems, Discrete Contin. Dyn. Syst. Ser. S 4 (2011), no. 3, 761-790. · Zbl 1213.93018 |
| [36] | J. Vancostenoble, Lipschitz stability in inverse source problems for singular parabolic equations, Comm. Partial Differential Equations 36 (2011), no. 8, 1287-1317. · Zbl 1229.35329 |
| [37] | J. Vancostenoble and E. Zuazua, Null controllability for the heat equation with singular inverse-square potentials, J. Funct. Anal. 254 (2008), no. 7, 1864-1902. · Zbl 1145.93009 |
| [38] | J. L. Vazquez and E. Zuazua, The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential, J. Funct. Anal. 173 (2000), no. 1, 103-153. · Zbl 0953.35053 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
