Degenerate non-autonomous parabolic equation in divergence form: well-posedness and controllability via Carleman estimates

Mohammad Akil; Genni Fragnelli; Sarah Ismail; Mohammad Akil; Genni Fragnelli; Sarah Ismail

doi:10.3934/era.2026295

Electronic Research Archive

2026, Volume 34, Issue 9: 6772-6806. doi: 10.3934/era.2026295

Previous Article Next Article

Research article Special Issues

Degenerate non-autonomous parabolic equation in divergence form: well-posedness and controllability via Carleman estimates

1.
Université Polytechnique Hauts-de-France, CÉRAMATHS/DEMAV, le Mont Houy Cedex 9, 59313 Valenciennes, France
2.
Università di Siena, Dipartimento di Ingegneria dell'Informazione e Scienze Matematiche, Via Roma 56, 53100 Siena, Italy
3.
Università degli Studi di Bari Aldo Moro, Dipartimento di Matematica, Via E. Orbona 4, 70125 Bari, Italy

Received: 22 June 2026 Revised: 29 July 2026 Accepted: 29 July 2026 Published: 11 August 2026

This paper addresses the null-controllability of a non-autonomous degenerate parabolic equation in divergence form, considering a diffusion coefficient $ a $ that degenerates in space and varies arbitrarily in time, thus generalizing the previously studied separable case in ^{[1, 2]}. Boundary conditions of Dirichlet or Robin type are imposed at the degeneracy point, while the control acts in the interior of the domain. After establishing well-posedness, we prove null-controllability by deriving new Carleman estimates for the associated non-homogeneous adjoint problem.
Dedicated to Professor Patrizia Pucci, with admiration and deep affection.
- null-controllability,
- degenerate equations,
- non-autonomous equations,
- Carleman estimates,
- operators in divergence form,
- Dirichlet and Robin boundary conditions
Citation: Mohammad Akil, Genni Fragnelli, Sarah Ismail. Degenerate non-autonomous parabolic equation in divergence form: well-posedness and controllability via Carleman estimates[J]. Electronic Research Archive, 2026, 34(9): 6772-6806. doi: 10.3934/era.2026295

Related Papers:

Abstract

This paper addresses the null-controllability of a non-autonomous degenerate parabolic equation in divergence form, considering a diffusion coefficient $ a $ that degenerates in space and varies arbitrarily in time, thus generalizing the previously studied separable case in ^{[1, 2]}. Boundary conditions of Dirichlet or Robin type are imposed at the degeneracy point, while the control acts in the interior of the domain. After establishing well-posedness, we prove null-controllability by deriving new Carleman estimates for the associated non-homogeneous adjoint problem.

Dedicated to Professor Patrizia Pucci, with admiration and deep affection.

References

[1]	M. Akil, G. Fragnelli, S. Ismail, Null-controllability and Carleman estimates for non-autonomous degenerate PDEs: A climatological application, J. Math. Anal. Appl., 543 (2025), 128984. https://doi.org/10.1016/j.jmaa.2024.128984 doi: 10.1016/j.jmaa.2024.128984
[2]	M. Akil, G. Fragnelli, S. Ismail, Non-autonomous degenerate parabolic equations with Robin boundary conditions: Carleman estimates and null-controllability, Appl. Math. Optim., 91 (2025), 35. https://doi.org/10.1007/s00245-025-10227-9 doi: 10.1007/s00245-025-10227-9
[3]	S. Aniţa, Analysis and Control of Age-Dependent Population Dynamics, Academic Publishers, Dordrecht, 2000. https://doi.org/10.1007/978-94-015-9436-3_2
[4]	I. Boutaayamou, G. Fragnelli, A degenerate population system: Carleman estimates and controllability, Nonlinear Anal., 195 (2020), 111742. https://doi.org/10.1016/j.na.2019.111742 doi: 10.1016/j.na.2019.111742
[5]	J. M. Buchot, J. P. Raymond, A linearized model for boundary layer equations, in Optimal control of complex structures, Int. Ser. Numer. Math., 139 (2001), 31–42. https://doi.org/10.1007/978-3-0348-8148-7_3 doi: 10.1007/978-3-0348-8148-7_3
[6]	M.I. Budyko, The effect of solar radiation variations on the climate of the Earth, Tellus, 21 (1969), 611–619. https://doi.org/10.3402/tellusa.v21i5.10109 doi: 10.3402/tellusa.v21i5.10109
[7]	P. Cannarsa, P. Martinez, J. Vancostenoble, Null controllability of the degenerate heat equations, Adv. Differ. Equations, 10 (2005), 153–190. https://doi.org/10.57262/ade/1355867887 doi: 10.57262/ade/1355867887
[8]	H. Emamirad, G. Goldstein, J. Goldstein, Chaotic solution for the Black-Scholes equation, Proc. Am. Math. Soc., 140 (2012), 2043–2052. https://doi.org/10.1090/S0002-9939-2011-11069-4 doi: 10.1090/S0002-9939-2011-11069-4
[9]	G. Fragnelli, Carleman estimates and null controllability for a degenerate population model, J. Math. Pures Appl., 115 (2018), 74–126. https://doi.org/10.1016/j.matpur.2018.01.003 doi: 10.1016/j.matpur.2018.01.003
[10]	G. Fragnelli, Null controllability for a degenerate population model in divergence form via Carleman estimates, Adv. Nonlinear Anal., 9 (2020), 1102–1129. https://doi.org/10.1515/anona-2020-0034 doi: 10.1515/anona-2020-0034
[11]	G. Fragnelli, M. Yamamoto, Carleman estimates and controllability for a degenerate structured population model, Appl. Math. Optim., 84 (2021), 999–1044. https://doi.org/10.1007/s00245-020-09669-0 doi: 10.1007/s00245-020-09669-0
[12]	R. M. Mininni, S. Romanelli, Martingale estimating functions for Feller diffusion processes generated by degenerate elliptic operators, J. Concr. Appl. Math., 1 (2003), 191–216.
[13]	W. D. Sellers, A global climatic model based on the energy balance of the earth-atmosphere system, J. Appl. Meteorol. Climatol., 8 (1969), 392–400. https://doi.org/10.1175/1520-0450(1969)008<0392:AGCMBO>2.0.CO;2 doi: 10.1175/1520-0450(1969)008<0392:AGCMBO>2.0.CO;2
[14]	N. Shimakura, Partial Differential Operators of Elliptic Type, American Mathematical Society, 1992. https://doi.org/10.1090/mmono/099
[15]	F. Alabau-Boussouira, P. Cannarsa, G. Fragnelli, Carleman estimates for degenerate parabolic operators with applications to null controllability, J. Evol. Equations, 6 (2006), 161–204. https://doi.org/10.1007/s00028-006-0222-6 doi: 10.1007/s00028-006-0222-6
[16]	P. Cannarsa, G. Fragnelli, D. Rocchetti, Controllability results for a class of one-dimensional degenerate parabolic problems in nondivergence form, J. Evol. Equations, 8 (2008), 583–616. https://doi.org/10.1007/s00028-008-0353-34 doi: 10.1007/s00028-008-0353-34
[17]	P. Cannarsa, G. Fragnelli, D. Rocchetti, Null controllability of degenerate parabolic operators with drift, Networks Heterogen. Media, 2 (2007), 695–715. https://doi.org/10.3934/nhm.2007.2.695 doi: 10.3934/nhm.2007.2.695
[18]	P. Martinez, J. P. Raymond, J. Vancostenoble, Regional null controllability for a linearized Crocco type equation, SIAM J. Control Optim., 42 (2003), 709–728. https://doi.org/10.1137/S0363012902403547 doi: 10.1137/S0363012902403547
[19]	P. Martinez, J. Vancostenoble, Carleman estimates for one-dimensional degenerate heat equations, J. Evol. Equations, 6 (2006), 325–362. https://doi.org/10.1007/s00028-006-0214-6 doi: 10.1007/s00028-006-0214-6
[20]	B. Allal, G. Fragnelli, Controllability of degenerate parabolic equation with memory, Math. Methods Appl. Sci., 44 (2021), 9163–9190. https://doi.org/10.1002/mma.7342 doi: 10.1002/mma.7342
[21]	B. Allal, G. Fragnelli, J. Salhi, On a general degenerate/singular parabolic equation with a nonlocal space term, Math. Methods Appl. Sci., 46 (2023), 12473–12504. https://doi.org/10.1002/mma.9190 doi: 10.1002/mma.9190
[22]	P. Cannarsa, P. Martinez, J. Vancostenoble, Global Carleman Estimates for Degenerate Parabolic Operators with Applications, 2016. https://doi.org/10.1090/memo/1133
[23]	R. Du, J. Eichhorn, Q. Liu, C. Wang, Carleman estimates and null controllability of a class of singular parabolic equations, Adv. Nonlinear Anal., 8 (2019), 1057–1082. https://doi.org/10.1515/anona-2016-0266 doi: 10.1515/anona-2016-0266
[24]	G. Fragnelli, D. Mugnai, Carleman estimates for singular parabolic equations with interior degeneracy and non-smooth coefficients, Adv. Nonlinear Anal., 6 (2017), 61–84. https://doi.org/10.1515/anona-2015-0163 doi: 10.1515/anona-2015-0163
[25]	C. Wang, R. Du, Carleman estimates and null controllability for a class of degenerate parabolic equations with convection terms, SIAM J. Control Optim., 52 (2014), 1457–1480. https://doi.org/10.1137/110820592 doi: 10.1137/110820592
[26]	C. Wang, Y. Zhou, Carleman estimate and null controllability for a degenerate parabolic equation with a slightly superlinear reaction term, Nonlinear Differ. Equations Appl., 30 (2023), 69. https://doi.org/10.1007/s00030-023-00881-9 doi: 10.1007/s00030-023-00881-9
[27]	C. Wang, Y. Zhou, R. Du, Q. Liu, Carleman estimate for solutions to a degenerate convection-diffusion equation, Discrete Contin. Dyn. Syst. - Ser. B, 23 (2018), 4207–4222. https://doi.org/10.3934/dcdsb.2018133 doi: 10.3934/dcdsb.2018133
[28]	R. Du, C. Wang, Null controllability of a class of systems governed by coupled degenerate equations, Appl. Math. Lett., 26 (2013), 113–119. https://doi.org/10.1016/j.aml.2012.04.005 doi: 10.1016/j.aml.2012.04.005
[29]	A. Hajjaj, L. Maniar, J. Salhi, Carleman estimates and null controllability of degenerate/singular parabolic systems, Electron. J. Differ. Equations, 2016 (2016), 1–25.
[30]	J. Xu, C. Wang, Y. Niem, Carleman estimate and null controllability of a cascade degenerate parabolic system with general convection terms, Electron. J. Differ. Equations, 2018 (2018), 1–20.
[31]	R. Du, C. Wang, Q. Zhou, Approximate controllability of a semilinear system involving a fully nonlinear gradient term, Appl. Math. Optim., 70 (2014), 165–183. https://doi.org/10.1007/s00245-014-9238-4 doi: 10.1007/s00245-014-9238-4
[32]	C. Wang, Approximate controllability of a class of semilinear systems with boundary degeneracy, J. Evol. Equations, 10 (2010), 163–193. https://doi.org/10.1007/s00028-009-0044-4 doi: 10.1007/s00028-009-0044-4
[33]	C. Wang, R. Du, Approximate controllability of a class of semilinear degenerate systems with convection term, J. Evol. Equations, 254 (2013), 3665–3689. https://doi.org/10.1016/j.jde.2013.01.038 doi: 10.1016/j.jde.2013.01.038
[34]	A. Benaissa, A. K. Mezadek, L. Maniar, Null controllability of degenerate nonautonomous parabolic equations, Facta Univ. Math. Inf., 34 (2019), 311–328. https://doi.org/10.22190/FUMI1902311B doi: 10.22190/FUMI1902311B
[35]	M. Akil, G. Fragnelli, S. Ismail, On a general degenerate non-autonomous parabolic equation in non-divergence form with Dirichlet or Robin boundary conditions, Discrete Contin. Dyn. Syst., 57 (2026), 53–94. https://doi.org/10.3934/dcds.2026120 doi: 10.3934/dcds.2026120
[36]	F. Alabau-Boussouira, P. Cannarsa, G. Leugering, Control and stabilization of degenerate wave equations, SIAM J. Control Optim., 55 (2017), 2052–2087. https://doi.org/10.1137/15M1020538 doi: 10.1137/15M1020538
[37]	M. Akil, G. Fragnelli, S. Ismail, Boundary stabilization for a general non-autonomous degenerate wave equation in divergence form, hal-05645320, 2026.
[38]	A. Camasta, G. Fragnelli, A degenerate operator in non divergence form, in Recent Advances in Mathematical Analysis, Springer International Publishing, (2023), 209–235. https://doi.org/10.1007/978-3-031-20021-2_11
[39]	K. J. Engel, R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer-Verlag, New York, Berlin, Heidelberg, 2000. https://doi.org/10.1007/b97696
[40]	T. Kato, Abstract Differential Equations and Nonlinear Mixed Problems, Accademia Nazionale dei Lincei, 1985.
[41]	T. Kato, Linear and quasi-linear equations of evolution of Hyperbolic type, in Hyperbolicity, (2011), 125–191. https://doi.org/10.1007/978-3-642-11105-1_4
[42]	D. Saba, G. Bayili, S. Nicaise, Polynomial stabilization of the wave equation with a time varying delay term in the dynamical control, 538 (2024), 128441. https://doi.org/10.1016/j.jmaa.2024.128441
[43]	J. L. Lions, E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Springer, New York, 1972. https://doi.org/10.1007/978-3-642-65161-8
[44]	P. Carvalho, R. Dermarque, J. Limaco, L. Viana, Null-controllability for 1-D degenerate quasilinear parabolic equations, Electron. J. Differ. Equations, 2025 (2025), 1–22. https://doi.org/10.58997/ejde.2025.15 doi: 10.58997/ejde.2025.15
[45]	J. C. Flores, L. De Teresa, Null controllability of one dimensional degenerate parabolic equations with first order terms, Discrete Contin. Dyn. Syst. - Ser. B, 25 (2020), 3963–3981. https://doi.org/10.3934/dcdsb.2020136 doi: 10.3934/dcdsb.2020136
[46]	G. Fragnelli, D. Mugnai, Control of Degenerate and Singular Parabolic Equations: Carleman Estimates and Observability, Springer Briefs in Mathematics, 2021. https://doi.org/10.1007/978-3-030-69349-7
[47]	G. Fragnelli, D. Mugnai, Corrigendum and improvements to "Carleman estimates, observability inequalities and null controllability for interior degenerate nonsmooth parabolic equations" and its consequences, Mem. Am. Math. Soc., 272 (2021), 1332. https://doi.org/10.1090/memo/1332 doi: 10.1090/memo/1332

Reader Comments

Your name:*

Email:*
© 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)