# American Institute of Mathematical Sciences

June  2022, 12(2): 495-530. doi: 10.3934/mcrf.2021032

## Existence and cost of boundary controls for a degenerate/singular parabolic equation

 1 Chair of Computational Mathematics, Fundación Deusto, Avda. de las Universidades 24, 48007 Bilbao, Basque Country, Spain, Facultad de Ingeniería, Universidad de Deusto, Avda. de las Universidades 24, 48007 Bilbao, Basque Country, Spain 2 Instituto de Matemáticas, Universidad Nacional Autónoma de México, Circuito Exterior, C.U., C.P. 04510 CDMX, Mexico 3 Institut de mathématiques de Toulouse, UMR5219; Université de Toulouse, CNRS, UPS IMT, F-31062 Toulouse Cedex 9, France

* Corresponding author: Umberto Biccari

Received  January 2020 Revised  December 2020 Published  June 2022 Early access  June 2021

Fund Project: This project has received funding from the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme (grant agreement NO. 694126-DyCon). The work of U. B. was partially supported by the Grant MTM2017-92996-C2-1-R COSNET of MINECO (Spain), by the Elkartek grant KK-2020/00091 CONVADP and by the Air Force Office of Scientific Research (AFOSR) under Award NO. FA9550-18-1-0242. The work of V. H.-S. was supported by the programme "Estancias posdoctorales por México" of CONACyT, Mexico

In this paper, we consider the following degenerate/singular parabolic equation
 \begin{align*} u_t -(x^\alpha u_{x})_x - \frac{\mu}{x^{2-\alpha}} u = 0, \qquad x\in (0,1), \ t \in (0,T), \end{align*}
where
 $0\leq \alpha <1$
and
 $\mu\leq (1-\alpha)^2/4$
are two real parameters. We prove the boundary null controllability by means of a
 $H^1(0,T)$
control acting either at
 $x = 1$
or at the point of degeneracy and singularity
 $x = 0$
. Besides we give sharp estimates of the cost of controllability in both cases in terms of the parameters
 $\alpha$
and
 $\mu$
. The proofs are based on the classical moment method by Fattorini and Russell and on recent results on biorthogonal sequences.
Citation: U. Biccari, V. Hernández-Santamaría, J. Vancostenoble. Existence and cost of boundary controls for a degenerate/singular parabolic equation. Mathematical Control and Related Fields, 2022, 12 (2) : 495-530. doi: 10.3934/mcrf.2021032
##### References:
 [1] F. Alabau-Boussouira, P. Cannarsa and G. Fragnelli, Carleman estimates for degenerate parabolic operators with applications to null controllability, J. Evol. Equ., 6 (2006), 161-204.  doi: 10.1007/s00028-006-0222-6. [2] F. Ammar Khodja, A. Benabdallah, M. González-Burgos and L. de Teresa, The Kalman condition for the boundary controllability of coupled parabolic systems. Bounds on biorthogonal families to complex matrix exponentials, J. Math. Pures Appl., 96 (2011), 555-590.  doi: 10.1016/j.matpur.2011.06.005. [3] G. Avalos and I. Lasiecka, Optimal blowup rates for the minimal energy null control of the strongly damped abstract wave equation, Ann. Sc. Norm. Super. Pisa Cl. Sci., 2 (2003), 601-616. [4] G. Avalos and I. Lasiecka, The null controllability of thermoelastic plates and singularity of the associated minimal energy function, J. Math. Anal. Appl., 294 (2004), 34-61.  doi: 10.1016/j.jmaa.2004.01.035. [5] A. Benabdallah, F. Boyer, M. González-Burgos and G. Olive, Sharp estimates of the one-dimensional boundary control cost for parabolic systems and applications to the $N$-dimensional boundary null controllability in cylindrical domains, SIAM J. Control Optim., 52 (2014), 2970-3001.  doi: 10.1137/130929680. [6] A. Bensoussan, G. Da Prato, M.C. Delfour and S.K. Mitter, Representation and Control of Infinite Dimensional Systems, Vol. I, Systems and Control: Foundations and Appliactions, Birkhauser-Boston, 1992. [7] 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), 2809-2853.  doi: 10.1016/j.jde.2016.05.019. [8] U. Biccari, Boundary controllability for a one-dimensional heat equation with a singular inverse-square potential, Math. Control Relat. Fields, 9 (2019), 191-219.  doi: 10.3934/mcrf.2019011. [9] P. Cannarsa, G. Fragnelli and D. Rocchetti, Controllability results for a class of one-dimensional degenerate parabolic problems in nondivergence form, J. Evol. Equ., 8 (2008), 583-616.  doi: 10.1007/s00028-008-0353-34. [10] P. Cannarsa, P. Martinez and J. Vancostenoble, Carleman estimates for a class of degenerate parabolic operators, SIAM J. Control Optim., 47 (2008), 1-19.  doi: 10.1137/04062062X. [11] P. Cannarsa, P. Martinez and J. Vancostenoble, Global carleman estimates for degenerate parabolic operators with applications, Memoirs of the American Mathematical Society, 239 (2016), ix+209 pp. doi: 10.1090/memo/1133. [12] P. Cannarsa, P. Martinez and J. Vancostenoble, The cost of controlling weakly degenerate parabolic equations by boundary controls, Math. Control Relat. Fields, 7 (2017), 171-211.  doi: 10.3934/mcrf.2017006. [13] P. Cannarsa, P. Martinez and J. Vancostenoble, Precise estimates for biorthogonal families under asymptotic gap conditions, Discrete Contin. Dyn. Syst. Ser. S., 13 (2020), 1441-1472.  doi: 10.3934/dcdss.2020082. [14] P. Cannarsa, P. Martinez and J. Vancostenoble, The cost of controlling strongly degenerate parabolic equations, ESAIM: Control Optim. Calc. Var., 26 (2020), 50pp. doi: 10.1051/cocv/2018007. [15] C. Cazacu, Controllability of the heat equation with an inverse-square potential localized on the boundary, SIAM J. Control Optim., 52 (2014), 2055-2089.  doi: 10.1137/120862557. [16] J. M. Coron and S. Guerrero, Singular optimal control: A linear 1-D parabolic-hyperbolic example, Asymp. Anal., 44 (2005), 237-257. [17] G. Da Prato, An Introduction to Infinite Dimensional Analysis, Scuola Normale Superiore, Pisa, 2001. [18] G. Da Prato and J. Zabczyk, Second Order Partial Differential Equations in Hilbert Spaces, London Mathematical Society Lecture Note Series, 293. Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511543210. [19] S. Ervedoza, Control and stabilization properties for a singular heat equation with an inverse-square potential, Comm. Partial Differential Equations, 33 (2008), 1996-2019.  doi: 10.1080/03605300802402633. [20] H. O. Fattorini and D. L. Russel, Exact controllability theorems for linear parabolic equations in one space dimension, Arch. Rat. Mech. Anal., 43 (1971), 272-292.  doi: 10.1007/BF00250466. [21] H. O. Fattorini and D. L. Russel, Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations, Quart. Appl. Math., 32 (1974), 45-69.  doi: 10.1090/qam/510972. [22] M. Fotouhi and L. Salimi, Null controllability of degenerate/singular parabolic equations, J. Dyn. Control Syst., 18 (2012), 573-602.  doi: 10.1007/s10883-012-9160-5. [23] O. Glass, A complex-analytic approach to the problem of uniform controllability of transport equation in the vanishing viscosity limit, J. Funct. Anal., 258 (2010), 852-868.  doi: 10.1016/j.jfa.2009.06.035. [24] F. Gozzi and P. Loreti, Regularity of the minimum time function and minimum energy problem: The linear case, SIAM J. Control Optim., 37 (1999), 1195-1221.  doi: 10.1137/S0363012996312763. [25] M. Gueye, Exact boundary controllability of 1-D parabolic and hyperbolic degenerate equations, SIAM J. Control Optim., 52 (2014), 2037-2054.  doi: 10.1137/120901374. [26] E. N. Güichal, A lower bound of the norm of the control operator for the heat equation, J. Math. Anal. Appl., 110 (1985), 519-527.  doi: 10.1016/0022-247X(85)90313-0. [27] A. Hajjaj, L. Maniar and J. Salhi, Carleman estimates and null controllability of degenerate/singular parabolic systems, Electron. J. Differential Equations, 2016 (2016), 25pp. [28] S. Hansen, Bounds on functions biorthogonal to sets of complex exponentials; control of damped elastic systems, J. Math. Anal. Appl., 158 (1991), 487-508.  doi: 10.1016/0022-247X(91)90252-U. [29] I. Lasiecka and T. I. Seidman, Blowup estimates for observability of a thermoelastic system, Asymptot. Anal., 50 (2006), 93-120. [30] P. Lissy, On the cost of fast controls for some families of dispersive or parabolic equations in one space dimension, SIAM J. Control Optim., 52 (2014), 2651-2676.  doi: 10.1137/140951746. [31] P. Lissy, Explicit lower bounds for the cost of fast controls for some 1-D parabolic or dispersive equations, and a new lower bound concerning the uniform controllability of the 1-D transport-diffusion equation, J. Differential Equations, 259 (2015), 5331-5352.  doi: 10.1016/j.jde.2015.06.031. [32] L. Miller, Geometric bounds on the growth rate of null-controllability cost for the heat equation in small time, J. Differential Equations, 204 (2004), 202-226.  doi: 10.1016/j.jde.2004.05.007. [33] V. Komornik and P. Loreti, Fourier Series in Control Theory, Springer Monographs in Mathematics. Springer-Verlag, New York, 2005. [34] N. N. Lebedev, Special Functions and Their Applications, Dover Publications, Inc., New York, 1972. [35] L. Lorch and M. E. Muldoon, Monotonic sequences related to zeros of Bessel functions, Numer. Algor., 49 (2008), 221-233.  doi: 10.1007/s11075-008-9189-4. [36] P. Martinez and J. Vancostenoble, Carleman estimates for one-dimensional degenerate heat equations, J. Evol. Eq., 6 (2006), 325-362.  doi: 10.1007/s00028-006-0214-6. [37] P. Martinez and J. Vancostenoble, The cost of boundary controllability for a parabolic equation with inverse square potential, Evol. Equ. Control Theory, 8 (2019), 397-422.  doi: 10.3934/eect.2019020. [38] C. K. Qu and R. Wong, "Best possible" upper and lower bounds for the zeros of the Bessel function $J_\nu(x)$, Trans. Amer. Math. Soc., 351 (1999), 2833-2859.  doi: 10.1090/S0002-9947-99-02165-0. [39] T. I. Seidman, How violent are fast controls?, Math. Control Signals Systems, 1 (1988), 89-95.  doi: 10.1007/BF02551238. [40] T. I. Seidman and J. Yong, How violent are fast controls? Ⅱ, Math. Control Signals Systems, 9 (1996), 327-340.  doi: 10.1007/BF01211854. [41] T. I. Seidman, S. A. Avdonin and S. A. Ivanov, The "window problem" for series of complex exponentials, J. Fourier Anal. Appl., 6 (2000), 233-254.  doi: 10.1007/BF02511154. [42] G. Tenenbaum and M. Tucsnak, New blow-up rates for fast controls of Schrödinger and heat equations, J. Differential Equations, 243 (2007), 70-100.  doi: 10.1016/j.jde.2007.06.019. [43] J. Vancostenoble, Improved Hardy-Poincaré inequalities and sharp Carleman estimates for degenerate/singular parabolic problems, Discr. Cont. Dyn. Syst., 4 (2011), 761-790.  doi: 10.3934/dcdss.2011.4.761. [44] J. Vancostenoble and E. Zuazua, Null controllability for the heat equation with singular inverse-square potentials, J. Funct. Anal., 254 (2008), 1864-1902.  doi: 10.1016/j.jfa.2007.12.015. [45] J. L. Vázquez and E. Zuazua, The hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential, J. Funct. Anal., 173 (2000), 103-153.  doi: 10.1006/jfan.1999.3556. [46] G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, England, The Macmillan Company, New York, 1944

show all references

##### References:
 [1] F. Alabau-Boussouira, P. Cannarsa and G. Fragnelli, Carleman estimates for degenerate parabolic operators with applications to null controllability, J. Evol. Equ., 6 (2006), 161-204.  doi: 10.1007/s00028-006-0222-6. [2] F. Ammar Khodja, A. Benabdallah, M. González-Burgos and L. de Teresa, The Kalman condition for the boundary controllability of coupled parabolic systems. Bounds on biorthogonal families to complex matrix exponentials, J. Math. Pures Appl., 96 (2011), 555-590.  doi: 10.1016/j.matpur.2011.06.005. [3] G. Avalos and I. Lasiecka, Optimal blowup rates for the minimal energy null control of the strongly damped abstract wave equation, Ann. Sc. Norm. Super. Pisa Cl. Sci., 2 (2003), 601-616. [4] G. Avalos and I. Lasiecka, The null controllability of thermoelastic plates and singularity of the associated minimal energy function, J. Math. Anal. Appl., 294 (2004), 34-61.  doi: 10.1016/j.jmaa.2004.01.035. [5] A. Benabdallah, F. Boyer, M. González-Burgos and G. Olive, Sharp estimates of the one-dimensional boundary control cost for parabolic systems and applications to the $N$-dimensional boundary null controllability in cylindrical domains, SIAM J. Control Optim., 52 (2014), 2970-3001.  doi: 10.1137/130929680. [6] A. Bensoussan, G. Da Prato, M.C. Delfour and S.K. Mitter, Representation and Control of Infinite Dimensional Systems, Vol. I, Systems and Control: Foundations and Appliactions, Birkhauser-Boston, 1992. [7] 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), 2809-2853.  doi: 10.1016/j.jde.2016.05.019. [8] U. Biccari, Boundary controllability for a one-dimensional heat equation with a singular inverse-square potential, Math. Control Relat. Fields, 9 (2019), 191-219.  doi: 10.3934/mcrf.2019011. [9] P. Cannarsa, G. Fragnelli and D. Rocchetti, Controllability results for a class of one-dimensional degenerate parabolic problems in nondivergence form, J. Evol. Equ., 8 (2008), 583-616.  doi: 10.1007/s00028-008-0353-34. [10] P. Cannarsa, P. Martinez and J. Vancostenoble, Carleman estimates for a class of degenerate parabolic operators, SIAM J. Control Optim., 47 (2008), 1-19.  doi: 10.1137/04062062X. [11] P. Cannarsa, P. Martinez and J. Vancostenoble, Global carleman estimates for degenerate parabolic operators with applications, Memoirs of the American Mathematical Society, 239 (2016), ix+209 pp. doi: 10.1090/memo/1133. [12] P. Cannarsa, P. Martinez and J. Vancostenoble, The cost of controlling weakly degenerate parabolic equations by boundary controls, Math. Control Relat. Fields, 7 (2017), 171-211.  doi: 10.3934/mcrf.2017006. [13] P. Cannarsa, P. Martinez and J. Vancostenoble, Precise estimates for biorthogonal families under asymptotic gap conditions, Discrete Contin. Dyn. Syst. Ser. S., 13 (2020), 1441-1472.  doi: 10.3934/dcdss.2020082. [14] P. Cannarsa, P. Martinez and J. Vancostenoble, The cost of controlling strongly degenerate parabolic equations, ESAIM: Control Optim. Calc. Var., 26 (2020), 50pp. doi: 10.1051/cocv/2018007. [15] C. Cazacu, Controllability of the heat equation with an inverse-square potential localized on the boundary, SIAM J. Control Optim., 52 (2014), 2055-2089.  doi: 10.1137/120862557. [16] J. M. Coron and S. Guerrero, Singular optimal control: A linear 1-D parabolic-hyperbolic example, Asymp. Anal., 44 (2005), 237-257. [17] G. Da Prato, An Introduction to Infinite Dimensional Analysis, Scuola Normale Superiore, Pisa, 2001. [18] G. Da Prato and J. Zabczyk, Second Order Partial Differential Equations in Hilbert Spaces, London Mathematical Society Lecture Note Series, 293. Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511543210. [19] S. Ervedoza, Control and stabilization properties for a singular heat equation with an inverse-square potential, Comm. Partial Differential Equations, 33 (2008), 1996-2019.  doi: 10.1080/03605300802402633. [20] H. O. Fattorini and D. L. Russel, Exact controllability theorems for linear parabolic equations in one space dimension, Arch. Rat. Mech. Anal., 43 (1971), 272-292.  doi: 10.1007/BF00250466. [21] H. O. Fattorini and D. L. Russel, Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations, Quart. Appl. Math., 32 (1974), 45-69.  doi: 10.1090/qam/510972. [22] M. Fotouhi and L. Salimi, Null controllability of degenerate/singular parabolic equations, J. Dyn. Control Syst., 18 (2012), 573-602.  doi: 10.1007/s10883-012-9160-5. [23] O. Glass, A complex-analytic approach to the problem of uniform controllability of transport equation in the vanishing viscosity limit, J. Funct. Anal., 258 (2010), 852-868.  doi: 10.1016/j.jfa.2009.06.035. [24] F. Gozzi and P. Loreti, Regularity of the minimum time function and minimum energy problem: The linear case, SIAM J. Control Optim., 37 (1999), 1195-1221.  doi: 10.1137/S0363012996312763. [25] M. Gueye, Exact boundary controllability of 1-D parabolic and hyperbolic degenerate equations, SIAM J. Control Optim., 52 (2014), 2037-2054.  doi: 10.1137/120901374. [26] E. N. Güichal, A lower bound of the norm of the control operator for the heat equation, J. Math. Anal. Appl., 110 (1985), 519-527.  doi: 10.1016/0022-247X(85)90313-0. [27] A. Hajjaj, L. Maniar and J. Salhi, Carleman estimates and null controllability of degenerate/singular parabolic systems, Electron. J. Differential Equations, 2016 (2016), 25pp. [28] S. Hansen, Bounds on functions biorthogonal to sets of complex exponentials; control of damped elastic systems, J. Math. Anal. Appl., 158 (1991), 487-508.  doi: 10.1016/0022-247X(91)90252-U. [29] I. Lasiecka and T. I. Seidman, Blowup estimates for observability of a thermoelastic system, Asymptot. Anal., 50 (2006), 93-120. [30] P. Lissy, On the cost of fast controls for some families of dispersive or parabolic equations in one space dimension, SIAM J. Control Optim., 52 (2014), 2651-2676.  doi: 10.1137/140951746. [31] P. Lissy, Explicit lower bounds for the cost of fast controls for some 1-D parabolic or dispersive equations, and a new lower bound concerning the uniform controllability of the 1-D transport-diffusion equation, J. Differential Equations, 259 (2015), 5331-5352.  doi: 10.1016/j.jde.2015.06.031. [32] L. Miller, Geometric bounds on the growth rate of null-controllability cost for the heat equation in small time, J. Differential Equations, 204 (2004), 202-226.  doi: 10.1016/j.jde.2004.05.007. [33] V. Komornik and P. Loreti, Fourier Series in Control Theory, Springer Monographs in Mathematics. Springer-Verlag, New York, 2005. [34] N. N. Lebedev, Special Functions and Their Applications, Dover Publications, Inc., New York, 1972. [35] L. Lorch and M. E. Muldoon, Monotonic sequences related to zeros of Bessel functions, Numer. Algor., 49 (2008), 221-233.  doi: 10.1007/s11075-008-9189-4. [36] P. Martinez and J. Vancostenoble, Carleman estimates for one-dimensional degenerate heat equations, J. Evol. Eq., 6 (2006), 325-362.  doi: 10.1007/s00028-006-0214-6. [37] P. Martinez and J. Vancostenoble, The cost of boundary controllability for a parabolic equation with inverse square potential, Evol. Equ. Control Theory, 8 (2019), 397-422.  doi: 10.3934/eect.2019020. [38] C. K. Qu and R. Wong, "Best possible" upper and lower bounds for the zeros of the Bessel function $J_\nu(x)$, Trans. Amer. Math. Soc., 351 (1999), 2833-2859.  doi: 10.1090/S0002-9947-99-02165-0. [39] T. I. Seidman, How violent are fast controls?, Math. Control Signals Systems, 1 (1988), 89-95.  doi: 10.1007/BF02551238. [40] T. I. Seidman and J. Yong, How violent are fast controls? Ⅱ, Math. Control Signals Systems, 9 (1996), 327-340.  doi: 10.1007/BF01211854. [41] T. I. Seidman, S. A. Avdonin and S. A. Ivanov, The "window problem" for series of complex exponentials, J. Fourier Anal. Appl., 6 (2000), 233-254.  doi: 10.1007/BF02511154. [42] G. Tenenbaum and M. Tucsnak, New blow-up rates for fast controls of Schrödinger and heat equations, J. Differential Equations, 243 (2007), 70-100.  doi: 10.1016/j.jde.2007.06.019. [43] J. Vancostenoble, Improved Hardy-Poincaré inequalities and sharp Carleman estimates for degenerate/singular parabolic problems, Discr. Cont. Dyn. Syst., 4 (2011), 761-790.  doi: 10.3934/dcdss.2011.4.761. [44] J. Vancostenoble and E. Zuazua, Null controllability for the heat equation with singular inverse-square potentials, J. Funct. Anal., 254 (2008), 1864-1902.  doi: 10.1016/j.jfa.2007.12.015. [45] J. L. Vázquez and E. Zuazua, The hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential, J. Funct. Anal., 173 (2000), 103-153.  doi: 10.1006/jfan.1999.3556. [46] G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, England, The Macmillan Company, New York, 1944
 [1] Brahim Allal, Abdelkarim Hajjaj, Jawad Salhi, Amine Sbai. Boundary controllability for a coupled system of degenerate/singular parabolic equations. Evolution Equations and Control Theory, 2021  doi: 10.3934/eect.2021055 [2] Morteza Fotouhi, Leila Salimi. Controllability results for a class of one dimensional degenerate/singular parabolic equations. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1415-1430. doi: 10.3934/cpaa.2013.12.1415 [3] Hiroshi Watanabe. Solvability of boundary value problems for strongly degenerate parabolic equations with discontinuous coefficients. Discrete and Continuous Dynamical Systems - S, 2014, 7 (1) : 177-189. doi: 10.3934/dcdss.2014.7.177 [4] Brahim Allal, Genni Fragnelli, Jawad Salhi*. Controllability for degenerate/singular parabolic systems involving memory terms. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022071 [5] Luisa Moschini, Guillermo Reyes, Alberto Tesei. Nonuniqueness of solutions to semilinear parabolic equations with singular coefficients. Communications on Pure and Applied Analysis, 2006, 5 (1) : 155-179. doi: 10.3934/cpaa.2006.5.155 [6] Alhabib Moumni, Jawad Salhi. Exact controllability for a degenerate and singular wave equation with moving boundary. Numerical Algebra, Control and Optimization, 2022  doi: 10.3934/naco.2022001 [7] Brahim Allal, Abdelkarim Hajjaj, Lahcen Maniar, Jawad Salhi. Null controllability for singular cascade systems of $n$-coupled degenerate parabolic equations by one control force. Evolution Equations and Control Theory, 2021, 10 (3) : 545-573. doi: 10.3934/eect.2020080 [8] Patrick Martinez, Judith Vancostenoble. The cost of boundary controllability for a parabolic equation with inverse square potential. Evolution Equations and Control Theory, 2019, 8 (2) : 397-422. doi: 10.3934/eect.2019020 [9] Mu-Ming Zhang, Tian-Yuan Xu, Jing-Xue Yin. Controllability properties of degenerate pseudo-parabolic boundary control problems. Mathematical Control and Related Fields, 2020, 10 (1) : 157-169. doi: 10.3934/mcrf.2019034 [10] Lahcen Maniar, Martin Meyries, Roland Schnaubelt. Null controllability for parabolic equations with dynamic boundary conditions. Evolution Equations and Control Theory, 2017, 6 (3) : 381-407. doi: 10.3934/eect.2017020 [11] Hiroshi Watanabe. Existence and uniqueness of entropy solutions to strongly degenerate parabolic equations with discontinuous coefficients. Conference Publications, 2013, 2013 (special) : 781-790. doi: 10.3934/proc.2013.2013.781 [12] Kenneth Hvistendahl Karlsen, Nils Henrik Risebro. On the uniqueness and stability of entropy solutions of nonlinear degenerate parabolic equations with rough coefficients. Discrete and Continuous Dynamical Systems, 2003, 9 (5) : 1081-1104. doi: 10.3934/dcds.2003.9.1081 [13] J. Carmelo Flores, Luz De Teresa. Null controllability of one dimensional degenerate parabolic equations with first order terms. Discrete and Continuous Dynamical Systems - B, 2020, 25 (10) : 3963-3981. doi: 10.3934/dcdsb.2020136 [14] El Mustapha Ait Ben Hassi, Mohamed Fadili, Lahcen Maniar. Controllability of a system of degenerate parabolic equations with non-diagonalizable diffusion matrix. Mathematical Control and Related Fields, 2020, 10 (3) : 623-642. doi: 10.3934/mcrf.2020013 [15] Fengyan Yang. Exact boundary null controllability for a coupled system of plate equations with variable coefficients. Evolution Equations and Control Theory, 2021  doi: 10.3934/eect.2021036 [16] Genni Fragnelli, Dimitri Mugnai. Singular parabolic equations with interior degeneracy and non smooth coefficients: The Neumann case. Discrete and Continuous Dynamical Systems - S, 2020, 13 (5) : 1495-1511. doi: 10.3934/dcdss.2020084 [17] Lin Yan, Bin Wu. Null controllability for a class of stochastic singular parabolic equations with the convection term. Discrete and Continuous Dynamical Systems - B, 2022, 27 (6) : 3213-3240. doi: 10.3934/dcdsb.2021182 [18] Mounim El Ouardy, Youssef El Hadfi, Aziz Ifzarne. Existence and regularity results for a singular parabolic equations with degenerate coercivity. Discrete and Continuous Dynamical Systems - S, 2022, 15 (1) : 117-141. doi: 10.3934/dcdss.2021012 [19] Umberto Biccari. Boundary controllability for a one-dimensional heat equation with a singular inverse-square potential. Mathematical Control and Related Fields, 2019, 9 (1) : 191-219. doi: 10.3934/mcrf.2019011 [20] Françoise Demengel, O. Goubet. Existence of boundary blow up solutions for singular or degenerate fully nonlinear equations. Communications on Pure and Applied Analysis, 2013, 12 (2) : 621-645. doi: 10.3934/cpaa.2013.12.621

2021 Impact Factor: 1.141