May  2020, 13(5): 1495-1511. doi: 10.3934/dcdss.2020084

Singular parabolic equations with interior degeneracy and non smooth coefficients: The Neumann case

1. 

Dipartimento di Matematica, Università di Bari "Aldo Moro", Via E. Orabona 4, 70125 Bari, Italy

2. 

Dipartimento di Scienze Ecologiche e Biologiche, Università della Tuscia, Largo dell'Università, 01100 Viterbo, Italy

* Corresponding author: Dimitri Mugnai

To Angelo on the occasion of his 70th birthday, with esteem

Received  March 2018 Revised  November 2018 Published  June 2019

Fund Project: The first author is a member of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). She is supported by the GNAMPA project 2017 Comportamento asintotico e controllo di equazioni di evoluzione non lineari and by the FFABR "Fondo per il finanziamento delle attività base di ricerca" 2017. The second author is a member of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM) and supported by the 2017 INdAM-GNAMPA Project Equazioni Differenziali Non Lineari. He is also supported by the Italian MIUR project Variational methods, with applications to problems in mathematical physics and geometry (2015KB9WPT_009) and by the FFABR "Fondo per il finanziamento delle attività base di ricerca" 2017

We establish Hardy - Poincaré and Carleman estimates for non-smooth degenerate/singular parabolic operators in divergence form with Neumann boundary conditions. The degeneracy and the singularity occur both in the interior of the spatial domain. We apply these inequalities to deduce well-posedness and null controllability for the associated evolution problem.

Citation: Genni Fragnelli, Dimitri Mugnai. Singular parabolic equations with interior degeneracy and non smooth coefficients: The Neumann case. Discrete & Continuous Dynamical Systems - S, 2020, 13 (5) : 1495-1511. doi: 10.3934/dcdss.2020084
References:
[1]

I. BoutaayamouG. Fragnelli and L. Maniar, Carleman estimates for parabolic equations with interior degeneracy and Neumann boundary conditions, J. Anal. Math., 135 (2018), 1-35.  doi: 10.1007/s11854-018-0030-2.  Google Scholar

[2]

P. CannarsaP. Martinez and J. Vancostenoble, Null controllability of the degenerate heat equations, Adv. Differential Equations, 10 (2005), 153-190.   Google Scholar

[3] T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, Clarendon Press, Oxford, 1998.   Google Scholar
[4]

G. Fragnelli, Interior degenerate/singular parabolic equations in nondivergence form: well-posedness and Carleman estimates, J. Differential Equations, 260 (2016), 1314-1371.  doi: 10.1016/j.jde.2015.09.019.  Google Scholar

[5]

G. Fragnelli and D. Mugnai, Carleman estimates and observability inequalities for parabolic equations with interior degeneracy, Advances in Nonlinear Analysis, 2 (2013), 339-378.  doi: 10.1515/anona-2013-0015.  Google Scholar

[6]

G. Fragnelli and D. Mugnai, Carleman estimates, observability inequalities and null controllability for interior degenerate non smooth parabolic equations, Memoirs Amer. Math. Soc., 242 (2016), v+84pp. doi: 10.1090/memo/1146.  Google Scholar

[7]

G. Fragnelli and D. Mugnai, Carleman estimates for singular parabolic equations with interior degeneracy and non smooth coefficients, Adv. Nonlinear Anal., 6 (2017), 61-84.  doi: 10.1515/anona-2015-0163.  Google Scholar

[8]

G. Fragnelli and D. Mugnai, Controllability of strongly degenerate parabolic problems with strongly singular potentials, Electron. J. Qual. Theory Differ. Equ., 2018 (2018), Paper No. 50, 11 pp.  Google Scholar

[9]

J. Le Rousseau and G. Lebeau, On carleman estimates for elliptic and parabolic operators. applications to unique continuation and control of parabolic equations, ESAIM Control Optim. Calc. Var., 18 (2012), 712-747.  doi: 10.1051/cocv/2011168.  Google Scholar

[10]

J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Vol. Ⅰ. Grundlehren Math. Wiss. 181. Springer-Verlag, New York-Heidelberg, 1972.  Google Scholar

[11]

S. Micu and E. Zuazua, On the lack of null controllability of the heat equation on the half-line, Trans. Amer. Math. Soc., 353 (2001), 1635-1659.  doi: 10.1090/S0002-9947-00-02665-9.  Google Scholar

[12]

M. Renardy and R. C. Rogers, An Introduction to Partial Differential Equations, Second edition., Texts in Applied Mathematics, 13. Springer-Verlag, New York, 2004.  Google Scholar

[13]

M. Yamamoto, Carleman estimates for parabolic equations and applications, Inverse Problems, 25 (2009), 123013 (75pp). doi: 10.1088/0266-5611/25/12/123013.  Google Scholar

show all references

References:
[1]

I. BoutaayamouG. Fragnelli and L. Maniar, Carleman estimates for parabolic equations with interior degeneracy and Neumann boundary conditions, J. Anal. Math., 135 (2018), 1-35.  doi: 10.1007/s11854-018-0030-2.  Google Scholar

[2]

P. CannarsaP. Martinez and J. Vancostenoble, Null controllability of the degenerate heat equations, Adv. Differential Equations, 10 (2005), 153-190.   Google Scholar

[3] T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, Clarendon Press, Oxford, 1998.   Google Scholar
[4]

G. Fragnelli, Interior degenerate/singular parabolic equations in nondivergence form: well-posedness and Carleman estimates, J. Differential Equations, 260 (2016), 1314-1371.  doi: 10.1016/j.jde.2015.09.019.  Google Scholar

[5]

G. Fragnelli and D. Mugnai, Carleman estimates and observability inequalities for parabolic equations with interior degeneracy, Advances in Nonlinear Analysis, 2 (2013), 339-378.  doi: 10.1515/anona-2013-0015.  Google Scholar

[6]

G. Fragnelli and D. Mugnai, Carleman estimates, observability inequalities and null controllability for interior degenerate non smooth parabolic equations, Memoirs Amer. Math. Soc., 242 (2016), v+84pp. doi: 10.1090/memo/1146.  Google Scholar

[7]

G. Fragnelli and D. Mugnai, Carleman estimates for singular parabolic equations with interior degeneracy and non smooth coefficients, Adv. Nonlinear Anal., 6 (2017), 61-84.  doi: 10.1515/anona-2015-0163.  Google Scholar

[8]

G. Fragnelli and D. Mugnai, Controllability of strongly degenerate parabolic problems with strongly singular potentials, Electron. J. Qual. Theory Differ. Equ., 2018 (2018), Paper No. 50, 11 pp.  Google Scholar

[9]

J. Le Rousseau and G. Lebeau, On carleman estimates for elliptic and parabolic operators. applications to unique continuation and control of parabolic equations, ESAIM Control Optim. Calc. Var., 18 (2012), 712-747.  doi: 10.1051/cocv/2011168.  Google Scholar

[10]

J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Vol. Ⅰ. Grundlehren Math. Wiss. 181. Springer-Verlag, New York-Heidelberg, 1972.  Google Scholar

[11]

S. Micu and E. Zuazua, On the lack of null controllability of the heat equation on the half-line, Trans. Amer. Math. Soc., 353 (2001), 1635-1659.  doi: 10.1090/S0002-9947-00-02665-9.  Google Scholar

[12]

M. Renardy and R. C. Rogers, An Introduction to Partial Differential Equations, Second edition., Texts in Applied Mathematics, 13. Springer-Verlag, New York, 2004.  Google Scholar

[13]

M. Yamamoto, Carleman estimates for parabolic equations and applications, Inverse Problems, 25 (2009), 123013 (75pp). doi: 10.1088/0266-5611/25/12/123013.  Google Scholar

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