DISCRETE AND CONTINUOUS doi:10.3934/dcdss.2020084 DYNAMICAL SYSTEMS SERIES S Volume 13, Number 5, May 2020 pp. 1495–1511 SINGULAR PARABOLIC EQUATIONS WITH INTERIOR DEGENERACY AND NON SMOOTH COEFFICIENTS: THE NEUMANN CASE Genni Fragnelli Dipartimento di Matematica Università di Bari “Aldo Moro” Via E. Orabona 4, 70125 Bari, Italy Dimitri Mugnai ∗ Dipartimento di Scienze Ecologiche e Biologiche Università della Tuscia Largo dell’Università, 01100 Viterbo, Italy To Angelo on the occasion of his 70th birthday, with esteem Abstract. We establish Hardy - Poincaré and Carleman estimates for non- smooth degenerate/singular parabolic operators in divergence form with Neu- mann 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. 1. Introduction. This paper deals with a class of degenerate and singular para- bolic operators with interior degeneracy and singularity of the form u t − (a(x)u x ) x − λ b(x) u, associated to Neumann boundary conditions and with (t,x) ∈ Q T := (0,T ) × (0, 1), T> 0 being a fxed number. Here λ ∈ R satisfes suitable assumptions and the functions a and b, that can be non-smooth, degenerate at the same interior point x 0 ∈ (0, 1). The fact that both a and b degenerate at the same point is actually the most complicated situation. Indeed, if a and b degenerated at diferent points, we could separate the problem in a purely singular one and in a purely degenerate 2010 Mathematics Subject Classifcation. Primary: 35Q93; Secondary: 93B05, 34H05, 35A23. Key words and phrases. Carleman estimates, singular/degenerate equations, non smooth coefcients, Hardy–Poincaré inequality, Caccioppoli inequality, observability inequality, null controllability. The frst 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 fnanziamento 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 Diferenziali Non Lin- eari. 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 fnanziamento delle attività base di ricerca” 2017. * Corresponding author: Dimitri Mugnai. 1495