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

Genni Fragnelli; Dimitri Mugnai

doi:10.3934/dcdss.2020084

Article Contents

2020, Volume 13, Issue 5: 1495-1511. Doi: 10.3934/dcdss.2020084

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

Genni Fragnelli^1, and
Dimitri Mugnai^2, ,

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
^* Corresponding author: Dimitri Mugnai

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

Received: February 28, 2018

Revised: October 31, 2018

Published: March 2020

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

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Abstract

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.

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Mathematics Subject Classification: Primary: 35Q93; Secondary: 93B05, 34H05, 35A23.

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References

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