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December  2020, 13(12): 3417-3426. doi: 10.3934/dcdss.2020128

## Operators of order 2$n$ with interior degeneracy

 1 Department of Mathematics, University of Bari Aldo Moro, Via E. Orabona 4, 70125 Bari, Italy 2 Department of Mathematical Sciences, University of Memphis, 373 Dunn Hall, Memphis, TN 38152-3240, USA

* Corresponding author: Rosa Maria Mininni

Dedicated to Gisèle Ruiz Goldstein, outstanding mathematician, with great admiration and friendship on her 60th birthday

Received  March 2019 Published  December 2020 Early access  November 2019

We consider a differential operator of order 2$n$ of the type $A_n u = (-1)^n (a u^{(n)})^{(n)}$, where $a(x)>0$ in $[0, 1]\setminus\{x_0\}$ and $a(x_0) = 0$. We show that, for any $n\in{\mathbb{N}}$, the operator $-A_n$ generates a contractive analytic semigroup of angle $\pi/2$ on $L^2 (0, 1)$. Note that the domain of $A_n$ depends on the type of degeneracy of $a$. Our theorems extend some previous results in [3] where $n = 1$.

Citation: Genni Fragnelli, Jerome A. Goldstein, Rosa Maria Mininni, Silvia Romanelli. Operators of order 2$n$ with interior degeneracy. Discrete and Continuous Dynamical Systems - S, 2020, 13 (12) : 3417-3426. doi: 10.3934/dcdss.2020128
##### References:
 [1] I. Boutaayamou, G. 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. [2] P. Cannarsa, P. Martinez and J. Vancostenoble, Null controllability of the degenerate heat equations, Adv. Diff. Equations, 10 (2005), 153-190. [3] G. Fragnelli, G. Ruiz Goldstein, J. A. Goldstein and S. Romanelli, Generators with interior degeneracy on spaces of L2 type, Electron. J. Differ. Equations, 2012 (2012), 30 pp. [4] G. Fragnelli, G. Ruiz Goldstein, J. A. Goldstein, R. M. Mininni and S. Romanelli, Generalized Wentzell boundary conditions for second order operators with interior degeneracy, Discrete Cont. Dyn. Systems-S, 9 (2016), 697-715.  doi: 10.3934/dcdss.2016023. [5] G. Fragnelli, G. Marinoschi, R. M. Mininni and S. Romanelli, Identification of a diffusion coefficient in strongly degenerate parabolic equations with interior degeneracy, J. Evol. Equ., 15 (2015), 27-51.  doi: 10.1007/s00028-014-0247-1. [6] G. Fragnelli, G. Marinoschi, R. M. Mininni and S. Romanelli, A control approach for an identification problem associated to a strongly degenerate parabolic system with interior degeneracy, in New Prospects in Direct, Inverse and Control Problems for Evolution Equations (eds. A. Favini, G. Fragnelli, R. M. Mininni), Springer INdAM Series 10 (2014), 121-139. doi: 10.1007/978-3-319-11406-4_7. [7] G. Fragnelli and D. Mugnai, Carleman estimates and observability inequalities for parabolic equations with interior degeneracy, Adv. Nonlinear Anal., 2 (2013), 339-378.  doi: 10.1515/anona-2013-0015. [8] G. Fragnelli and D. Mugnai, Carleman estimates, observability inequalities and null controllability for interior degenerate non smooth parabolic equations, Mem. Amer. Math. Soc., 242 (2016). doi: 10.1090/memo/1146. [9] G. Fragnelli and D. Mugnai, Corrigendum and improvements to "Carleman estimates, observability inequalities and null controllability for interior degenerate non smooth parabolic equations", and its consequences, Mem. Amer. Math. Soc., to appear [10] J. A. Goldstein, Semigroups of Linear Operators and Applications, 2nd edition, Dover Publications, Mineola, New York, 2017.

show all references

##### References:
 [1] I. Boutaayamou, G. 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. [2] P. Cannarsa, P. Martinez and J. Vancostenoble, Null controllability of the degenerate heat equations, Adv. Diff. Equations, 10 (2005), 153-190. [3] G. Fragnelli, G. Ruiz Goldstein, J. A. Goldstein and S. Romanelli, Generators with interior degeneracy on spaces of L2 type, Electron. J. Differ. Equations, 2012 (2012), 30 pp. [4] G. Fragnelli, G. Ruiz Goldstein, J. A. Goldstein, R. M. Mininni and S. Romanelli, Generalized Wentzell boundary conditions for second order operators with interior degeneracy, Discrete Cont. Dyn. Systems-S, 9 (2016), 697-715.  doi: 10.3934/dcdss.2016023. [5] G. Fragnelli, G. Marinoschi, R. M. Mininni and S. Romanelli, Identification of a diffusion coefficient in strongly degenerate parabolic equations with interior degeneracy, J. Evol. Equ., 15 (2015), 27-51.  doi: 10.1007/s00028-014-0247-1. [6] G. Fragnelli, G. Marinoschi, R. M. Mininni and S. Romanelli, A control approach for an identification problem associated to a strongly degenerate parabolic system with interior degeneracy, in New Prospects in Direct, Inverse and Control Problems for Evolution Equations (eds. A. Favini, G. Fragnelli, R. M. Mininni), Springer INdAM Series 10 (2014), 121-139. doi: 10.1007/978-3-319-11406-4_7. [7] G. Fragnelli and D. Mugnai, Carleman estimates and observability inequalities for parabolic equations with interior degeneracy, Adv. Nonlinear Anal., 2 (2013), 339-378.  doi: 10.1515/anona-2013-0015. [8] G. Fragnelli and D. Mugnai, Carleman estimates, observability inequalities and null controllability for interior degenerate non smooth parabolic equations, Mem. Amer. Math. Soc., 242 (2016). doi: 10.1090/memo/1146. [9] G. Fragnelli and D. Mugnai, Corrigendum and improvements to "Carleman estimates, observability inequalities and null controllability for interior degenerate non smooth parabolic equations", and its consequences, Mem. Amer. Math. Soc., to appear [10] J. A. Goldstein, Semigroups of Linear Operators and Applications, 2nd edition, Dover Publications, Mineola, New York, 2017.
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