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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  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 & Continuous Dynamical Systems - S, 2020, 13 (12) : 3417-3426. doi: 10.3934/dcdss.2020128
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