On the basis of the Carleman estimate for the parabolic equation, we prove a Carleman estimate for the integro-differential operator $\partial_t-\triangle+\int_0^t K(x,t,r)\triangle\ dr$ where the integral kernel has a behaviour like a weakly singular one. In the proof we consider the integral term as a perturbation. The crucial point is a special choice of the time factor of the weight function.

Carleman estimates for integro-differential parabolic equations with singular memory kernels / Loreti, Paola; Sforza, Daniela; Yamamoto, Masahiro. - In: JOURNAL OF ELLIPTIC AND PARABOLIC EQUATIONS. - ISSN 2296-9020. - ELETTRONICO. - 3:1-2(2017), pp. 53-64. [10.1007/s41808-017-0004-z]

### Carleman estimates for integro-differential parabolic equations with singular memory kernels

#### Abstract

On the basis of the Carleman estimate for the parabolic equation, we prove a Carleman estimate for the integro-differential operator $\partial_t-\triangle+\int_0^t K(x,t,r)\triangle\ dr$ where the integral kernel has a behaviour like a weakly singular one. In the proof we consider the integral term as a perturbation. The crucial point is a special choice of the time factor of the weight function.
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parabolic equations; integro-differential equations; fading memory; Carleman estimate
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Carleman estimates for integro-differential parabolic equations with singular memory kernels / Loreti, Paola; Sforza, Daniela; Yamamoto, Masahiro. - In: JOURNAL OF ELLIPTIC AND PARABOLIC EQUATIONS. - ISSN 2296-9020. - ELETTRONICO. - 3:1-2(2017), pp. 53-64. [10.1007/s41808-017-0004-z]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/1114888