Speaker: Joel E. Restrepo, Center for Research and Advanced Studies, Av. IPN 2508, 07360, Mexico City, Mexico

Time : 18:00 - 19:00 CEST (Rome/Paris)

Hosted at: SISSA, International School of Advanced Studies, Trieste, Italy

Zoom : A Zoom link will appear here, an hour before the talk

Organizers : Pavan Pranjivan Mehta* (pavan.mehta@sissa.it) and Arran Fernandez** (arran.fernandez@emu.edu.tr)

* SISSA, International School of Advanced Studies, Italy

** Eastern Mediterranean University, Northern Cyprus

Keywords: almost sectorial operators, abstract Volterra equations of wave type, linear and semilinear partial integro-differential equations, nonlocal operators, bounded and unbounded operators, non-dense domains or range

Abstract: We discuss some open problems on the regularity of the solutions for semilinear, linear and homogeneous abstract equations with almost sectorial operators (in space) and non-local operators (in time). We study the classical solutions for those equations in the case of the non-local operator being of wave-type. Note that the solution operators in the heat-type case could be rewritten by the classical C₀ semigroup which estimates are in many cases known and even sharp [1]. While, in the wave-type, the latter strategy is not possible, and it gives an extra difficulty to handle the problem mentioned. From 2012, there is not any advance with respect to this direction since the many obstacles that needed to be avoided. The main difficulties were the construction of the solutions operators and how to prove directly the classical solutions. Nevertheless, in this talk, we show that using a functional calculus for almost sectorial operators [2] (as an alternative way) can help to avoid many of those troubles in the study of these equations [3]

Biography: Joel E. Restrepo is a Colombian mathematician and is a researcher in the Department of Mathematics of Center for Research and Advanced Studies of the National Polytechnic Institute, Mexico City, Mexico. Previously, he was a postdoctoral researcher at different research centers in Belgium, Russia and Kazakhstan. He has carried out mathematical research in a variety of topics connecting different areas of mathematics. His research links several techniques in complex analysis, operator theory, harmonic analysis, PDEs, etc. He also has participated in many international conferences, seminars and other related events and has been involved in the organisation of several of these scientific projects as well.

Bibliography

[1] R-N. Wang, D-H. Chen, T-J. Xiao. Abstract fractional Cauchy problems with almost sectorial operators. J. Differ. Equ. 252(1), (2012), 202–235.
[2] F. Periago, B. Straub. A functional calculus for almost sectorial operators and applications to abstract evolution equations. J. Evol. Equ., 2, (2002), 41–68.
[3] J.E. Restrepo. Abstract Volterra integral equations of wave type with almost sectorial operators. https://doi.org/10.48550/arXiv.2509.05229, (2025)