Existence and qualitative properties of global solutions to abstract differential equations with state-dependent delay.

• Grant number: 25/00201-3
• Support Opportunities: Scholarships abroad - Research
• Start date: June 23, 2025
• End date: September 29, 2025
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Analysis
• Principal Investigator: Michelle Fernanda Pierri Hernandez
• Grantee: Michelle Fernanda Pierri Hernandez
• Host Investigator: Jianhong Wu
• Host Institution: Faculdade de Filosofia, Ciências e Letras de Ribeirão Preto (FFCLRP). Universidade de São Paulo (USP). Ribeirão Preto , SP, Brazil
• Institution abroad: York University, Canada

Abstract

This project aims to study the existence and qualitative properties of global solutions of differential equations with state-dependent memory. We will study two specific models of differential equations that are currently being studied by Jianhong Wu. We will have a special interest in studying the existence and uniqueness of global solutions and the existence of solutions with periodicity properties, in particular, quasi-periodic, asymptotically quasi-periodic and S-asymptotically w-periodic solutions. To develop the project, it will be necessary to generalize the ideas and results in:(a) Hernandez, E., Wu, J. Explicit abstract neutral differential equations with state-dependent delay: Existence,uniqueness and local well-posedness. Journal of Differential Equations, v.365, (2023), 750-811.(b) Hernandez, Eduardo., Pierri, Michelle. On Explicit Abstract Neutral Differential Equations with State-Dependent Delay. Applied Mathematics & Optimization, 89 (2024) no. 3, 1-23.(c) Hernandez, E., Michelle Pierri., J. Wu. C^{1+\alpha}-strict solutions and well-posedness of abstract differential equations with state dependent delay. J. Differential Equations 261, (2016) 12, 6856-6882.(d) E., Hernandez, J. Wu. Existence, uniqueness and qualitative properties of global solutions of abstract differential equations with state dependent delay. Proc. Edinb. Math. Soc. (2) 62 (2019), no. 3, 771-788.The project will be developed in collaboration with Jianhong Wu, one of the most important researchers in the area of ¿¿differential equations with memory. Part of my visit to Canada will be financed by Jianhong Wu and this project aims to request additional financial support from Fapesp to cover the entire period of the visit.