Partial Differential Equations & Ordinary Differential Equations: Theory and Applications

Abstract

In our presentation, we spotlight the theory of C0 semigroups as a fundamental tool for analyzing evolution equations in Banach spaces. We begin by reviewing the general properties of C0 semigroups and their role in describing the time evolution of systems governed by linear operators. The discussion then moves to the abstract Cauchy problem, emphasizing conditions for the existence, uniqueness, and regularity of solutions using semigroup methods. We extend these ideas to nonlinear evolution equations, highlighting techniques that address the challenges posed by nonlinearity. Special attention is given to delay differential equations (DFEs), where the system’s future state depends on its history. After providing an overview of delayed systems, we discuss two major cases: abstract linear delayed equations and semi-linear delayed equations, showing how semigroup theory adapts to account for memory effectsL’objectif de cet exposé est de prouver l’existence et l’unicité d’une solution d’un problème de Goursat non linéaire dans la classe des fonctions quasi-analytiques de type Denjoy- Carleman, plus précisément dans l’ensemble des fonctions continues Denjoy-Carleman. L’idée est de transformer le problème integro-diferentiel à un problème de point fixe appliqué dans une boule fermée dans une algèbre de Banach définie par une série formelle et une suite numérique logarithméquement convexe convenablement choisieA problem of initial value problem for a nonlinear Caputo fractional differential equation on an unbounded interval is considered. Based on some fractional calculus and the Krasnoselskii’s fixed point theorem, we prove our main results (existence and asymptotic stability). Then we give an example to illustrate our study.s..It is our pleasure to cordially invite all professors and students to participate in an Open Study Day dedicated to

Ordinary Differential Equations (ODE) & Partial Differential Equations (PDE).