## Section: New Results

### Differential and partial differential equations with delay

#### Stability and Hopf bifurcation for a first-order linear delay differential equation with distributed delay

Participant : Fabien Crauste.

F. Crauste published in Complex-Time Delay Systems (Ed F. Atay, Springer), a chapter on the stability and the existence of a Hopf bifurcation for delay differential equations with distributed delay [19] . This class of equations is widely used in many research fields such as automatic, economic, and, for our purpose, in biological modelling because it can be associated with problems in which it is important to take into account some history of the state variable (e.g., gestation period, cell cycle durations or incubation time). When few data are available, this history is usually assumed to be discrete (so one gets a discrete delay equation, well studied in the literature). Yet, in most cases very few is known about it, and how it can be distributed, so very abstract assumptions lead to equations with distributed delay. The paper focused on stability properties of such equations, that is under which conditions on the parameters do all solutions converge toward zero? And, as a consequence, how is it possible to destabilize the equation? Can oscillating or periodic solutions appear? All these questions arise from needs to understand how many systems can be destabilized, or how can they stay stable for a long time.

In this chapter, F. Crauste presented a state of the art on the topic and the most recent advances in the stability analysis of differential equations with distributed delay. It is noticeable that only partial results have been proved up to now. Mainly, only sufficient conditions for the stability - which is sometimes enough - have been obtained.

**Keywords:** *time-delay systems, asymptotic stability, delay-dependent
characteristic equation, stability switch, Hopf bifurcation*.

#### Extrapolation spaces and partial neutral functional Differential Equations with infinite delay

Participant : Mostafa Adimy.

*In collaboration with M. Alia and K. Ezzinbi (University of Marrakech, Morocco)*.

We studied in [4] the existence regularity and stability of solutions for some nonlinear partial functional differential equations with infinite delay. We supposed that the linear term was a Hille-Yosida operator on a Banach space and the nonlinear function took its values on some spaces larger than the initial Banach space, namely the extrapolated Favard class corresponding to the semigroup generated by the linear part. Our approach was based on the theory of the extrapolation. We gave also a linearized stability principle to study the behavior of solutions near the equilibriums of the model.

**Keywords:** *partial functional differential
equations, infinite delay, extrapolation spaces, semigroup,
linearized stability*.