This paper outlines the manner in which Thom’s theory of catastrophes fits into the Hamilton-Jacobi theory of partial differential equations. The representation of solutions of a first order partial differential equation as lagrangian manifolds allows one to study the local structure of their singularities. The structure of generic singularities is closely related to Thom’s concept of the elementary catastrophe associated to a singularity. Three concepts of the stability of a singularity are discussed....

We study the growth rate of a cell population that follows an age-structured PDE with time-periodic coefficients. Our motivation comes from the comparison between experimental tumor growth curves in mice endowed with intact or disrupted circadian clocks, known to exert their influence on the cell division cycle. We compare the growth rate of the model controlled by a time-periodic control on its coefficients with the growth rate of stationary models of the same nature, but with averaged coefficients....

Let (x,y,z) ∈ ℂ³. In this paper we shall study the solvability of singular first order partial differential equations of nilpotent type by the following typical example: $Pu(x,y,z):=(y{\partial }_x-z{\partial }_y)u(x,y,z)=f(x,y,z){\in }_{x,y,z}$, where $P=y{\partial }_x-z{\partial }_y{:}_{x,y,z}{\to }_{x,y,z}$. For this equation, our aim is to characterize the solvability on ${}_{x,y,z}$ by using the Im P, Coker P and Ker P, and we give the exact forms of these sets.