Romain Joly

Generic Morse–Smale property for the parabolic equation on the circle

Abstract In this paper, we show that, for scalar reaction–diffusion equations u t = u x x + f ( x , u , u x ) on the circle S 1 , the Morse–Smale property is generic with respect to the non-linearity f. In Czaja and Rocha (2008) [13] , Czaja and Rocha have proved that any connecting orbit, which connects two hyperbolic periodic orbits, is transverse and that there does not exist any homoclinic orbit, connecting a hyperbolic periodic orbit to itself. In Joly and Raugel (2010) [31] , we have shown that, generically with respect to the non-linearity f, all the equilibria and periodic orbits are hyperbolic. Here we complete these results by showing that any connecting orbit between two hyperbolic equilibria with distinct Morse indices or between a hyperbolic equilibrium and a hyperbolic periodic orbit is automatically transverse. We also show that, generically with respect to f, there does not exist any connection between equilibria with the same Morse index. The above properties, together with the existence of a compact global attractor and the Poincare–Bendixson property, allow us to deduce that, generically with respect to f, the non-wandering set consists in a finite number of hyperbolic equilibria and periodic orbits. The main tools in the proofs include the lap number property, exponential dichotomies and the Sard–Smale theorem. The proofs also require a careful analysis of the asymptotic behavior of solutions of the linearized equations along the connecting orbits.

Exponential decay for the damped wave equation in unbounded domains

We study the decay of the semigroup generated by the damped wave equation in an unbounded domain. We first prove under the natural geometric control condition the exponential decay of the semigroup. Then we prove under a weaker condition the logarithmic decay of the solutions (assuming that the initial data are smoother). As corollaries, we obtain several extensions of previous results of stabilization and control.

Generic hyperbolicity of equilibria and periodic orbits of the parabolic equation on the circle

In this paper, we show that, for scalar reaction-diffusion equations on the circle S 1 , the property of hyperbolicity of all equilibria and periodic orbits is generic with respect to the non-linearity. In other words, we prove that in an appropriate functional space of non-linear terms in the equation, the set of functions, for which all equilibria and periodic orbits are hyperbolic, is a countable intersection of open dense sets. The main tools in the proof are the property of the lap number and the Sard-Smale Theorem.

Adaptation of the Generic PDE's Results to the Notion of Prevalence ∗

Many generic results have been proved, especially concerning the qualitative behaviour of solutions of partial differential equations. Recently, a new notion of “almost always”, the prevalence, has been developed for vectorial spaces. This notion is interesting since, for example, prevalence sets are equivalent to the full Lebesgue measure sets in finite dimensional spaces. The purpose of this article is to adapt the generic PDE’s results to the notion of prevalence. In particular, we consider the cases where Sard–Smale theorems or arguments of analytic perturbations of the parameters are used.

Asymptotic profiles for a travelling front solution of a biological equation

We are interested in the existence of depolarization waves in the human brain. These waves propagate in the grey matter and are absorbed in the white matter. We consider a two-dimensional model ut = �u + f(u) |y|≤R − �u |y|>R, with f a bistable nonlinearity taking effect only on the domain R × [−R,R], which represents the grey matter layer. We study the existence, the stability and the energy of non-trivial asymptotic profiles of the possible travelling fronts. For this purpose, we present dynamical systems techniques and graphic criteria based on Sturm-Liouville theory and apply them to the above equation. This yields three different behaviours of the solution u after stimulation, depending of the thickness R of the grey matter. This may partly explain the difficulties to observe depolarization waves in the human brain and the failure of several therapeutic trials.

Observation and inverse problems in coupled cell networks

A coupled cell network is a model for many situations such as food webs in ecosystems, cellular metabolism and economic networks. It consists in a directed graph G, each node (or cell) representing an agent of the network and each directed arrow representing which agent acts on which. It yields a system of differential equations , where the component i of f depends only on the cells xj(t) for which the arrow j → i exists in G. In this paper, we investigate the observation problems in coupled cell networks: can one deduce the behaviour of the whole network (oscillations, stabilization, etc) by observing only one of the cells? We show that the natural observation properties hold for almost all the interactions f.

A Note on the Semiglobal Controllability of the Semilinear Wave Equation

We study the internal controllability of the semilinear wave equation

A STRIKING CORRESPONDENCE BETWEEN THE DYNAMICS GENERATED BY THE VECTOR FIELDS AND BY THE SCALAR PARABOLIC EQUATIONS

v_{tt}(x,t)-\Delta v(x,t) + f(x,v(x,t))=\mathbbm{1}_{\omega} u(x,t)

How opening a hole affects the sound of a flute

for some nonlinearities

Global stability of travelling fronts for a damped wave equation with bistable nonlinearity

f