Jean-Michel Roquejoffre

Variational problems with free boundaries for the fractional Laplacian

We discuss properties (optimal regularity, non-degeneracy, smoothness of the free boundary...) of a variational interface problem involving the fractional Laplacian; Due to the non-locality of the Dirichlet problem, the task is nontrivial. This difficulty is by-passed by an extension formula, discovered by the first author and Silvestre, which reduces the study to that of a co-dimension 2 (degenerate) free boundary.

Eventual monotonicity and convergence to travelling fronts for the solutions of parabolic equations in cylinders

The paper is concerned with the long-time behaviour of the solutions of a certain class of semilinear parabolic equations in cylinders, which contains as a particular case the multidimensional thermo-diffusive model in combustion theory. We prove, under minimal conditions on the initial values, that the solutions eventually become monotone in the direction of the axis of the cylinder on every compact subset; this implies convergence to travelling fronts. This result is applied to propagation versus extinction problems: given a compactly supported initial datum, sufficient conditions ensuring that the solution will either converge to 0 or to a pair of travelling fronts are given. Additional information on the corresponding equations in finite cylinders is also obtained.

Convergence to steady states or periodic solutions in a class of Hamilton-Jacobi equations

Abstract The purpose of this paper is to prove long-time behaviour results for Hamilton–Jacobi equations. For autonomous equations, we give an alternative proof of a convergence theorem obtained by A. Fathi when the equations are posed on a manifold, then extend it to Dirichlet boundary conditions on an open subset. When the equations are time-periodic we prove the convergence in several nontrivial special cases.

Existence and Non-Existence of Fisher-KPP Transition Fronts

We consider Fisher-KPP-type reaction–diffusion equations with spatially inhomogeneous reaction rates. We show that a sufficiently strong localized inhomogeneity may prevent existence of transition-front-type global-in-time solutions while creating a global-in-time bump-like solution. This is the first example of a medium in which no reaction–diffusion transition front exists. A weaker localized inhomogeneity leads to the existence of transition fronts, but only in a finite range of speeds. These results are in contrast with both Fisher-KPP reactions in homogeneous media as well as ignition-type reactions in inhomogeneous media.

The Influence of Fractional Diffusion in Fisher-KPP Equations

We study the Fisher-KPP equation where the Laplacian is replaced by the generator of a Feller semigroup with power decaying kernel, an important example being the fractional Laplacian. In contrast with the case of the standard Laplacian where the stable state invades the unstable one at constant speed, we prove that with fractional diffusion, generated for instance by a stable Lévy process, the front position is exponential in time. Our results provide a mathematically rigorous justification of numerous heuristics about this model.

A short proof of the logarithmic Bramson correction in Fisher-KPP equations

In this paper, we explain in simple PDE terms a famous result of Bramson about the loga- rithmic delay of the position of the solutions u(t, x) of Fisher-KPP reaction-diffusion equations in R, with respect to the position of the travelling front with minimal speed. Our proof is based on the comparison of u to the solutions of linearized equations with Dirichlet boundary conditions at the position of the minimal front, with and without the logarithmic delay. Our analysis also yields the large-time convergence of the solutions u along their level sets to the profile of the minimal travelling front.

Stability of Generalized Transition Fronts

We study the qualitative properties of the generalized transition fronts for the reaction–diffusion equations with the spatially inhomogeneous nonlinearity of the ignition type. We show that transition fronts are unique up to translation in time and are globally exponentially stable for the solutions of the Cauchy problem. The results hold for reaction rates that have arbitrary spatial variations provided that the rate is uniformly positive and bounded from above.

Stability of travelling fronts in a model for flame propagation Part II: Nonlinear stability

This paper is concerned with the nonlinear stability of the travelling-wave solutions of the multidimensional thermodiffusive model for flame propagation, with unit Lewis number. The model consists in a semilinear parabolic equation in an infinite cylinder, with Neumann boundary conditions. We prove that any solution which is initially close to a travelling wave will converge to a translate of that wave.

Stability of travelling fronts in a model for flame propagation part I: Linear analysis

We investigate the stability of travelling wave solutions of the multidimensional thermodiffusive model for flame propagation with unit Lewis number. This model consists in a system of two nonlinear parabolic equations posed in an infinite cylinder, with Neumann boundary conditions. In this paper, we prove that every travelling wave solution of this model is linearly stable. Our tools are exponential decay estimates for solutions of elliptic equations in a cylinder, and the Maximum Principle for parabolic equations.

Ergodic Type Problems and Large Time Behaviour of Unbounded Solutions of Hamilton–Jacobi Equations

We study the large time behavior of Lipschitz continuous, possibly unbounded, viscosity solutions of Hamilton–Jacobi Equations in the whole space ℝ N . The associated ergodic problem has Lipschitz continuous solutions if the analogue of the ergodic constant is larger than a minimal value λmin. We obtain various large-time convergence and Liouville type theorems, some of them being of completely new type. We also provide examples showing that, in this unbounded framework, the ergodic behavior may fail, and that the asymptotic behavior may also be unstable with respect to the initial data.