Eventual monotonicity and convergence to travelling fronts for the solutions of parabolic equations in cylinders
A nonlinear oblique derivative boundary value problem for the heat equation. Part 1 : basic results
A nonlinear oblique derivative boundary value problem for the heat equation. Part 2 : singular self-similar solutions
A nonlinear oblique derivative boundary value problem for the heat equation : analogy with the porous medium equation
Numerical simulation of a point-source initiated flame ball with heat losses
This article is devoted to the numerical study of a flame ball model, derived by Joulin, which obeys to a singular integro-differential equation. The numerical scheme that we analyze here, is based upon a one step method, and we are interested in its long-time behaviour. We recover the same dynamics as in the continuous case: quenching, or stabilization of the flame, depending on heat losses, and an energy input parameter.
Stability of travelling waves in a model for conical flames in two space dimensions
Travelling graphs for the forced mean curvature motion in an arbitrary space dimension
We construct travelling wave graphs of the form , , , solutions to the -dimensional forced mean curvature motion () with prescribed asymptotics. For any -homogeneous function , viscosity solution to the eikonal equation , we exhibit a smooth concave solution to the forced mean curvature motion whose asymptotics is driven by . We also describe in terms of a probability measure on .
Speed-up of reaction-diffusion fronts by a line of fast diffusion
In these notes, we discuss a new model, proposed by H. Berestycki, J.-M. Roquejoffre and L. Rossi, to describe biological invasions in the plane when a strong diffusion takes place on a line. This model seems relevant to account for the effects of roads on the spreading of invasive species. In what follows, the diffusion on the line will either be modelled by the Laplacian operator, or the fractional Laplacian of order less than 1. Of interest to us is the asymptotic speed of spreading in the direction...
Existence of pulsating waves in a model of flames in sprays
A one-dimensional system describing the propagation of low Mach number flames in sprays is studied. We show that pulsating waves may exist when the droplet distribution in the unburnt region is spatially periodic. The range of possible propagation speeds may be either bounded or unbounded, depending on the threshold temperatures of the burning and vaporization rates.
Variational problems with free boundaries for the fractional Laplacian
We discuss properties (optimal regularity, nondegeneracy, smoothness of the free boundary etc.) of a variational interface problem involving the fractional Laplacian; due to the nonlocality of the Dirichlet problem, the task is nontrivial. This difficulty is bypassed by an extension formula, discovered by the first author and Silvestre, which reduces the study to that of a codimension 2 (degenerate) free boundary.
The logarithmic delay of KPP fronts in a periodic medium
We extend, to parabolic equations of the KPP type in periodic media, a result of Bramson which asserts that, in the case of a spatially homogeneous reaction rate, the time lag between the position of an initially compactly supported solution and that of a traveling wave grows logarithmically in time.
Numerical simulation of a point-source initiated flame ball with heat losses
This article is devoted to the numerical study of a flame ball model, derived by Joulin, which obeys to a singular integro-differential equation. The numerical scheme that we analyze here, is based upon a one step method, and we are interested in its long-time behaviour. We recover the same dynamics as in the continuous case: quenching, or stabilization of the flame, depending on heat losses, and an energy input parameter.
Traveling waves with paraboloid like interfaces for balanced bistable dynamics
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