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Jean-Michel Roquejoffre

Publications and source records attributed to Jean-Michel Roquejoffre.

45 records · Page 3Linked to original sources
arXiv2012

Traveling Wave Solutions of Advection-Diffusion Equations with Nonlinear Diffusion

We study the existence of particular traveling wave solutions of a nonlinear parabolic degenerate diffusion equation with a shear flow. Under some assumptions we prove that such solutions exist at least for propagation speeds c {\in}]c*, +{\infty}, where c* > 0 is explicitly computed but may not be optimal. We also prove that a free boundary hy- persurface separates a region where u = 0 and a region where u > 0, and that this free boundary can be globally parametrized as a Lipschitz continuous graph under some additional non-degeneracy hypothesis; we investigate solutions which are, in the region u > 0, planar and linear at infinity in the propagation direction, with slope equal to the propagation speed.

math.AP↗
arXiv2011

Transition between linear and exponential propagation in Fisher-KPP type reaction-diffusion equations

We study the Fisher-KPP equation with a fractional laplacian of order α \in (0, 1). We know that the stable state invades the unstable one at constant speed for α = 1, and at an exponential in time velocity for α \in (0, 1). The transition between these two different speeds is examined in this paper. We prove that during a time of the order - ln(1 - α), the propagation is linear and then it is exponential.

math.AP↗
arXiv2011

Travelling graphs for the forced mean curvature motion in an arbitrary space dimension

We construct travelling wave graphs of the form $z=-ct+ϕ(x)$, $ϕ: x \in \mathbb{R}^{N-1} \mapsto ϕ(x)\in \mathbb{R}$, $N \geq 2$, solutions to the $N$-dimensional forced mean curvature motion $V_n=-c_0+κ$ ($c\geq c_0$) with prescribed asymptotics. For any 1-homogeneous function $ϕ_{\infty}$, viscosity solution to the eikonal equation $|Dϕ_{\infty}|=\sqrt{(c/c_0)^2-1}$, we exhibit a smooth concave solution to the forced mean curvature motion whose asymptotics is driven by $ϕ_{\infty}$. We also describe $ϕ_{\infty}$ in terms of a probability measure on $\mathbb{S}^{N-2}$.

math.AP↗
arXiv2010

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 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.

math.AP↗
arXiv2009

Front propagation in Fisher-KPP equations with fractional diffusion

We study in this note the Fisher-KPP equation where the Laplacian is replaced by the generator of a Feller semigroup with slowly decaying kernel, an important example being the fractional Laplacian. Contrary to what happens in the standard Laplacian case, where the stable state invades the unstable one at constant speed, we prove here that invasion holds at an exponential in time velocity. These results provide a mathematically rigorous justification of numerous heuristics about this model.

math.AP↗
arXiv2006

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 $λ_{min}$. 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.

math.AP↗