717 research outputs found
Robustness for a Liouville type theorem in exterior domains
We are interested in the robustness of a Liouville type theorem for a
reaction diffusion equation in exterior domains. Indeed H. Berestycki, F. Hamel
and H. Matano (2009) proved such a result as soon as the domain satisfies some
geometric properties. We investigate here whether their result holds for
perturbations of the domain. We prove that as soon as our perturbation is close
to the initial domain in the topology the result remains true
while it does not if the perturbation is not smooth enough
KPP reaction-diffusion equations with a non-linear loss inside a cylinder
We consider in this paper a reaction-diffusion system in presence of a flow
and under a KPP hypothesis. While the case of a single-equation has been
extensively studied since the pioneering Kolmogorov-Petrovski-Piskunov paper,
the study of the corresponding system with a Lewis number not equal to 1 is
still quite open. Here, we will prove some results about the existence of
travelling fronts and generalized travelling fronts solutions of such a system
with the presence of a non-linear spacedependent loss term inside the domain.
In particular, we will point out the existence of a minimal speed, above which
any real value is an admissible speed. We will also give some spreading results
for initial conditions decaying exponentially at infinity
Asymptotic Implied Volatility at the Second Order with Application to the SABR Model
We provide a general method to compute a Taylor expansion in time of implied
volatility for stochastic volatility models, using a heat kernel expansion.
Beyond the order 0 implied volatility which is already known, we compute the
first order correction exactly at all strikes from the scalar coefficient of
the heat kernel expansion. Furthermore, the first correction in the heat kernel
expansion gives the second order correction for implied volatility, which we
also give exactly at all strikes. As an application, we compute this asymptotic
expansion at order 2 for the SABR model.Comment: 27 pages; v2: typos fixed and a few notation changes; v3: published
version, typos fixed and comments added. in Large Deviations and Asymptotic
Methods in Finance, Springer (2015) 37-6
Solitary wave dynamics in time-dependent potentials
We rigorously study the long time dynamics of solitary wave solutions of the
nonlinear Schr\"odinger equation in {\it time-dependent} external potentials.
To set the stage, we first establish the well-posedness of the Cauchy problem
for a generalized nonautonomous nonlinear Schr\"odinger equation. We then show
that in the {\it space-adiabatic} regime where the external potential varies
slowly in space compared to the size of the soliton, the dynamics of the center
of the soliton is described by Hamilton's equations, plus terms due to
radiation damping. We finally remark on two physical applications of our
analysis. The first is adiabatic transportation of solitons, and the second is
Mathieu instability of trapped solitons due to time-periodic perturbations.Comment: 38 pages, some typos corrected, one reference added, one remark adde
Dynamics of soliton-like solutions for slowly varying, generalized gKdV equations: refraction vs. reflection
In this work we continue the description of soliton-like solutions of some
slowly varying, subcritical gKdV equations.
In this opportunity we describe, almost completely, the allowed behaviors:
either the soliton is refracted, or it is reflected by the potential, depending
on its initial energy. This last result describes a new type of soliton-like
solution for gKdV equations, also present in the NLS case.
Moreover, we prove that the solution is not pure at infinity, unlike the
standard gKdV soliton.Comment: 51 pages, submitte
On a functional satisfying a weak Palais-Smale condition
In this paper we study a quasilinear elliptic problem whose functional
satisfies a weak version of the well known Palais-Smale condition. An existence
result is proved under general assumptions on the nonlinearities.Comment: 18 page
Positive solutions to indefinite Neumann problems when the weight has positive average
We deal with positive solutions for the Neumann boundary value problem
associated with the scalar second order ODE where is positive on and is an indefinite weight. Complementary to previous
investigations in the case , we provide existence results
for a suitable class of weights having (small) positive mean, when
at infinity. Our proof relies on a shooting argument for a suitable equivalent
planar system of the type with
a continuous function defined on the whole real line.Comment: 17 pages, 3 figure
Pulsating Front Speed-up and Quenching of Reaction by Fast Advection
We consider reaction-diffusion equations with combustion-type non-linearities
in two dimensions and study speed-up of their pulsating fronts by general
periodic incompressible flows with a cellular structure. We show that the
occurence of front speed-up in the sense ,
with the amplitude of the flow and the (minimal) front speed, only
depends on the geometry of the flow and not on the reaction function. In
particular, front speed-up happens for KPP reactions if and only if it does for
ignition reactions. We also show that the flows which achieve this speed-up are
precisely those which, when scaled properly, are able to quench any ignition
reaction.Comment: 16p
Maximal and minimal spreading speeds for reaction diffusion equations in nonperiodic slowly varying media
This paper investigates the asymptotic behavior of the solutions of the
Fisher-KPP equation in a heterogeneous medium, associated with a compactly supported initial datum. A typical
nonlinearity we consider is , where
is a 1-periodic function and is a increasing function
that satisfies and . Although quite specific, the choice of such a reaction
term is motivated by its highly heterogeneous nature. We exhibit two different
behaviors for for large times, depending on the speed of the convergence of
at infinity. If grows sufficiently slowly, then we prove that the
spreading speed of oscillates between two distinct values. If grows
rapidly, then we compute explicitly a unique and well determined speed of
propagation , arising from the limiting problem of an infinite
period. We give a heuristic interpretation for these two behaviors
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