66 research outputs found
KPP reaction-diffusion systems with loss inside a cylinder: convergence toward the problem with Robin boundary conditions
We consider in this paper a reaction-diffusion system under a KPP hypothesis
in a cylindrical domain in the presence of a shear flow. Such systems arise in
predator-prey models as well as in combustion models with heat losses.
Similarly to the single equation case, the existence of a minimal speed c* and
of traveling front solutions for every speed c > c* has been shown both in the
cases of heat losses distributed inside the domain or on the boundary. Here, we
deal with the accordance between the two models by choosing heat losses inside
the domain which tend to a Dirac mass located on the boundary. First, using the
characterizations of the corresponding minimal speeds, we will see that they
converge to the minimal speed of the limiting problem. Then, we will take
interest in the convergence of the traveling front solutions of our
reaction-diffusion systems. We will show the convergence under some assumptions
on those solutions, which in particular can be satisfied in dimension 2
Varying the direction of propagation in reaction-diffusion equations in periodic media
We consider a multidimensional reaction-diffusion equation of either ignition
or monostable type, involving periodic heterogeneity, and analyze the
dependence of the propagation phenomena on the direction. We prove that the
(minimal) speed of the underlying pulsating fronts depends continuously on the
direction of propagation, and so does its associated profile provided it is
unique up to time shifts. We also prove that the spreading properties
\cite{Wein02} are actually uniform with respect to the direction
A KPP road-field system with spatially periodic exchange terms
We take interest in a reaction-diffusion system which has been recently
proposed [11] as a model for the effect of a road on propagation phenomena
arising in epidemiology and ecology. This system consists in coupling a
classical Fisher-KPP equation in a half-plane with a line with fast diffusion
accounting for a straight road. The effect of the line on spreading properties
of solutions (with compactly supported initial data) was investigated in a
series of works starting from [11]. We recover these earlier results in a more
general spatially periodic framework by exhibiting a threshold for road
diffusion above which the propagation is driven by the road and the global
speed is accelerated. We also discuss further applications of our approach,
which will rely on the construction of a suitable generalized principal
eigenvalue, and investigate in particular the spreading of solutions with
exponentially decaying initial data.Comment: Updated version, minor typos and details fixe
Convergence to pulsating traveling waves with minimal speed in some KPP heterogeneous problems
The notion of traveling wave, which typically refers to some particular
spatio-temporal con- nections between two stationary states (typically, entire
solutions keeping the same profile's shape through time), is essential in the
mathematical analysis of propagation phenomena. They provide insight on the
underlying dynamics, and an accurate description of large time behavior of
large classes of solutions, as we will see in this paper. For instance, in an
homogeneous framework, it is well-known that, given a fast decaying initial
datum (for instance, compactly supported), the solution of a KPP type
reaction-diffusion equation converges in both speed and shape to the traveling
wave with minimal speed. The issue at stake in this paper is the gener-
alization of this result to some one-dimensional heterogeneous environments,
namely spatially periodic or converging to a spatially periodic medium. This
result fairly improves our under- standing of the large-time behavior of
solutions, as well as of the role of heterogeneity, which has become a crucial
challenge in this field over the past few years
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