152 research outputs found
Fujita blow up phenomena and hair trigger effect: the role of dispersal tails
We consider the nonlocal diffusion equation in
the whole of . We prove that the Fujita exponent dramatically depends on
the behavior of the Fourier transform of the kernel near the origin, which
is linked to the tails of . In particular, for compactly supported or
exponentially bounded kernels, the Fujita exponent is the same as that of the
nonlinear Heat equation . On the other hand,
for kernels with algebraic tails, the Fujita exponent is either of the Heat
type or of some related Fractional type, depending on the finiteness of the
second moment of . As an application of the result in population dynamics
models, we discuss the hair trigger effect for $\partial \_t
u=J*u-u+u^{1+p}(1-u)
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
Replicator-mutator equations with quadratic fitness
This work completes our previous analysis on models arising in evolutionary
genetics. We consider the so-called replicator-mutator equation, when the
fitness is quadratic. This equation is a heat equation with a harmonic
potential, plus a specific nonlocal term. We give an explicit formula for the
solution, thanks to which we prove that when the fitness is non-positive
(harmonic potential), solutions converge to a universal stationary Gaussian for
large time, whereas when the fitness is non-negative (inverted harmonic
potential), solutions always become extinct in finite time.Comment: 12 page
Convergence to a propagating front in a degenerate Fisher-KPP equation with advection
We consider a Fisher-KPP equation with density-dependent diffusion and
advection, arising from a chemotaxis-growth model. We study its behavior as a
small parameter, related to the thickness of a diffuse interface, tends to
zero. We analyze, for small times, the emergence of transition layers induced
by a balance between reaction and drift effects. Then we investigate the
propagation of the layers. Convergence to a free-boundary limit problem is
proved and a sharp estimate of the thickness of the layers is provided
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