296 research outputs found
The fisher-kpp equation with nonlinear fractional diffusion
Abstract.We study the propagation properties of nonnegative and bounded solutions of theclass of reaction-diffusion equations with nonlinear fractional diffusion:ut+(−Δ)s(um)=f(u). Forall 0mc=(N−2s)+/N, we consider the solution of the initial-value problemwith initial data having fast decay at infinity and prove that its level sets propagate exponentiallyfast in time, in contrast to the traveling wave behavior of the standard KPP case, which correspondsto puttings=1,m=1,andf(u)=u(1−u). The proof of this fact uses as an essential ingredientthe recently established decay properties of the self-similar solutions of the purely diffusive equation,ut+(−Δ)sum=
The influence of fractional diffusion in Fisher-KPP equations
We study the Fisher-KPP equation where the Laplacian is replaced by the
generator of a Feller semigroup with power decaying kernel, an important
example being the fractional Laplacian. In contrast with the case of the stan-
dard Laplacian where the stable state invades the unstable one at constant
speed, we prove that with fractional diffusion, generated for instance by a
stable L\'evy process, the front position is exponential in time. Our results
provide a mathe- matically rigorous justification of numerous heuristics about
this model
The Fisher-KPP problem with doubly nonlinear "fast" diffusion
The famous Fisher-KPP reaction diffusion model combines linear diffusion with
the typical Fisher-KPP reaction term, and appears in a number of relevant
applications. It is remarkable as a mathematical model since, in the case of
linear diffusion, it possesses a family of travelling waves that describe the
asymptotic behaviour of a wide class solutions of the
problem posed in the real line. The existence of propagation wave with finite
speed has been confirmed in the cases of "slow" and "pseudo-linear" doubly
nonlinear diffusion too, see arXiv:1601.05718. We investigate here the
corresponding theory with "fast" doubly nonlinear diffusion and we find that
general solutions show a non-TW asymptotic behaviour, and exponential
propagation in space for large times. Finally, we prove precise bounds for the
level sets of general solutions, even when we work in with spacial dimension . In particular, we show that location of the level sets is
approximately linear for large times, when we take spatial logarithmic scale,
finding a strong departure from the linear case, in which appears the famous
Bramson logarithmic correction.Comment: 42 pages, 6 figure
Bistable reaction equations with doubly nonlinear diffusion
Reaction-diffusion equations appear in biology and chemistry, and combine
linear diffusion with different kind of reaction terms. Some of them are
remarkable from the mathematical point of view, since they admit families of
travelling waves that describe the asymptotic behaviour of a larger class of
solutions of the problem posed in the real line. We
investigate here the existence of waves with constant propagation speed, when
the linear diffusion is replaced by the "slow" doubly nonlinear diffusion. In
the present setting we consider bistable reaction terms, which present
interesting differences w.r.t. the Fisher-KPP framework recently studied in
\cite{AA-JLV:art}. We find different families of travelling waves that are
employed to describe the wave propagation of more general solutions and to
study the stability/instability of the steady states, even when we extend the
study to several space dimensions. A similar study is performed in the critical
case that we call "pseudo-linear", i.e., when the operator is still nonlinear
but has homogeneity one. With respect to the classical model and the
"pseudo-linear" case, the travelling waves of the "slow" diffusion setting
exhibit free boundaries. \\ Finally, as a complement of \cite{AA-JLV:art}, we
study the asymptotic behaviour of more general solutions in the presence of a
"heterozygote superior" reaction function and doubly nonlinear diffusion
("slow" and "pseudo-linear").Comment: 42 pages, 11 figures. Accepted version on Discrete Contin. Dyn. Sys
Finite and infinite speed of propagation for porous medium equations with fractional pressure
We study a porous medium equation with fractional potential pressure: for
, and . To be specific, the problem is posed for
, , and . The initial data is assumed
to be a bounded function with compact support or fast decay at infinity. We
establish existence of a class of weak solutions for which we determine
whether, depending on the parameter , the property of compact support is
conserved in time or not, starting from the result of finite propagation known
for . We find that when the problem has infinite speed of
propagation, while for it has finite speed of propagation.
Comparison is made with other nonlinear diffusion models where the results are
widely different.Comment: 6 pages, 1 figur
Propagation in a kinetic reaction-transport equation: travelling waves and accelerating fronts
In this paper, we study the existence and stability of travelling wave
solutions of a kinetic reaction-transport equation. The model describes
particles moving according to a velocity-jump process, and proliferating thanks
to a reaction term of monostable type. The boundedness of the velocity set
appears to be a necessary and sufficient condition for the existence of
positive travelling waves. The minimal speed of propagation of waves is
obtained from an explicit dispersion relation. We construct the waves using a
technique of sub- and supersolutions and prove their \eb{weak} stability in a
weighted space. In case of an unbounded velocity set, we prove a
superlinear spreading. It appears that the rate of spreading depends on the
decay at infinity of the velocity distribution. In the case of a Gaussian
distribution, we prove that the front spreads as
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