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 $0\leq u(x,t)\leq 1$ 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 $N \geq 1$. 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 $0\leq u(x,t)\leq 1$ 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: $$\partial_t u= \nabla \cdot (u^{m-1} \nabla p), \quad p=(-\Delta)^{-s}u,$$ for $m>1$, $0<s<1$ and $u(x,t)\ge 0$. To be specific, the problem is posed for $x\in \mathbb{R}^N$, $N\geq 1$, and $t>0$. The initial data $u(x,0)$ 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 $m$, the property of compact support is conserved in time or not, starting from the result of finite propagation known for $m=2$. We find that when $m\in [1,2)$ the problem has infinite speed of propagation, while for $m\in [2,\infty)$ 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 $L^2$ 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 $t^{3/2}$