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

On the nonlocal Fisher-KPP equation: steady states, spreading speed and global bounds

We consider the Fisher-KPP equation with a non-local interaction term. We establish a condition on the interaction that allows for existence of non-constant periodic solutions, and prove uniform upper bounds for the solutions of the Cauchy problem, as well as upper and lower bounds on the spreading rate of the solutions with compactly supported initial data

Discontinuous Transition in a Boundary Driven Contact Process

The contact process is a stochastic process which exhibits a continuous, absorbing-state phase transition in the Directed Percolation (DP) universality class. In this work, we consider a contact process with a bias in conjunction with an active wall. This model exhibits waves of activity emanating from the active wall and, when the system is supercritical, propagating indefinitely as travelling (Fisher) waves. In the subcritical phase the activity is localised near the wall. We study the phase transition numerically and show that certain properties of the system, notably the wave velocity, are discontinuous across the transition. Using a modified Fisher equation to model the system we elucidate the mechanism by which the the discontinuity arises. Furthermore we establish relations between properties of the travelling wave and DP critical exponents.Comment: 14 pages, 9 figure

The Bramson delay in the non-local Fisher-KPP equation

We consider the non-local Fisher-KPP equation modeling a population with individuals competing with each other for resources with a strength related to their distance, and obtain the asymptotics for the position of the invasion front starting from a localized population. Depending on the behavior of the competition kernel at infinity, the location of the front is either $2t - ({3}/2)\log t + O(1)$, as in the local case, or $2t - O(t^\beta)$ for some explicit $\beta \in (0,1)$. Our main tools here are a local-in-time Harnack inequality and an analysis of the linearized problem with a suitable moving Dirichlet boundary condition. Our analysis also yields, for any $\beta\in(0,1)$, examples of Fisher-KPP type non-linearities $f_\beta$ such that the front for the local Fisher-KPP equation with reaction term $f_\beta$ is at $2t - O(t^\beta)$

The Bramson delay in the non-local Fisher-KPP equation

We consider the non-local Fisher-KPP equation modeling a population with individuals competing with each other for resources with a strength related to their distance, and obtain the asymptotics for the position of the invasion front starting from a localized population. Depending on the behavior of the competition kernel at infinity, the location of the front is either $2t - ({3}/2)\log t + O(1)$, as in the local case, or $2t - O(t^\beta)$ for some explicit $\beta \in (0,1)$. Our main tools here are alocal-in-time Harnack inequality and an analysis of the linearized problem with a suitable moving Dirichlet boundary condition. Our analysis also yields, for any $\beta\in(0,1)$, examples of Fisher-KPP type non-linearities $f\_\beta$ such that the front for the localFisher-KPP equation with reaction term $f\_\beta$ is at $2t - O(t^\beta)$

The effect on Fisher-KPP propagation in a cylinder with fast diffusion on the boundary

In this paper we consider a reaction-diffusion equation of Fisher-KPP type inside an infinite cylindrical domain in $\mathbb{R}^{N+1}$, coupled with a reaction-diffusion equation on the boundary of the domain, where potentially fast diffusion is allowed. We will study the existence of an asymptotic speed of propagation for solutions of the Cauchy problem associated with such system, as well as the dependence of this speed on the diffusivity at the boundary and the amplitude of the cylinder. When $N=1$ the domain reduces to a strip between two straight lines. This models the effect of two roads with fast diffusion on a strip-shaped field bounded by them.Comment: 31 pages, 3 figure