We consider the nonlinear Stefan problem $$
\left \{ \begin{array} {ll}
-d Δu=a u-b u^2 \;\; & \mbox{for } x \in Ω(t), \; t>0,
\\ u=0 \mbox{ and } u_t=μ|\nabla_x u |^2 \;\;&\mbox{for } x \in \partialΩ(t), \; t>0, \\ u(0,x)=u_0 (x) \;\; & \mbox{for } x \in Ω_0, \end{array}\right. $$
where $Ω(0)=Ω_0$ is an unbounded smooth domain in $\mathbb R^N$, $u_0>0$ in $Ω_0$ and $u_0$ vanishes on $\partialΩ_0$. When $Ω_0$ is bounded, the long-time behavior of this problem has been rather well-understood by \cite{DG1,DG2,DLZ, DMW}. Here we reveal some interesting different behavior for certain unbounded $Ω_0$. We also give a unified approach for a weak solution theory to this kind of free boundary problems with bounded or unbounded $Ω_0$.
In this paper we prove the existence and uniqueness of the almost periodic semi- wave (traveling wave) solutions of KPP-Fisher equations with free boundaries in spatially almost periodic media.
We use a new method in the study of Fisher-KPP reaction-diffusion equations to prove existence of transition fronts for inhomogeneous KPP-type non-linearities in one spatial dimension. We also obtain new estimates on entire solutions of some KPP reaction-diffusion equations in several spatial dimensions. Our method is based on the construction of sub- and super-solutions to the non-linear PDE from solutions of its linearization at zero.
We establish the logarithmic Bramson correction to the position of solutions to the Fisher--KPP equation with nonlocal diffusion. Solutions with step-like initial data typically resemble a front at position $c_{*} t - \frac{3}{2 λ_{*}} \log t + \mathcal{O}(1)$ for explicit constants $c_{*}$ and $λ_{*}$. However, certain singular diffusions exhibit more exotic behavior.
This paper is concerned with non-cooperative parabolic reaction--diffusion systems which share structural similarities with the scalar Fisher--KPP equation. In a previous paper, we established that these systems admit traveling wave solutions. It is then natural to investigate their asymptotic distributions. This is the main object of this paper. Non-cooperative KPP systems can model various phenomena where the following three mechanisms occur: local diffusion in space, linear cooperation and superlinear competition.
We establish a new property of Fisher-KPP type propagation in a plane, in the presence of a line with fast diffusion. We prove that the line enhances the asymptotic speed of propagation in a cone of directions. Past the critical angle given by this cone, the asymptotic speed of propagation coincides with the classical Fisher-KPP invasion speed. Several qualitative properties are further derived, such as the limiting behaviour when the diffusion on the line goes to infinity.
We consider the Fisher-KPP reaction-diffusion equation in the whole space. We prove that if a solution has, to main order and for all times (positive and negative), the same exponential decay as a planar traveling wave with speed larger than the minimal one at its leading edge, then it has to coincide with the aforementioned traveling wave.
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.
The aim of this paper is to study the generalized Fisher-KPP equation with nonlocal diffusion. In specific we prove the existence of a critical speed so that traveling front type solutions exist up to this critical speed and non-existence of traveling fronts below this critical value. Moreover, we obtain uniqueness, up to translation, and decay estimates of these traveling fronts.
We consider Fisher-KPP-type reaction-diffusion equations with spatially inhomogeneous reaction rates. We show that a sufficiently strong localized inhomogeneity may prevent existence of transition-front-type global in time solutions while creating a global in time bump-like solution. This is the first example of a medium in which no reaction-diffusion transition front exists. A weaker localized inhomogeneity leads to existence of transition fronts but only in a finite range of speeds. These results are in contrast with both Fisher-KPP reactions in homogeneous media as well as ignition-type rea…
We consider the non-local Fisher-KPP equation on a bounded domain with Neu-mann boundary conditions. Thanks to a Lyapunov function, we prove that under a general hypothesis on the Kernel involved in the non-local term, the homogenous steady state 1 is globally asymptotically stable. This assumption happens to be linked to some conditions given in the literature, which ensure that travelling waves link 0 to 1.
We prove existence of and construct transition fronts for a class of reaction- diffusion equations with spatially inhomogeneous Fisher-KPP type reactions and non-local diffusion. Our approach is based on finding these solutions as perturbations of appropriate solutions to the linearization of the PDE at zero. Our work extends a method introduced by one of us to study such questions in the case of classical diffusion.
We study in this note the Fisher-KPP equation where the Laplacian is replaced by the generator of a Feller semigroup with slowly decaying kernel, an important example being the fractional Laplacian. Contrary to what happens in the standard Laplacian case, where the stable state invades the unstable one at constant speed, we prove here that invasion holds at an exponential in time velocity. These results provide a mathematically rigorous justification of numerous heuristics about this model.
We are interested in the time asymptotic location of the level sets of solutions to Fisher-KPP reaction-diffusion equations with fractional diffusion in periodic media. We show that the speed of propagation is exponential in time, with a precise exponent depending on a periodic principal eigenvalue, and that it does not depend on the space direction. This is in contrast with the Freidlin-Gärtner formula for the standard Laplacian.
In this paper, we investigate the asymptotic dynamics of Fisher-KPP equations with nonlocal dispersal operator and nonlocal reaction term in time periodic and space heterogeneous media. We first show the global existence and boundedness of nonnegative solutions, and then obtain some sufficient conditions ensuring the uniform persistence. In particular, we study the existence, uniqueness and global stability of positive time periodic solutions under several different conditions.
We study the existence, bifurcations, and stability of stationary solutions for the doubly-nonlocal Fisher-KPP equation. We prove using Lyapunov-Schmidt reduction that under suitable conditions on the parameters, a bifurcation from the non-trivial homogeneous state can occur. The kernel of the linearized operator at the bifurcation is two-dimensional and periodic stationary patterns are generated. Then we prove that these patterns are, again under suitable conditions, locally asymptotically stable. We also compare our results to previous work on the nonlocal Fisher-KPP equation containing a lo…
We study the Fisher-KPP equation with a fractional laplacian of order α \in (0, 1). We know that the stable state invades the unstable one at constant speed for α = 1, and at an exponential in time velocity for α \in (0, 1). The transition between these two different speeds is examined in this paper. We prove that during a time of the order - ln(1 - α), the propagation is linear and then it is exponential.
We revisit Wu and Zou non-standard quasi-monotonicity approach for proving existence of monotone wavefronts in monostable reaction-diffusion equations with delays. This allows to solve the problem of existence of monotone wavefronts in a neutral KPP-Fisher equation. In addition, using some new ideas proposed recently by Solar et al., we establish the uniqueness (up to a translation) of these monotone wavefronts.
This paper is concerned with non-cooperative parabolic reaction--diffusion systems which share structural similarities with the scalar Fisher--KPP equation. These similarities make it possible to prove, among other results, an extinction and persistence dichotomy and, when persistence occurs, the existence of a positive steady state, the existence of traveling waves with a half-line of possible speeds and a positive minimal speed and the equality between this minimal speed and the spreading speed for the Cauchy problem. Non-cooperative KPP systems can model various phenomena where the followin…
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 Lévy process, the front position is exponential in time. Our results provide a mathe- matically rigorous justification of numerous heuristics about this model.
The purpose of the current paper is to unveil the key mechanism which is responsible for the occurrence of {\it Turing-type instability} for a non-local Fisher-KPP type model. In particular, we prove that the solution of the considered non-local Fisher-KPP equation in the neighbourhood of a constant stationary solution, is destabilized via a {\it diffusion-driven blow-up}. It is also shown that the observed {\it diffusion-driven blow-up} is complete, whilst its blow-up rate is completely classified. Finally, the detected {\it diffusion-driven instability} results in the formation of unstable b…
This paper is concerned with the stability of transition waves and strictly positive entire solutions of random and nonlocal dispersal evolution equations of Fisher-KPP type with general time and space dependence, including time and space periodic or almost periodic dependence as special cases. We first show the existence, uniqueness, and stability of strictly positive entire solutions of such equations. Next, we show the stability of uniformly continuous transition waves connecting the unique strictly positive entire solution and the trivial solution zero and satisfying certain decay property…
We address the uniqueness of the nonzero stationary state for a reaction-diffusion system of Fisher-KPP type that does not satisfy the comparison principle. Although the uniqueness is false in general, it turns out to be true under biologically natural assumptions on the parameters. This Liouville-type result is then used to characterize the wake of traveling waves. All results are extended to an analogous nonlocal reaction-diffusion equation that contains as a particular case the cane toads equation with bounded traits.
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.
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 $2t - O(t^β)$ for some explicit $β\in (0,1)$. Our main tools here are alocal-in-time Harnack inequality and an analysis of the linearized problem with a suitable moving Diri…
We study the large time behaviour of the Fisher-KPP equation $\partial$ t u = $Δ$u + u -- u 2 in spatial dimension N , when the initial datum is compactly supported. We prove the existence of a Lipschitz function s of the unit sphere, such that u(t, x) converges, as t goes to infinity, to U c * |x| -- c * t + N + 2 c * lnt + s $\infty$ x |x| , where U c * is the 1D travelling front with minimal speed c * = 2. This extends an earlier result of G{ä}rtner.
We propose a simple alternative proof of a famous result of Gallay regarding the nonlinear asymptotic stability of the critical front of the Fisher-KPP equation which shows that perturbations of the critical front decay algebraically with rate $t^{-3/2}$ in a weighted $L^\infty$ space. Our proof is based on pointwise semigroup methods and the key remark that the faster algebraic decay rate $t^{-3/2}$ is a consequence of the lack of an embedded zero of the Evans function at the origin for the linearized problem around the critical front.
Incorporating free boundary into time-delayed reaction-diffusion equations yields a compatible condition that guarantees the well-posedness of the initial value problem. With the KPP type nonlinearity we then establish a vanishing-spreading dichotomy result. Further, when the spreading happens, we show that the spreading speed and spreading profile are nonlinearly determined by a delay-induced nonlocal semi-wave problem. It turns out that time delay slows down the spreading speed.
This article is concerned with the rigorous validation of anomalous spreading speeds in a system of coupled Fisher-KPP equations of cooperative type. Anomalous spreading refers to a scenario wherein the coupling of two equations leads to faster spreading speeds in one of the components. The existence of these spreading speeds can be predicted from the linearization about the unstable state. We prove that initial data consisting of compactly supported perturbations of Heaviside step functions spreads asymptotically with the anomalous speed. The proof makes use of a comparison principle and the …
We study the long time behavior of solutions of periodic Fisher-KPP type equations in $\mathbb{R}^n$ that arise from compactly supported initial data. We prove that propagation along a fixed direction $e\in\mathbb{S}^{n-1}$ is completely determined by an associated minimizing direction $e'$ which is unique and in a bijective correspondence with $e$. This correspondence allows us to determine the correct Bramson shift in higher dimensions and show that that the propagation along each $e$ occurs asymptotically at the speed $w_*(e)t-\frac{n/2+1}{λ_*(e')\,e\cdot e'}\log t+O(1)$.
In this paper, we study gradient decay estimates for solutions to the multi-dimensional Fisher-KPP equation with fractional diffusion. It is known that this equation exhibits exponentially advancing level sets with strong qualitative upper and lower bounds on the solution. However, little has been shown concerning the gradient of the solution. We prove that, under mild conditions on the initial data, the first and second derivatives of the solution obey a comparative exponential decay in time. We then use this estimate to prove a symmetrization result, which shows that the reaction front flatt…
In this paper we study the entire solutions of the Fisher-KPP equation $u_t=u_{xx}+f(u)$ on the half line $[0,\infty)$ with Dirichlet boundary condition at $x=0$. (1). For any $c\geq 2\sqrt{f'(0)}$, we show the existence of an entire solution $\mathcal{U}^c(x,t)$ which connects the traveling wave solution $ϕ^c(x+ct)$ at $t=-\infty$ and the unique positive stationary solution $V(x)$ at $t=+\infty$; (2). We also construct an entire solution $\mathcal{U}(x,t)$ which connects the solution of $η_t =f(η)$ at $t=-\infty$ and $V(x)$ at $t=+\infty$.
We study entire solutions to homogeneous reaction-diffusion equations in several dimensions with Fisher-KPP reactions. Any entire solution $0<u<1$ is known to satisfy \[ \lim_{t\to -\infty} \sup_{|x|\le c|t|} u(t,x) = 0 \qquad \text{for each $c<2\sqrt{f'(0)}\,$,} \] and we consider here those satisfying \[ \lim_{t\to -\infty} \sup_{|x|\le c|t|} u(t,x) = 0 \qquad \text{for some $c>2\sqrt{f'(0)}\,$.} \] When $f$ is $C^2$ and concave, our main result provides an almost complete characterization of transition fronts as well as transition solutions with bounded width within this class of solutions.
We investigate the influence of a general non-local advection term of the form K * u to propagation in the one-dimensional Fisher-KPP equation. This model is a generalization of the Keller-Segel-Fisher system. When K $\in$ L 1 (R), we obtain explicit upper and lower bounds on the propagation speed which are asymptotically sharp and more precise than previous works. When K $\in$ L p (R) with p > 1 and is non-increasing in (--$\infty$, 0) and in (0, +$\infty$), we show that the position of the "front" is of order O(t 1/p) if p < $\infty$ and O(e $λ$t) for some $λ$ > 0 if p = $\infty$ and K(+$\in…
We consider a one-dimensional reaction-diffusion equation of Fisher-Kolmogoroff-Petrovsky-Piscounoff type. We investigate the effect of the interaction between the nonlinear diffusion coefficient and the reaction term on the existence and nonexistence of travelling waves. Our diffusion coefficient is allowed to be degenerate or singular at both equilibrium points, 0 and 1, while the reaction term need not be differentiable. These facts influence the existence and qualitative properties of travelling waves in a substantial way.
This paper is concerned with transition fronts for reaction-diffusion equations of the Fisher-KPP type. Basic examples of transition fronts connecting the unstable steady state to the stable one are the standard traveling fronts, but the class of transition fronts is much larger and the dynamics of the solutions of such equations is very rich. In the paper, we describe the class of transition fronts and we study their qualitative dynamical properties. In particular, we characterize the set of their admissible asymptotic past and future speeds and their asymptotic profiles and we show that the …
We study the large time asymptotics of a solution of the Fisher-KPP reaction-diffusion equation, with an initial condition that is a compact perturbation of a step function. A well-known result of Bramson states that, in the reference frame moving as $2t - ({3}/{2}) \log t +x_\infty$, the solution of the equation converges as $t\to+\infty$ to a translate of the traveling wave corresponding to the minimal speed~$c_*=2$. The constant $x_\infty$ depends on the initial condition $u(0,x)$. The proof is elaborate, and based on probabilistic arguments. The purpose of this paper is to provide a simple…
We introduce and study a class of free boundary models with "nonlocal diffusion", which are natural extensions of the free boundary models in Du and Lin [17] and elsewhere, where "local diffusion" is used to describe the population dispersal, with the free boundary representing the spreading front of the species. We show that this nonlocal problem has a unique solution defined for all time, and then examine its long-time dynamical behavior when the growth function is of Fisher-KPP type. We prove that a spreading-vanishing dichotomy holds, though for the spreading-vanishing criteria significant…
We study the asymptotic behavior of solutions to a monostable integro-differential Fisher-KPP equation , that is where the standard Laplacian is replaced by a convolution term, when the dispersal kernel is fat-tailed. We focus on two different regimes. Firstly, we study the long time/long range scaling limit by introducing a relevant rescaling in space and time and prove a sharp bound on the (super-linear) spreading rate in the Hamilton-Jacobi sense by means of sub-and super-solutions. Secondly, we investigate a long time/small mutation regime for which, after identifying a relevant rescaling …
This paper is a continuation of [2] where a new model of biological invasions in the plane directed by a line was introduced. Here we include new features such as transport and reaction terms on the line. Their interaction with the pure diffusivity in the plane is quantified in terms of enhancement of the propagation speed. We establish conditions that determine whether the spreading speed exceeds the standard Fisher KPP invasion speed. These conditions involve the ratio of the diffusivities on the line and in the field, the transport term and the reactions. We derive the asymptotic behaviour …
The Fisher-KPP equation is a model for population dynamics that has generated a huge amount of interest since its introduction in 1937. The speed with which a population spreads has been computed quite precisely when the initial data decays exponentially. More recently, though, the case when the initial data decays more slowly has been studied. Building on the results of Hamel and Roques '10, in this paper we improve their precision for a broader class of initial data and for a broader class of equations. In particular, our approach yields the explicit highest order term in the location of the…
We investigate the singular limit, as $\ep \to 0$, of the Fisher equation $\partial_t u=\ep Δu + \ep ^{-1}u(1-u)$ in the whole space. We consider initial data with compact support plus perturbations with {\it slow exponential decay}. We prove that the sharp interface limit moves by a constant speed, which dramatically depends on the tails of the initial data. By performing a fine analysis of both the generation and motion of interface, we provide a new estimate of the thickness of the transition layers.
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…
We give an iterative method to estimate the disturbance of semi-wavefronts of the equation: $\dot{u}(t,x) = u''(t,x) +u(t,x)(1-u(t-h,x)),$ $x \in \mathbb{R},\ t >0;$ where $h>0.$ As a consequence, we show the exponential stability, with an unbounded weight, of semi-wavefronts with speed $c>2\sqrt{2}$ and $h>0$. Under the same restriction of $c$ and $h$, the uniqueness of semi-wavefronts is obtained.
We study the asymptotic behaviour, as time goes to infinity, of the Fisher-KPP equation $\partial_t u=Δu +u-u^2$ in spatial dimension $2$, when the initial condition looks like a Heaviside function. Thus the solution is, asymptotically in time, trapped between two planar critical waves whose positions are corrected by the Bramson logarithmic shift. The issue is whether, in this reference frame, the solutions will converge to a travelling wave, or will exhibit more complex behaviours. We prove here that both convergence and nonconvergence may happen: the solution may converge towards one transl…
We establish in this paper the logarithmic Bramson correction for Fisher-KPP equations on the lattice $\mathbb{Z}$. The level sets of solutions with step-like initial conditions are located at position $c_*t-\frac{3}{2λ_*}\ln t+\mathcal{O}(1)$ as $t\rightarrow+\infty$ for some explicit positive constants $c_*$ and $λ_*$. This extends a well-known result of Bramson in the continuous setting to the discrete case using only PDE arguments. A by-product of our analysis also gives that the solutions approach the family of logarithmically shifted traveling front solutions with minimal wave speed $c_*…
In this article, we study the growth and spread of a new species in a river network with two or three branches via the Fisher-KPP advection-diffusion equation over some simple graphs with every edge a half infinite line. We obtain a rather complete description of the long-time dynamical behavior for every case under consideration, which can be loosely described by a trichotomy (see Remark 1.7), including two different kinds of persistence states as parameters vary. The phenomenon of "persistence below carrying capacity" revealed here appears new, which does not occur in related models of the e…
We investigate the singular limit, as $\ep \to 0$, of the Fisher equation $\partial_t u=\ep Δu + \ep ^{-1}u(1-u)$ in the whole space. We consider initial data with compact support plus, possibly, perturbations very small as $\Vert x \Vert \to \infty$. By proving both generation and motion of interface properties, we show that the sharp interface limit moves by a constant speed, which is the minimal speed of some related one-dimensional travelling waves. We obtain an estimate of the thickness of the transition layers. We also exhibit initial data "not so small" at infinity which do not allow th…
The solution h to the Fisher-KPP equation with a steep enough initial condition develops into a front moving at velocity 2, with logarithmic corrections to its position. In this paper we investigate the value h(c t, t) of the solution ahead of the front, at time t and position c t, with c > 2. That value goes to zero exponentially fast with time, with a well-known rate, but the prefactor depends in a non-trivial way of c, the initial condition and the non-linearity in the equation. We compute an asymptotic expansion of that prefactor for velocities c close to 2. The expansion is surprisingly e…
While there is a long history of employing moving boundary problems in physics, in particular via Stefan problems for heat conduction accompanied by a change of phase, more recently such approaches have been adapted to study biological invasion. For example, when a logistic growth term is added to the governing partial differential equation in a Stefan problem, one arrives at the Fisher-Stefan model, a generalisation of the well-known Fisher-KPP model, characterised by a leakage coefficient $κ$ which relates the speed of the moving boundary to the flux of population there. This Fisher-Stefan m…
This paper deals with nonnegative solutions of the Neumann initial-boundary value problem for the fully parabolic chemotaxis-growth system $ (u_{\varepsilon})_t$ $=Δu_{\varepsilon} - \varepsilon \nabla \cdot ( u_\varepsilon \nabla v_\varepsilon) + μu_\varepsilon(1 - u_\varepsilon)$, $ (v_{\varepsilon})_t=Δv_\varepsilon -v_\varepsilon+u_\varepsilon,$ with positive small parameter $\varepsilon>0$ in a bounded convex domain $Ω\subset\mathbb{R}^n$ ($n\geq 1$) with smooth boundary. The solutions converge to the solution $u$ to the Fisher-KPP equation as $\varepsilon\to 0$. It is shown that for all …
This paper investigates the existence of generalized transition fronts for Fisher-KPP equations in one-dimensional, almost periodic media. Assuming that the linearized elliptic operator near the unstable steady state admits an almost periodic eigenfunction, we show that such fronts exist if and only if their average speed is above an explicit threshold. This hypothesis is satisfied in particular when the reaction term does not depend on x or (in some cases) is small enough. Moreover, except for the threshold case, the fronts we construct and their speeds are almost periodic, in a sense. When o…
