Spreading properties for non-autonomous Fisher-KPP equations with nonlocal diffusion

In this paper we study spreading properties for the solutions of the following non-autonomous and nonlocal one-dimensional equation

$$\partial_{t}u(t,x)=\int_{\mathbb{R}}K(y)\left[u(t,x-y)-u(t,x)\right]\mathrm{d}y+u(t,x)f\left(t,u(t,x)\right),$$

(1.1)

posed for time $t\geq 0$ and $x\in\mathbb{R}$. This evolution problem is supplemented with an appropriated initial data, that will be discussed below. Here $K=K(y)$ is a nonnegative dispersal kernel with thin-tailed (see Assumption 1.3 below). Let us set $F(t,u):=uf(t,u)$. At the same time, $F=F(t,u)$ stands for the nonlinear growth term, which depends on time $t$ and that will be assumed in this note to be of the Fisher-KPP type (see Assumption 1.5). The above problem typically describes the spatial invasion of a population (see for instance [6, 34] and the references therein) with the following features:
1) individuals exhibit long distance dispersal according to the kernel $K$, in other words the quantity $K(x-y)$ corresponds to the probability for individuals to jump from $y$ to $x$;
2) time varying birth and death processes modeled by the nonlinear Fisher-KPP type function $F(t,u)$. The time variations may stand for seasonality and/or external events (see [24]).

When local diffusion is considered, the Fisher-KPP equation posed in a time homogeneous medium reads as

$$\partial_{t}u(t,x)=\partial_{xx}u(t,x)+F(u(t,x)).$$

(1.2)

As mentioned above, this problem arises as a basic model in many different fields, in biology and ecology in particular. It can be used for instance to describe the spatio-temporal evolution of an invading species into an empty environment. The above equation (1.2) was introduced separately by Fisher [20] and Kolmogorov, Petrovsky and Piskunov [27], when the nonlinear function $F$ satisfies the Fisher-KPP conditions. Recall that a typical example of such Fisher-KPP nonlinearity is given by the logistic function $F(u)=u(1-u)$.

There is a large amount of literature related to (1.2) and its generalizations. To study propagation phenomena generated by reaction diffusion equations, in addition to the existence of travelling wave solution, the asymptotic speed of spread (or spreading speed) was introduced and studied by Aronson and Weinberger in [4]. Roughly speaking if $u_{0}$ is a nontrival and nonnegative initial data with compact support, then the solution of (1.2) associated with this initial data $u_{0}$ spreads with the speed $c^{*}>0$ (the minimal wave speed of the travelling waves) in the sense that

$$\lim_{t\to\infty}\sup_{|x|\leq ct}|u(t,x)-1|=0,\;\forall\;c\in[0,c^{*})\;\text{ and }\;\lim_{t\to\infty}\sup_{|x|\geq ct}u(t,x)=0,\;\forall\;c>c^{*}.$$

This concept of spreading speed has been further developed by several researchers in the last decades from different view points including PDE’s argument, dynamical systems theory, probability theory, mathematical biology, etc. Spreading speeds of KPP-type reaction diffusion equations in homogeneous and periodic media have been extensively studied (see [8, 17, 29, 30, 44, 45] and the references cited therein). There is also an extensive literature on spreading phenomena for reaction diffusion systems. We refer for instance [3, 15, 22] and the references cited therein.

Recently spreading properties for KPP-type reaction-diffusion equations in more general environments have attracted a lot of attention, see [7, 9, 37, 40] and the references cited therein. In particular, Nadin and Rossi [37] studied spreading properties for KPP equation with local diffusion and general time heterogeneities. Furthermore, they obtained a definite spreading speed when the coefficients share some averaging properties.

The spreading properties of nonlocal diffusion equation as (1.1) has attracted a lot of interest in the last decades. Since the semiflow generated by nonlocal diffusion equations does not enjoy any regularization effects, this brings additional difficulties. Fisher-KPP equations or monostable problems in homogeneous environments have been studied from various point of views: wave front propagation (see [12, 39] and the references cited therein), hair trigger effect and spreading speed (see [2, 10, 13, 18, 34, 47] and the references cited therein). For the thin-tailed kernel, we refer for instance to [34] and the recent work [47] where a new sub-solution has been constructed to provide a lower bound of the spreading speed. Note also that the aforementioned work deals with possibly non-symmetric kernel where the propagation speed on the left and the right-hand side of the domain can be different. For the fat-tailed dispersion kernels the propagating behaviour of the solutions can be very different from the one observed with thin-tailed kernel. Acceleration may occur. We refer to [19, 21] for fat-tailed kernel and to [10] for fractional Laplace type dispersion.

Recently, wave propagation and spreading speeds for nonlocal diffusion problem with time and/or space heterogeneities have been considered. Existence and nonexistence of generalized travelling wave solutions have been discussed in [16, 25, 32, 41] and the references cited therein. For spreading speed results, we refer the reader to [24, 25, 31, 42] and the references cited therein. We also refer to [5, 46, 48] for the analysis of the spreading speed for systems with nonlocal diffusion.

As far as monotone problem is concerned, one may apply the well developed monotone semiflow method to study the spreading speed for nonlocal diffusion problems. We refer the reader to [30, 44] and to [24, 25] for time periodic systems.
In this work, we provide a new approach which is based on the construction of suitable propagating paths (namely, functions $t\mapsto X(t)$ with $\liminf_{t\to\infty}u(t,X(t))>0$) coupled with what we call a persistence lemma (see Lemma 2.6 below) for uniformly continuous solutions, to obtain lower estimate for the propagating set. This lemma roughly states that controlling from below the solution at $x=0$ and $X(t)$ for $t\gg 1$ implies a control of the solution $u=u(t,x)$ from below on the whole interval $x\in[0,kX(t)]$ for some $k\in(0,1)$ and $t\gg 1$. The proof of this lemma does not make use of the properties of the tail of the kernel, so that we expect our key persistence lemma to be applied for the study of acceleration phenomena for fat tailed dispersal kernel. However, the uniform continuity property for the solutions is important for our proof and this remains complicated to check. For the regularity results of some specific time global solutions to nonlocal diffusion equations, we refer the reader to [11, 32] for spatial heterogeneous case and to [16, 41] for time heterogeneous media. Here we are able to prove such a property for some specific initial data and logistic type nonlinearities.

Note that in [28] the authors consider the regularity problem. They show that when the nonlinear term satisfies $F_{u}(u)<\overline{K}$ for any $u\geq 0$, where $\overline{K}=\int_{\mathbb{R}}K(y)\mathrm{d}y$, then solutions of the homogeneous problem inherit the Lipschitz continuity property from those of their initial data, with a control of the Lipschitz constant for all time $t\gg 1$. In this note, we prove the uniform continuity of some solutions when the above condition fails (see Assumption 1.5 $(f4)$). This point is studied in Section 3.1, where we provide a class of initial data for which the solutions (of the nonlocal logistic equation) are uniformly continuous on $[0,\infty)\times\mathbb{R}$.

Now to state our results, we first introduce some notations and present our main assumptions. Let us define the important notion of the least mean for a bounded function.

If $h$ admits a mean value $\langle h\rangle$, that is, there exists

$$\langle h\rangle:=\lim\limits_{T\to+\infty}\frac{1}{T}\int_{0}^{T}h(t+s)\mathrm{d}t,\text{ uniformly with respect to }s\geq 0.$$

(1.4)

Then $\lfloor h\rfloor=\langle h\rangle$. Particularly, the time periodic, almost periodic and uniquely ergodic coefficients have the mean value. Here recall that a bounded and uniformly continuous function $f:\mathbb{R}\to\mathbb{R}$ is called uniquely ergodic if, for any continuous map $G:\mathcal{H}_{f}\to\mathbb{R}$, the following limit exists uniformly in $s\in\mathbb{R}$:

$$\lim\limits_{T\to+\infty}\frac{1}{T}\int_{s}^{s+T}G(f(\cdot+\tau))\mathrm{d}\tau,$$

where $\mathcal{H}_{f}:={\rm cl}\{f(\cdot+\tau),\;\tau\in\mathbb{R}\}$ is the closure of the translation set of $f$ under the local uniform topology.

Periodic, almost periodic and compactly supported functions are specific subclass of uniquely ergodic functions. A celebrated example of uniquely ergodic function is constructed from the Penrose tiling. For more examples and properties of almost periodic and uniquely ergodic functions, we refer the reader to [9, 33, 36].

An equivalent and useful characterization for the least mean of the function, as above, is given in the next lemma.

We are now able to present the main assumptions that will be needed in this note. First we assume that the kernel $K=K(y)$ enjoys the following set of properties:

Now we discuss our Fisher-KPP assumptions for the nonlinear term $F(t,u)=uf(t,u)$.

Let us now define some notations related to the speed function that will be used in the following. We define $\sigma(K)$, the abscissa of convergence of $K$, by

$$\sigma\left(K\right):=\sup\left\{\gamma>0:\;\int_{\mathbb{R}}K(y)e^{\gamma y}dy<\infty\right\}.$$

Assumption 1.3 $(ii)$ yields that $\sigma(K)\in(0,\infty]$. We set

$$L(\lambda):=\int_{\mathbb{R}}K(y)[e^{\lambda y}-1]\mathrm{d}y,\;\lambda\in\left[0,\sigma(K)\right),$$

(1.7)

as well for $\lambda\in(0,\sigma(K))$ and $t\geq 0$,

$$c(\lambda)(t):=\lambda^{-1}L(\lambda)+\lambda^{-1}\mu(t).$$

(1.8)

For a given function $a\in W^{1,\infty}(0,\infty)$, denote $c_{\lambda,a}$ the function given by

$$c_{\lambda,a}(t):=c(\lambda)(t)+a^{\prime}(t),\;\lambda\in(0,\sigma(K)),\;t\geq 0.$$

(1.9)

Obviously, it follows from Definition 1.1 that $\lfloor c_{\lambda,a}(\cdot)\rfloor=\lfloor c(\lambda)(\cdot)\rfloor$ for each $\lambda\in(0,\sigma(K))$. Next note that

$$\lfloor c(\lambda)(\cdot)\rfloor=\lambda^{-1}L(\lambda)+\lambda^{-1}\lfloor\mu\rfloor.$$

Now we state some properties of $\lfloor c(\lambda)(\cdot)\rfloor$ in the following proposition.

The above Proposition 1.8 has been mostly proved in [16] (see Proposition 2.8 in [16]) with a more general kernel which depends on $t$.

Here we only explain that $c_{r}^{*}>0$. To see this, note that for $\lambda\in(0,\sigma(K))$ one has

$$\lambda c(\lambda)(t)=\int_{\mathbb{R}}K(y)e^{\lambda y}\mathrm{d}y+\mu(t)-\overline{K},\;\forall t\geq 0.$$

Next due to Assumption 1.5 $(f4)$ and Lemma 1.2, there exists some function $a\in W^{1,\infty}(0,\infty)$ such that $\mu(t)-\overline{K}+a^{\prime}(t)\geq 0$ for all $t\geq 0$. This yields for all $\lambda\in(0,\sigma(K))$ and $t\geq 0$,

$$\lambda c(\lambda)(t)+a^{\prime}(t)=\int_{\mathbb{R}}K(y)e^{\lambda y}\mathrm{d}y+\mu(t)-\overline{K}+a^{\prime}(t)\geq\int_{\mathbb{R}}K(y)e^{\lambda y}\mathrm{d}y>0,$$

that rewrites $c_{r}^{*}>0$ since $\lfloor a^{\prime}\rfloor=0$. The result follows.

To state our spreading result, we impose in the following that the condition discussed in the previous remark is satisfied, that means $\lambda_{r}^{*}$ is different from the convergence abscissa.

Using the above properties for the speed function $c(\lambda)(\cdot)$ and its least mean value, we are now able to state our main results.

For the lower estimates of the propagation set, we first state our result for a specific function $f=f(t,u)$ of the form $f(t,u)=\mu(t)(1-u).$ In other words, we are considering the following non-autonomous logistic equation

$$\partial_{t}u(t,x)=\int_{\mathbb{R}}K(y)\left[u(t,x-y)-u(t,x)\right]\mathrm{d}y+\mu(t)u(t,x)\left(1-u(t,x)\right).$$

(1.12)

To enter the framework of Assumption 1.5, we assume that the function $\mu$ satisfies following conditions:

$$\begin{split}&t\mapsto\mu(t)\text{ is uniformly continuous and bounded with }\inf_{t\geq 0}\mu(t)>0,\\ &\text{ and the least mean of }\mu(\cdot)\text{ satisfies }\lfloor\mu\rfloor>\overline{K}.\end{split}$$

(1.13)

For this problem, our lower estimate of propagation set reads as follows.

Next as a consequence of the comparison principle, one obtains the following lower estimates of the propagation set to the right-hand side for more general nonlinearity satisfying Assumption 1.5.

In the above result we only consider the propagation to the right-hand side of the real line and obtain a propagation result on some interval of the form $[0,ct]$ for suitable speed $c$ and for $t\gg 1$. Note that the kernel is not assumed to be even, so that the propagation behaviours on the right and the left-hand sides can be different. For instance, different spreading speeds may arise at right and left-hand sides when the kernel is thin-tailed on both sides. To study the propagation behaviour of the left-hand side, it is sufficient to change $x$ to $-x$ in the above results.

The results stated in this section and more precisely the lower bounds for the propagation follows from the derivation of suitable regularity estimates for the solution. Here we show that the solutions of (1.12) with suitable initial data are uniformly continuous. Next Theorem 1.12 follows from the application of a general persistence lemma (see Lemma 2.6) for uniformly continuous solutions. This key lemma roughly ensures that if there is a uniformly continuous solution $u=u(t,x)$ admitting a propagating path $t\mapsto X(t)$, then $[0,kX(t)]$ with any $k\in(0,1)$ is a propagating interval, that is $u$ stays uniformly far from $0$ on this interval, in the large time. The idea of the proof of this lemma comes from the uniform persistence theory for dynamical systems for which we refer the reader to [23, 35, 43, 49] and references cited therein.

This paper is organized as follows. In Section 2, we recall comparison principles and derive our general key persistence Lemma. Section 3 is devoted to the derivation of some regularity estimates for the solutions of (1.12) with suitable initial data. With all these materials, we conclude the proofs of theorems and the corollary.

This section is devoted to the statement of the comparison principle and a key lemma that will be used to prove the inner propagation theorem, namely Theorem 1.12.