Abstract
Accelerated propagation is a new phenomenon associated with nonlocal diffusion problems. In this paper, we determine the exact rate of accelerated propagation in the Fisher-KPP equation with nonlocal diffusion and free boundaries, where the nonlocal diffusion operator is given by \(\displaystyle \int _{\mathbb {R}}J(x-y)u(t,y)dy-u(t,x)\), and the kernel function J(x) behaves like a power function near infinity, namely \(\lim _{|x|\rightarrow \infty } J(x)|x|^{\alpha }=\lambda >0\) for some \(\alpha \in (1,2]\). This is the precise range of \(\alpha \) where accelerated spreading can happen for such kernels. By constructing subtle upper and lower solutions, we prove that the location of the free boundaries \(x=h(t)\) and \(x=g(t)\) goes to infinity at exactly the following rates:
Here \(\mu >0\) is a given parameter in the free boundary condition. Accelerated propagation can also happen when \(\lim _{|x|\rightarrow \infty }J(x)|x|(\ln |x|)^\beta =\lambda >0\) for some \(\beta >1\). For this case, we prove that
These results considerably sharpen the corresponding ones in [20], and the techniques developed here open the door for obtaining similar precise results for other problems. A crucial technical point is that such precise conclusions on the propagation are achievable by finding the correct improvements on the form of the lower solutions used in [20], even though the precise long-time profile of the density function u(t, x) is still lacking.
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1 Introduction
In this paper we determine the exact rate of acceleration for the spreading behaviour governed by the Fisher-KPP equation with nonlocal diffusion and free boundaries considered in [10, 16, 20], which has the form:
where \(x=g(t)\) and \(x=h(t)\) are the moving boundaries to be determined together with u(t, x), which is always assumed to be identically 0 for \(x\in \mathbb {R}\setminus [g(t), h(t)]\)Footnote 1; d and \(\mu \) are given positive constants.
The initial function \(u_0(x)\) satisfies
The basic assumptions on the kernel function \(J: {\mathbb {R}}\rightarrow {\mathbb {R}}\) are
The nonlinear term f(u) is assumed to be a Fisher-KPP function, namely it satisfies
The nonlocal free boundary problem (1.1) may be viewed as a model describing the spreading of a new or invasive species with population density u(t, x), whose population range [g(t), h(t)] expands according to the free boundary conditions
that is, the expanding rate of the range [g(t), h(t)] is proportional to the outward flux of the population across the boundary of the range. Such a free boundary condition was proposed independently in [10, 12]; [12] assumes \(f(u)\equiv 0\), and hence the long-time dynamics of the model there is completely different from the Fisher-KPP case studied in [10] and here.
Problem (1.1) is a “nonlocal diffusion" version of the following free boundary problem with “local diffusion”:
where \(u_0\) is a \(C^2\) function which is positive in \((-h_0, h_0)\) and vanishes at \(x=\pm h_0\). For a special Fisher-KPP type of f(u), (1.3) was first studied in [17] (see [18] for more general f), as a model for the spreading of a new or invasive species with population density u(t, x), whose population range [g(t), h(t)] expands through its boundaries \(x=g(t)\) and \(x=h(t)\) according to the Stefan conditions \(g'(t)=-\mu u_x(t, g(t)),\; h'(t)=-\mu u_x(t,h(t))\). A deduction of these conditions based on some ecological assumptions can be found in [8].
By [17, 18], problem (1.3) admits a unique solution (u(t, x), g(t), h(t)) defined for all \(t>0\), and its long-time dynamical behaviour is characterised by a “spreading-vanishing dichotomy”: Either (g(t), h(t)) is contained in a bounded set of \(\mathbb {R}\) for all \(t>0\) and \(u(t,x)\rightarrow 0\) uniformly as \(t\rightarrow \infty \) (called the vanishing case), or (g(t), h(t)) expands to \(\mathbb {R}\) and u(t, x) converges to 1 locally uniformly in \(x\in \mathbb {R}\) as \(t\rightarrow \infty \) (the spreading case). Moreover, when spreading occurs,
and \(k_0\) is uniquely determined by a semi-wave problem associated to (1.3) (see [8, 18]).
Problem (1.3) is closely related to the corresponding Cauchy problem
Indeed, it follows from [15] that the unique solution (u, g, h) of (1.3) and the unique solution U of (1.4) are related in the following way: For any fixed \(T>0\), as \(\mu \rightarrow \infty \), \((g(t), h(t))\rightarrow \mathbb {R}\) and \(u(t,x)\rightarrow U(t,x)\) locally uniformly in \((t,x)\in (0, T]\times \mathbb {R}\). Thus (1.4) may be viewed as the limiting problem of (1.3) (as \(\mu \rightarrow \infty \)).
Problem (1.4) with \(U_0\) a nonnegative function having nonempty compact support has long been used to describe the spreading of a new or invasive species; see, for example, classical works of Fisher [24], Kolmogorov-Petrovski-Piscunov (KPP) [29] and Aronson-Weinberger [2].
In both (1.3) and (1.4), the dispersal of the species is described by the diffusion term \(d u_{xx}\), widely called a “local diffusion” operator, which is obtained from the assumption that individual members of the species move in space according to the rule of Brownian motion. One advantage of the nonlocal problem (1.1) over the local problem (1.3) is that the nonlocal diffusion term
in (1.1) is capable to include spatial dispersal strategies of the species beyond random diffusion modelled by the term \(d u_{xx}\) in (1.3). Here \(J(x-y)\) may be interpreted as the probability that an individual of the species moves from x to y in a unit of time.
The long-time dynamical behaviour of (1.1), similar to that of (1.3), is determined by a “spreading-vanishing dichotomy" (see Theorem 1.2 in [10]): As \(t\rightarrow \infty \), either
-
(i)
Spreading: \(\lim _{t\rightarrow +\infty } (g(t), h(t))=\mathbb {R}\) and \(\lim _{t \rightarrow +\infty }u(t,x)=1\) locally uniformly in \({\mathbb {R}}\), or
-
(ii)
Vanishing: \(\lim _{t\rightarrow +\infty } (g(t), h(t))=(g_\infty , h_\infty )\) is a finite interval and \(\lim _{t \rightarrow +\infty }u(t,x)=0\) uniformly for \(x\in [g(t),h(t)]\).
Criteria for spreading and vanishing were also obtained in [10] (see Theorem 1.3 there). In particular, if the size of the initial population range \(2h_0\) is larger than a certain critical number determined by an associated eigenvalue problem, then spreading always happens.
A new phenomenon for the nonlocal Fisher-KPP model (1.1), in comparison with (1.3), is that when spreading is successful, “accelerated spreading" may happen; namely one may have
It was shown in [16] that whether this new phenomenon happens is determined by the following threshold condition on the kernel function J:
More precisely, we have
Theorem A
([16]). Suppose that (J) and (f) are satisfied, and spreading happens to the unique solution (u, g, h) of (1.1). Then
As usual, when (J1) holds, we call \(c_0\) the spreading speed of (1.1), which is determined by the semi-wave solutions to (1.1); see [16] for details.
When (J1) is not satisfied, and hence accelerated spreading happens, the rate of growth of h(t) (and \(-g (t)\)) was investigated in [20] for kernel functions satisfying, for some \(\alpha >0\),
namely,
For such kernel functions, clearly condition \((\textbf{J})\) is satisfied only if \(\alpha >1\), and \(\mathbf {(J1)}\) is satisfied only if \(\alpha >2\). Thus accelerated spreading can happen exactly when \(\alpha \in (1,2]\). The following result was proved in [20]:
Theorem B
([20]). In Theorem A, if additionally the kernel function satisfies (1.5) for some \(\alpha \in (1, 2]\), then for \(t\gg 1\),
One naturally asks:
This question was left unanswered in [20]. Let us note that in the case of finite speed propagation, the speed can be determined via a traveling wave problem, where the wave profile determines the long-time profile of the density function u(t, x), which provides crucial information for the construction of suitable upper and lower solutions to yield the precise propagation speed. However, in the case of accelerated spreading, it is unknown whether the density function u(t, x) converges in some sense to a definite profile function, which makes the determination of the precise rate of acceleration particularly challenging.
The main purpose of this paper is to give a complete answer to the above question, although a precise asymptotic profile of u(t, x) is still lacking. Moreover, we will also treat a new case not considered in [20], namely
By finding the right improvements on the form of the lower solutions used in [20], we are able to prove the following result:
Theorem 1.1
Let the assumptions in Theorem A be satisfied.
-
(i)
If
$$\begin{aligned} \lim _{|x|\rightarrow \infty }J(x)|x|^\alpha =\lambda \in (0,\infty )\ \mathrm{for\ some }\ \alpha \in (1,2], \end{aligned}$$then
$$\begin{aligned} {\left\{ \begin{array}{ll}\displaystyle \lim _{t\rightarrow \infty }\frac{h(t)}{t\ln t}=\lim _{t\rightarrow \infty }\frac{-g (t)}{t\ln t}=\mu \lambda ,&{} \hbox { when } \alpha =2,\\ \displaystyle \lim _{t\rightarrow \infty }\frac{h(t)}{ t^{1/(\alpha -1)}}= \lim _{t\rightarrow \infty }\frac{-g (t)}{ t^{1/(\alpha -1)}}=\frac{2^{2-\alpha }}{2-\alpha }\mu \lambda , &{} \hbox { when } \alpha \in (1,2). \end{array}\right. } \end{aligned}$$ -
(ii)
If
$$\begin{aligned} \lim _{|x|\rightarrow \infty }J(x)|x| (\ln |x|)^\beta =\lambda \in (0,\infty )\ \mathrm{for\ some }\ \beta \in (1,\infty ), \end{aligned}$$then
$$\begin{aligned} \displaystyle \lim _{t\rightarrow \infty }\frac{\ln h(t)}{t^{1/\beta }}= \lim _{t\rightarrow \infty }\frac{\ln [-g (t)]}{t^{1/\beta }}= \left( \frac{2\beta \mu \lambda }{\beta -1}\right) ^{1/\beta }, \end{aligned}$$namely,
$$\begin{aligned} -g (t), h(t)=\exp \Big \{\Big [\Big (\frac{2\beta \mu \lambda }{\beta -1}\Big )^{1/\beta }+o(1)\Big ]t^{1/\beta }\Big \} \hbox { as } t\rightarrow \infty . \end{aligned}$$
Before ending this section, let us mention some further related works. Similar to the relationship between the local diffusion problems (1.3) and (1.4), problem (1.1) is closely related to the following nonlocal version of (1.4):
It was proved in [16] (see Theorem 5.3 there) that as \(\mu \rightarrow \infty \), the limiting problem of (1.1) is (1.6). Problem (1.6) and its many variations have been extensively studied in the literature; see, for example, [1, 3,4,5,6, 11, 13, 14, 22, 23, 25, 27, 28, 30, 31, 33, 36] and the references therein. In particular, if (J) and (f) are satisfied, and if the nonnegative initial function \(u_0\) has non-empty compact support, then the basic long-time dynamical behaviour of (1.6) is given by
Similar to (1.4), the nonlocal Cauchy problem (1.6) does not give a finite population range when \(t>0\). To understand the spreading behaviour of (1.6), one examines the level set
by considering the large time behaviour of
As \(t\rightarrow \infty \), \(|x^{\pm }_\lambda (t)|\) may go to \(\infty \) linearly in t or super-linearly in t, depending on whether the following threshold condition is satisfied by the kernel function, apart from (J),
Yagisita [36] has proved the following result on traveling wave solutions to (1.6):
Theorem C
([36]). Suppose that f satisfies (f) and J satisfies (J). If additionally J satisfies (J2), then there is a constant \(c_*>0\) such that (1.6) has a traveling wave solution with speed c if and only if \(c\ge c_*\).
Condition (J2) is often called a “thin tail" condition for J. When f satisfies (f), and J satisfies (J) and (J2), it is well known (see, for example, [22, 34]) that
with \(c_*\) given by Theorem C. On the other hand, if (f) and (J) hold but (J2) is not satisfied, then it follows from Theorem 6.4 of [34] that \(|x_\lambda ^{\pm }(t)|\) grows faster than any linear function of t as \(t\rightarrow \infty \), namely, accelerated spreading happens:
We refer to [1, 6, 7, 9, 21, 23, 25, 26, 32, 35] and references therein for further progress on accelerated spreading for (1.6) and related problems.
It is easily seen that (J2) implies (J1), but the reverse is not true; for example, \(J(x)=C(1+x^2)^{-\sigma }\) with \(\sigma >1\) satisfies (J1) (for some suitable \(C>0\)) but not (J2). Therefore accelerated propagation is more likely to happen in (1.6) than in the corresponding free boundary model (1.1).
The relationship between \(c_0=c_0(\mu )\) in Theorem A and \(c_*\) in (1.7) is given in the following result (see Theorems 5.1 and 5.2 of [16]):
Theorem D
([16]). Suppose that (J), (J1) and (f) hold. Then \(c_0(\mu )\) increases to \(c_*\) as \(\mu \rightarrow \infty \), where we define \(c_*=\infty \) when (J2) does not hold.
Finally we briefly describe the organisation of the paper. Throughout the remainder of this paper, unless otherwise specified, we assume that J satisfies (J) and either
or
We will prove some sharp estimates (see Lemmas 3.1, 3.2 and 4.1) under the above assumptions for J; Theorem 1.1 is a direct consequence of these more general results.
To be precise, in Sect. 2, we give some crucial preparatory results which will be used in the later sections; Lemma 2.1 contains key ingredients of the strategy of estimates toward the precise values of the limits in Theorem 1.1, while Lemma 2.2 reveals the right structure the lower solution should take in order to obtain these precise rates. In Sect. 3, we obtain sharp lower bounds for h(t) and \(-g(t)\), which constitute Lemma 3.1 (for the case (1.8) holds with \(\alpha \in (1,2)\) and for the case that (1.9) is satisfied) and Lemma 3.2 (for the case that (1.8) holds with \(\alpha =2\)); the proofs are based on subtle constructions of lower solutions of (1.1), which turns out to possess the right improvements on those used in [20]. In Sect. 4, we prove the sharp upper bounds for h(t) and \(-g(t)\), which is much less demanding technically.
2 Some preparatory results
We prove two lemmas in this section, which will play a crucial role in Sects. 3 and 4. The first contains important information on the strategy of estimating the key terms to reach the precise limiting values in Theorem 1.1, while the second is a rather general result, where only (J) is needed for the kernel function J, neither (1.8) nor (1.9) is required. The function \(\phi (t,x)\) in Lemma 2.2 dictates the structure of the lower solutions to be used to obtain the desired precise rates in the main results.
Lemma 1.2
For \(k>1\), \(\delta \in [0,1)\), define
Then
Proof
Case 1: (1.8) holds with \(\alpha \in (1,2)\).
Denote
A direct calculation gives
Moreover,
and by (J),
Clearly,
We have
By this and (1.8), we deduce
Thus,
Similarly,
Case 2: (1.9) holds.
Let \(A_1\) and \(A_2\) be as in Case 1. Clearly, \(0\le A_1\le 2\). A simple calculation gives
By (1.9), there exists \(C>0\) such that for all large \(k>0\),
and
where \(o_k(1)\rightarrow 0\) as \(k\rightarrow \infty \). Hence,
Similarly,
Case 3: (1.8) holds with \(\alpha =2\).
By direct calculation,
and by (J),
By (1.8), we have
where \(o_k(1)\rightarrow 0\) as \(k\rightarrow \infty \). Therefore,
Similarly,
The proof is finished. \(\square \)
Lemma 1.3
Suppose that J satisfies (J) but neither (1.8) nor (1.9) is required. Let \(1<\xi (t)<L(t)\) be functions in \(C([0,\infty ))\), \(\rho \ge 2\) a constant, and define
Then, for any \(\epsilon \in (0,1)\), there exists a constant \(\theta ^*=\theta ^*(\epsilon ,J)>1\), such that
provided that
Proof
Since \(||J||_{L^1}=1\), there is \(L_0>0\) depending only on J and \(\epsilon \) such that
Define
We note that \(\rho \ge 2\) implies that \(\psi (t,x)\) is a convex function of x when
Clearly
It is also easy to check that
which implies
Since \(\psi (t, x)>0\) for \(x\in (-1,1)\), \(\psi (t,\pm 1)=0\), and \(\psi (t,x)\) is convex in x for \(x\in [-1, -1+1/\xi (t)]\) and for \(x\in [1-1/\xi (t), 1]\), if we extend \(\psi (t,x)\) by \(\psi (t,x)=0\) for \(|x|>1\), then
We now verify (2.2) for \(x\in [0, L(t)]\); the proof for \(x\in [-L(t), 0]\) is parallel and will be omitted. We will divide the proof into two cases:
Case (a). For
a direct calculation gives
where \(L_0\) is given by (2.4) and we have used
Clearly
Then from the above calculations we obtain, for \( x\in [0, (1-\frac{1}{2\xi (t)})L(t)]\),
provided that
Case (b). For
we have, using \(-L(t)-x<-L_0\) and \(\phi (t, x)=0\) for \(x\ge L(t)\),
Since \(\phi (t,s)\) is convex in s for \(s\ge L(t)[1-\xi (t)^{-1}]\), and for \(x\in \left[ (1-\frac{1}{2\xi (t)})L(t), L(t)\right] \), \(y\in [0, L_0]\), we have
Then, we can use the convexity of \(\phi (t,\cdot )\) and (2.4) to obtain
Thus
Summarising, from the above conclusions in cases (a) and (b), we see that (2.2) holds if \(L(t)\ge \theta ^* \xi (t)\) for all \(t\ge 0\) with \(\theta ^*:=\frac{2^{\rho +1}L_0\rho }{\epsilon }>2L_0\). The proof is finished. \(\square \)
3 Lower bounds
Recall that J(x) satisfies (J) and either (1.8) or (1.9). The case (1.8) holds with \(\alpha \in (1,2)\) and the case (1.9) holds will be considered in subsection 3.1, while the case that (1.8) holds with \(\alpha =2\) will be considered in subsection 3.2.
From now on, in all our stated results, we will only list the conclusions for h(t); the corresponding conclusions for \(-g (t)\) follow directly by considering the problem with initial function \(u_0(-x)\), whose unique solution is given by \(({\tilde{u}}(t,x), {\tilde{g}} (t), {\tilde{h}}(t))=(u(t,-x), -h(t), -g (t))\).
3.1 The case (1.8) holds with \(\alpha \in (1,2)\) and the case (1.9) holds
Lemma 1.4
Assume that J satisfies (J) and either (1.8) with \(\alpha \in (1,2)\) or (1.9), f satisfies (f), and spreading happens to (1.1). Then
Proof
We construct a suitable lower solution to (1.1), which will lead to the desired estimate by the comparison principle.
Let \(\rho > 2\) be a large constant to be determined. For any given small \(\epsilon >0\), define for \(t\ge 0\),
and
where
It is easily seen that \(\underline{u}(t,x)\equiv K_2\) for \(|x|\le (1-\epsilon )\underline{h}(t)\). Moreover, \(\underline{u}\) is continuous, and \(\underline{u}_t\) exists and is continuous except when \(|x|=(1-\epsilon )\underline{h}(t)\), where \(\underline{u}_t\) has a jumping discontinuity. In what follows, we check that \((\underline{u}, \underline{g}, \underline{h})\) defined above forms a lower solution to (1.1). We will do this in three steps.
Step 1. We prove the inequality
which immediately gives
due to \(\underline{u}(t,x)=\underline{u}(t,-x)\) and \(J(x)=J(-x)\).
Using the definition of \(\underline{u}\), we have
Using Lemma 2.1, we obtain for large \(\underline{h}\) (guaranteed by \(\theta \gg 1\)),
and
Therefore, by the definition of \(K_1\), when (1.8) holds with \(\alpha \in (1,2)\), we have
and when (1.9) holds, we have
This proves (3.1).
Step 2. We prove the following inequality for \(t>0\) and \(|x|\in [0, \underline{h}(t)]\setminus \{(1-\epsilon )\underline{h}(t)\}\),
From the definition of \(\underline{u}\), we see that
and for \( (1-\epsilon )\underline{h}(t)<|x|<\underline{h}(t)\), if (1.8) holds with \(\alpha \in (1,2)\), then
where we have used \(\underline{h}'=\frac{K_1}{\alpha -1}\underline{h}^{2-\alpha }\); and if (1.9) holds, then
where we have utilized \(\underline{h}'=\frac{K_1^\beta }{\beta }\underline{h} (\ln \underline{h})^{1-\beta }\).
Claim. There is \(C_1=C_1(\epsilon )>0\) such that for \(x\in [-\underline{h}(t),\underline{h}(t)]\) and \(t\ge 0\),
The definition of \(\underline{u}\) indicates \(0\le \underline{u}(t,x)\le K_2=1-\epsilon <1\). By the properties of f, there exists \( \widetilde{C}_1:= \widetilde{C}_1(\epsilon )\in (0, d)\) such that
Using Lemma 2.2 with
for any given small \(\delta >0\), we can find large \(h_*=h_*(\delta ,\epsilon )\) such that for \(\underline{h}\ge h_*\) and \(|x|\le \underline{h}\),
Hence, due to \(d>\widetilde{C}_1\),
provided that \(\delta =\delta (\epsilon )>0\) is sufficiently small. Thus the claim holds with \(C_1=\widetilde{C}_1/3\).
To verify (3.2), it remains to prove
Since \(\underline{u}(x,t)\equiv 1-\epsilon \) for \(|x|<(1-\epsilon )\underline{h}(t)\), (3.4) holds trivially for such x. Hence we only need to consider the case of \((1-\epsilon )\underline{h}(t)<|x|<\underline{h}(t)\).
Since \(\theta \gg 1\) and \(0<\epsilon \ll 1\), for \(x\in [7\underline{h}(t)/8,\underline{h}(t)]\supset [(1-\epsilon )\underline{h}(t), \underline{h}(t)]\), we have
Hence, when (1.8) holds with \(\alpha \in (1,2)\), we obtain
and when (1.9) holds, we have
Similar estimates hold for \(x\in [-\underline{h}(t),-7\underline{h}(t)/8]\).
Now, if (1.8) holds with \(\alpha \in (1,2)\), then for \(|x|\in [(1-C_\epsilon )\underline{h}(t), \underline{h}(t)]\) with
we have
and for \((1-\epsilon )\underline{h}(t)<|x|\le (1-C_\epsilon )\underline{h}(t)\), using the definition of \(\underline{u}\), we obtain
since \(\theta \gg 1\) and \(\underline{h}(t)\ge \theta ^{1/(\alpha -1)}\), \(1-\alpha <0\). We have thus proved (3.4).
We next deal with the case that (1.9) holds. If |x| satisfies
with
then \(|x|\in [7\underline{h}(t)/8,\underline{h}(t)]\) and
For \((1-\epsilon )\underline{h}<|x|\le [1-\frac{\widetilde{C}_\epsilon }{(\ln \underline{h})^{1/(\rho -1)}}]\underline{h}\), from the definition of \(\underline{u}\), we deduce
since \(\underline{h}(t)\ge e^{K_1\theta ^{1/\beta }}\gg 1\) and we may choose \(\rho \) large enough such that \(1-\beta +(\rho -1)^{-1}<0\). The desired inequality (3.4) is thus proved.
Step 3. Completion of the proof by the comparison principle.
Since spreading happens, there is \(t_0>0\) large enough such that \([g(t_0),h(t_0)]\supset [-\underline{h}(0),\underline{h}(0)]\), and also
Moreover, from the definition of \(\underline{u}\), we see \(\underline{u}(x,t)=0\) for \(x=\pm \underline{h}(t)\) and \(t\ge 0\). Thus we are in a position to apply the comparison principle (see Theorem 3.1 in [10] and Remark 2.4 in [19], the latter explains why the jumping discontinuity of \(\underline{u}_t\) along \(|x|=(1-\epsilon ) \underline{h}(t)\) does not affect the conclusion) to conclude that
The desired conclusions then follow from the arbitrariness of \(\epsilon >0\) and the fact that \(D_{\epsilon /(2-\epsilon )}\rightarrow D_0\) as \(\epsilon \rightarrow 0\). The proof is finished. \(\square \)
3.2 The case that (1.8) holds with \({\alpha }=2\)
Lemma 1.5
If the conditions in Lemma 3.1 are satisfied except that J satisfies (1.8) with \({\alpha }=2\), then
Proof
For fixed \(\rho \ge 2\), \(0<\epsilon \ll 1\), \(0<\tilde{\epsilon }\ll 1\) and \(\theta \gg 1\), define
where
Obviously, for any \(t> 0\), \(\partial _t \underline{u}(t,x)\) exists for \(x\in [-\underline{h}(t), \underline{h}(t)]\) except when \(|x|=\underline{h}(t)-(t+\theta )^{\tilde{\epsilon }}\). However, the one-sided partial derivates \(\partial _t\underline{u}(t\pm 0, x)\) always exist.
Step 1. We show that for \(\theta \gg 1\),
which clearly implies, due to \(\underline{u}(t,x)=\underline{u}(t,-x)\) and \({J}(x)={J}(-x)\), that
Making use of the definition of \(\underline{u}\) and
for \(\theta \gg 1\), we obtain
Thanks to Lemma 2.1, for large \(\underline{h}\) (which is guaranteed by \(\theta \gg 1\)),
Hence, with \(\theta \gg 1\), we have
which proves (3.6).
Step 2. We show that for \(t>0\) and \(x\in [-\underline{h}(t),\underline{h}(t)]\) with \(|x|\not =\underline{h}(t)-(t+\theta )^{\tilde{\epsilon }}\),
for \(\theta \gg 1\).
From the definition of \(\underline{u}\), we obtain by direct calculation that, for \(t>0\),
Making use of Lemma 2.2 with
for any given small \(\delta >0\), we can find a large \(\theta _*=\theta _*(\delta ,\epsilon )\) such that for \(\theta \ge \theta _*\) and \(|x|\le \underline{h}(t)\),
Then, a similar analysis as in the proof of Lemma 3.1 shows that there exists \(C_1>0\), depending on \(\epsilon \) and \(\delta \), such that for \(\theta \gg 1\), \(x\in [-\underline{h}(t),\underline{h}(t)]\) and \(t\ge 0\),
Hence, to verify (3.7), we only need to show that
Clearly, (3.9) holds trivially for \(0\le |x|<\underline{h}(t)-(t+\theta )^{\tilde{\epsilon }}\) due to \(\underline{u}_t=0\) for such x. We next consider the remaining case \(\underline{h}(t)-(t+\theta )^{\tilde{\epsilon }}<|x|<\underline{h}(t)\).
Denote \(\eta =\eta (t):=(t+\theta )^{{\tilde{\epsilon }}}\). Using \(\theta \gg 1\) and (1.8), we obtain, for \(x\in [\underline{h}(t)-\eta (t),\underline{h}(t)]\),
The same estimate also holds for \(x\in [-\underline{h}(t),-\underline{h}(t)+\eta (t)]\). Therefore, for \(|x|\in [\underline{h}(t)-\eta (t),\underline{h}(t)]\), due to \(\rho >2\) and \(0<\tilde{\epsilon }\ll 1\), we have
if |x| further satisfies
On the other hand, for \(\underline{h}(t)-(t+\theta )^{\tilde{\epsilon }}<|x|<\underline{h}(t)-C_3{(t+\theta )^{{\tilde{\epsilon }}}}/[\ln (t+\theta )]^{1/(\rho -1)}\), using (3.8) and \(0<\tilde{\epsilon }\ll 1\), \(\theta \gg 1\), we deduce
Hence, (3.9) holds true. This concludes Step 2.
Step 3. We finally prove (3.5).
The definition of \(\underline{u}\) clearly gives \(\underline{u}(t,\pm \underline{h}(t))=0\) for \(t\ge 0\). Since spreading happens for (u, g, h) and \(K_2=1-\epsilon <1\), there is a large constant \(t_0>0\) such that
By Remark 2.4 in [19], we see that the comparison principle (Theorem 3.1 in [10]) applies to our situation here, even though \( \underline{u}_t(t,x)\) has a jumping discontinuity at \(|x|=\underline{h}(t)-(t+\theta )^{\tilde{\epsilon }}\). It follows that
which implies
Since \(\epsilon >0\) and \({\tilde{\epsilon }}>0\) can be arbitrarily small, we thus obtain (3.5) by letting \(\epsilon \rightarrow 0\) and \({\tilde{\epsilon }}\rightarrow 0\). This completes the proof of the lemma. \(\square \)
4 Upper bounds
Recall that we will only state and prove the conclusions for h(t), as the corresponding conclusion for \(-g (t)\) follows directly by considering the problem with initial function \(u_0(-x)\).
Lemma 1.6
Assume that J satisfies (J) and one of the conditions (1.8) and (1.9), f satisfies (f), and spreading happens to (1.1). Then
Proof
For any given small \(\epsilon >0\), define, for \(t\ge 0\),
where \(\theta \gg 1\) and
We verify that for \(t>0\),
which clearly implies
since \(\overline{u}(t,x)=\overline{u}(t,-x)\) and \(J(x)=J(-x)\).
Using \({\bar{u}}=1+\epsilon \), we have
By Lemma 2.1 with \(\delta =0\), we see that for large \({\bar{h}}\), which is guaranteed by \(\theta \gg 1\),
Therefore, when (1.8) holds with \(\alpha \in (1,2)\), by the definition of K, we have
When (1.8) holds with \({\alpha }=2\), we similarly obtain, due to \(\theta \gg 1\),
Finally, when (1.9) holds, we have
Thus (4.3) always holds.
Recalling that \(\overline{u}\ge 1\) is a constant, we get, for \(t>0\), \(x\in [-{\bar{h}}(t),{\bar{h}}(t)]\),
Note that condition (f) implies, by simple comparison with ODE solutions,
hence there is \(t_0>0\) such that
with the last part holding for large \(\theta \).
We are now in a position to use the comparison principle (Theorem 3.1 in [10]) to conclude that
By the arbitrariness of \(\epsilon >0\), we get (4.1). The proof is finished. \(\square \)
Data availability
Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
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Notes
Therefore \(\displaystyle \int _{\mathbb {R}}J(x-y)u(t,y)dy=\int _{g(t)}^{h(t)}J(x-y)u(t,y)dy\).
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Du, Y., Ni, W. Exact rate of accelerated propagation in the Fisher-KPP equation with nonlocal diffusion and free boundaries. Math. Ann. 389, 2931–2958 (2024). https://doi.org/10.1007/s00208-023-02706-7
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DOI: https://doi.org/10.1007/s00208-023-02706-7