This paper is concerned with the radially symmetric Fisher-KPP nonlocal diffusion equation with free boundary in dimension 3. For arbitrary dimension N≥2, in [
Citation: Yihong Du, Wenjie Ni. The Fisher-KPP nonlocal diffusion equation with free boundary and radial symmetry in R3[J]. Mathematics in Engineering, 2023, 5(2): 1-26. doi: 10.3934/mine.2023041
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This paper is concerned with the radially symmetric Fisher-KPP nonlocal diffusion equation with free boundary in dimension 3. For arbitrary dimension N≥2, in [
We consider a radially symmetric Fisher-KPP nonlocal diffusion equation with free boundary in RN (N≥2) of the form
{ut=d∫Bh(t)J(|x−y|)u(t,|y|)dy−du(t,|x|)+f(u),t>0,x∈Bh(t),u(t,|x|)=0,t>0,x∈∂Bh(t),h′(t)=μ|∂Bh(t)|∫Bh(t)∫RN∖Bh(t)J(|x−y|)u(t,|x|)dydx,t>0,h(0)=h0, u(0,|x|)=u0(|x|),x∈¯Bh0, | (1.1) |
where Bh(t):={x∈RN:|x|<h(t)}, with h(t) an unknown function to be determined with the density function u(t,|x|).
The basic assumptions on the kernel function J(|x|) are
(J): J∈C(R+)∩L∞(R+) is nonnegative, J(0)>0, ∫RNJ(|x|)dx=1.
Here and throughout the paper, R+=[0,∞).
The nonlinear function f is of Fisher-KPP type, namely, it satisfies
(f): {fisC1,f(0)=0<f′(0),thereexistsu∗>0suchthatf(u∗)=0>f′(u∗)and(u∗−u)f(u)>0foru∈(0,∞)∖{u∗},f(u)/uisnon−increasingforu>0.
The initial function u0 is required to satisfy
u0∈C(¯Bh0) is radially symmetric, u0=0 on ∂Bh0 andu0>0 in Bh0. | (1.2) |
For r:=|x| with x∈RN and ρ>0, denote
˜J(r,ρ)=˜J(|x|,ρ):=∫∂BρJ(|x−y|)dSy. |
Then (1.1) can be rewritten into the equivalent form
{ut(t,r)=d∫h(t)0˜J(r,ρ)u(t,ρ)dρ−du(t,r)+f(u),t>0,r∈[0,h(t)),u(t,h(t))=0,t>0,h′(t)=μhN−1(t)∫h(t)0∫+∞h(t)˜J(r,ρ)rN−1u(t,r)dρdr,t>0,h(0)=h0, u(0,r)=u0(r),r∈[0,h0]. | (1.3) |
(Here a universal constant is absorbed by μ.)
Problem (1.1) may be used to model the spreading of a new or invasive species, whose population density is given by u(t,|x|), and whose population range is the evolving ball Bh(t), where the spatial dispersal of the species is assumed to obey a nonlocal diffusion law governed by the kernel function J. The one dimensional case of (1.3) was studied in [7,14,17] (see also [8] for the case f(u)≡0).
In Du and Ni [18], we have shown that problem (1.1), or equivalently (1.3), admits a unique positive solution (u,h) defined for all t>0. Moreover, the long-time dynamical behaviour of (1.1) follows a spreading-vanishing dichotomy:
Proposition 1.1 (Spreading-vanishing dichotomy [18]). Suppose (J), (f) and (1.2) are satisfied. Let (u,h) be the solution of (1.1). Then one of the following alternatives must occur :
(i) (Spreading) limt→∞h(t)=∞ and
limt→∞u(t,|x|)=u∗locallyuniformlyinRN, |
(ii) (Vanishing) limt→∞h(t)=h∞<∞ and
limt→∞u(t,|x|)=0uniformlyforx∈Bh(t). |
Apart from giving the precise criteria which govern the spreading-vanishing dichotomy, the spreading speed is also determined in [18], which depends on the function J∗ given by
J∗(l):=∫RN−1J(|(l,x′)|)dx′,l∈R, | (1.4) |
where x′=(x2,...,xN)∈RN−1.
It is easy to see that (J) implies
{J∗∈C(R)∩L∞(R) is nonnegative, even, J∗(0)>0,∫RJ∗(l)dl=∫RNJ(|x|)dx=1. |
Moreover, it was shown in [18] that
J∗(l)=ωN−1∫∞|l|J(r)r(r2−l2)(N−3)/2dr, | (1.5) |
where ωk denotes the area of the unit sphere in Rk,
˜J(r,ρ)=ωN−123−NρrN−2∫ρ+r|ρ−r|([(ρ+r)2−η2][η2−(ρ−r)2])N−32ηJ(η)dη ∀r, ρ>0, | (1.6) |
and
∫∞0J∗(l)ldl=ωN−1N−1∫∞0J(r)rNdr. | (1.7) |
The threshold condition for (1.1) to have a finite spreading speed is
(J1): ∫∞0J(r)rNdr<+∞.
By [14,Theorem 1.2] and (1.7), we have the following conclusions about the associated one-dimensional semi-wave problem.
Proposition 1.2 (Semi-wave[14]). Suppose (J) and (f) hold. Then the following equations
{d∫0−∞J∗(x−y)ϕ(y)dy−dϕ(x)+cϕ′(x)+f(ϕ(x))=0,x<0,ϕ(−∞)=u∗, ϕ(0)=0,c=μ∫0−∞∫∞0J∗(x−y)ϕ(x)dydx, |
admit a solution pair (c,ϕ) with c>0 and ϕ′≤0 if and only if (J1) is satisfied. Moreover, when (J1) holds, the solution pair is unique, which we denote by (c0,ϕ0), and it has the property that ϕ′0(x)<0 for x≤0.
The result on the spreading speed in [18] is the following:
Proposition 1.3 (Spreading speed [18]). Assume the conditions in Proposition 1.1 are satisfied, and spreading happens to (1.1). Then
limt→∞h(t)t={c0if (J1) is satisfied,∞if (J1) is not satisfied, |
where c0 is given by Proposition 1.2.
More accurate estimates than that in Proposition 1.3 have been obtained in Du and Ni [19] for a natural class of kernel functions, namely those satisfying
J(|x|)∼|x|−βfor|x|≫1inRN. | (1.8) |
It is easily seen that β∈(N,N+1] is the exact range of β such that (J1) is not satisfied by such kernels while (J) holds. Therefore for such β accelerated spreading may happen according to Proposition 1.3. The following result on the precise rate of spreading is proved in [19].
Proposition 1.4 (Rate of accelerated spreading [19]). Suppose the conditions in Proposition 1.1 hold, and the kernel function satisfies (1.8) with β∈(N,N+1]. If spreading happens, then for t≫1,
{h(t)∼t1/(β−N)if β∈(N,N+1),h(t)∼tlntif β=N+1. |
Recall that ξ(s)∼η(s) means c1η(s)≤ξ(s)≤c2η(s) for some positive constants c1,c2 and all s in the specified range.
If β>N+1, then condition (J1) is automatically satisfied, and so by Proposition 1.3, the spreading has a finite speed c0. The following result of [19] describes how c0t−h(t) behaves as t→∞, where a slightly more general class of kernel functions than (1.8) is allowed, namely one only requires
J(|x|)=O(|x|−β)for|x|≫1inRN. | (1.9) |
Proposition 1.5 (Rate of shift [19]). Suppose the conditions in Proposition 1.1 hold, and moreover f is C2 and J satisfies (1.9) with β>N+1. If spreading happens, then for t≫1,
{c0t−h(t)∼lntifβ>N+2,|c0t−h(t)|=O(lnt)ifβ=N+2,|c0t−h(t)|=O(tN+2−β)ifβ∈(N+1,N+2). |
The purpose of this paper is to give a more accurate description of the spreading behaviour described in Proposition 1.5 when N=3.
Let us note that when N=3, (1.5) and (1.6) are reduced to, respectively,
J∗(l)=ω2∫∞|l|rJ(r)dr, | (1.10) |
and
˜J(r,ρ)=ω2ρr∫ρ+r|ρ−r|ηJ(η)dη ∀r, ρ>0. | (1.11) |
These allow considerable simplifications in the estimates of [19], and enable us to obtain more precise spreading rate when N=3. The following theorem is the main result of this paper.
Theorem 1.6. Suppose the conditions in Proposition 1.1 hold, J satisfies (1.8), f is C2 and
[f(u)/u]′<0foru>0. | (1.12) |
If spreading happens and N=3, then for t≫1,
{c0t−h(t)∼lntif β=N+2=5,c0t−h(t)∼tN+2−β=t5−βif β∈(N+1,N+2)=(4,5), |
The above results in Propositions 1.4, 1.5 and Theorem 1.6 reveal a striking difference of the behaviour of (1.1) from the pattern exhibited in the corresponding one dimension case, when β crosses the value N+2. More precisely, when β>N+2, which guarantees finite speed spreading, Proposition 1.5 shows that logarithmic shifting occurs, while in dimension one, no such shifting happens for this kind of J according to [17]. When β∈(N+1,N+2], where finite speed spreading still holds, Proposition 1.5 and Theorem 1.6 exhibit similar shifting behaviour to the N=1 case in [17]. When β∈(N,N+1] (which is the exact range of β that accelerated spreading may happen with such kernel functions), Proposition 1.4 gives the exact rate of the accelerated spreading, which is again in agreement with the pattern observed in the case N=1 in [17].
Let us now comment on the difficulty in treating the high dimensional radially symmetric problem (1.3) (for a general N≥2). To obtain sharp estimates for the spreading profile, the main difficulty arises from the fact that the kernel function in (1.3) is given by
˜J(r,ρ)=˜J(|x|,ρ):=∫∂BρJ(|x−y|)dSy, |
which inherits the properties of the original kernel function J(|x|) in a rather implicit way; see (1.6). On the other hand, the kernel function which determines the spreading speed of (1.3) is given by J∗ in (1.5). Therefore the spreading behaviour of (1.3) involves the complicated interplays between J, ˜J and J∗, among other things. Note that in dimension 1, J=J∗ and ˜J is not needed.
Many of the difficulties here do not occur in the local diffusion counterpart of (1.1), which was examined in [12,16]. It follows from [12] that the long-time dynamics of the local diffusion problem is roughly the same as that for the one dimension case considered in [15], and when spreading happens, limt→∞h(t)/t=c∗ for some c∗>0 determined by the semi-wave problem associated to the one dimensional model. So accelerated spreading never happens to the local diffusion problem. Moreover, by [16], there exists another constant ˆc>0 independent of the dimension N such that
limt→∞[h(t)−c∗t+(N−1)ˆclnt]=C |
for some constant C depending on the initial function u0.
It was shown in [13] that when μ→∞, the limiting problem of the local diffusion version of (1.1) is the corresponding Cauchy problem
{ut=dΔu+f(u) for (t,x)∈R+×RN,u(0,x)=u0(x) for x∈RN, | (1.13) |
which, since the pioneering works of Fisher [22] and Kolmogorov, Peterovski and Piskunov [24], has been extended and used to describe the propagation phenomena arising from invasion ecology and other problems. Similarly, the argument in [14] for the one dimension case can be easily extended to show that when μ→∞, the limiting problem of (1.1) is the nonlocal Cauchy problem
{ut=d[∫RNJ(|x−y|)u(t,y)dy−u(t,x)]+f(u) for (t,x)∈R+×RN,u(0,x)=u0(x) for x∈RN. | (1.14) |
As a nonlocal extension of (1.13), problem (1.14) and its numerous variations have been extensively studied in the last three decades (see, e.g., [1,2,3,4,5,9,10,20,21,23,25,26,27,29,30,31] and the references therein). If f satisfies (f), then the long-time behaviour of (1.14) with a compactly supported initial function u0 is roughly the same as (1.13), namely
limt→∞u(t,x)=u∗ locally uniformly for x∈RN, | (1.15) |
where u∗ is the unique positive zero of f(u) given in (f). A striking difference between (1.14) and (1.13) arises in the spreading speed, where accelerated spreading can happen to (1.14) when the kernel function J is fat-tailed (see, e.g., [5,23] for space dimension 1), while (1.13) always spreads with a finite speed, determined by the minimal speed of its traveling wave solutions.
For fractional Laplacian type nonlocal diffusion operators in any dimension N≥1, it was shown in [6,28] that the rate of accelerated spreading is given by e[c+o(1)]t for some c>0 depending on N and the fractional Laplacian. It should be noted that our basic condition (J) here is not satisfied by the corresponding kernel function of the fractional Laplacian (−Δ)s, which is given by
J(|x|)=|x|−(N+2s) (0<s<1). |
It would be interesting to see what happens to (1.1) if the kernel function J is allowed to behave like the kernel function of the fractional Laplacian. A related work with f≡0 can be found in [11].
Note that as a population model, (1.1) provides additional information. For example, it gives the precise spreading front of the species via the free boundaries, while (1.14) does not, since its solution u(t,x) is positive for all x∈RN once t>0; moreover, (1.14) predicts consistent success of spreading (see (1.15)), but the long-time dynamics of (1.1) is governed by a spreading-vanishing dichotomy, which seems more natural.
The rest of the paper is organised as follows. In Section 2, for convenience of the reader, we collect several results from previous works, which will be used later in the paper. Section 3 is devoted to the proof of Theorem 1.6, by constructing subtle upper and lower solutions, based on careful estimates involving the connections of ˜J and J∗.
In this section, we recall some basic facts from [17,18] for convenience of later use in the paper. Here, only N≥2 is required.
Lemma 2.1 (Maximum principle [18]). Let T>0, d>0, and g,h∈C([0,T]) satisfy g(0)≤h(0) and g(t)<h(t) for t∈(0,T]. Denote DT:={(t,x):t∈(0,T],g(t)<x<h(t)} and suppose that ϕ, ϕt∈C(¯DT), c∈L∞(DT), and
{ϕt≥d∫h(t)g(t)P(x,y)ϕ(t,y)dy+c(t,x)ϕ,(t,x)∈DT,ϕ(t,g(t))≥0,t∈Σgmin,ϕ(t,h(t))≥0,t∈Σhmax,ϕ(0,x)≥0,x∈[g(0),h(0)], |
where
{Σgmin={t∈(0,T]:Thereexistsϵ>0suchthatg(t)<g(s)fors∈[t−ϵ,t)},Σhmax={t∈(0,T]:Thereexistsϵ>0suchthath(t)>h(s)fors∈[t−ϵ,t)}, |
and the kernel function P satisfies
P∈C(R2)∩L∞(R2), P≥0, P(x,x)>0foralmostallx∈R. |
Then ϕ≥0 on ¯DT, and if additionally ϕ(0,x)≢0 in [g(0),h(0)], then ϕ>0 in DT.
Lemma 2.2 (Comparison principle [18]). Suppose (J) and (f) hold, and (u,h) solves (1.3) for t∈[0,T] with some T>0. For convenience we extend u by u(t,r)=0 for t∈[0,T] and r>h(t). Let r∗,h∗∈C([0,T]) be nondecreasing functions satisfying 0≤r∗(t)<h∗(t), and
ΩT:={(t,r):t∈(0,T],r∈(0,h∗(t))}, ΘT:={(t,r):t∈(0,T],r∈(r∗(t),h∗(t))}. |
Suppose v∈C(¯ΩT) is nonnegative with vt∈C(¯ΘT), and
ˆv(t,r):={u(t,r)forr∈[0,r∗(t)], t∈[0,T],v(t,r)forr∈(r∗(t),h∗(t)], t∈[0,T]. |
(i) If (v,r∗,h∗) satisfy h∗(0)≥h(0),
{v(0,r)≥u(0,r), r∈[0,h∗(0)],v(t,r)≥u(t,r), t∈[0,T],r∈[0,r∗(t)] |
and
{vt≥d[∫h∗(t)0˜J(r,ρ)ˆv(t,ρ)dρ−v(t,r)]+f(v),t∈(0,T],r∈(r∗(t),h∗(t)),v(t,h∗(t))≥0,t∈(0,T],h′∗(t)≥μhN−1∗(t)∫h∗(t)0∫+∞h∗(t)˜J(r,ρ)rN−1v(t,r)dρdr,t∈[0,T], |
then
h∗(t)≥h(t), v(t,r)≥u(t,r) for t∈(0,T],r∈[0,h(t)]. |
(ii) If (v,r∗,h∗) satisfy h∗(0)≤h(0),
{v(0,r)≤u(0,r), r∈[0,h(0)],v(t,r)≤u(t,r), t∈[0,T],r∈[0,r∗(t)] |
and
{vt≤d[∫h∗(t)0˜J(r,ρ)ˆv(t,ρ)dρ−v(t,r)]+f(v),t∈(0,T],r∈(r∗(t),h∗(t)),v(t,h∗(t))≤0,t∈(0,T],h′∗(t)≤μhN−1∗(t)∫h∗(t)0∫+∞h∗(t)˜J(r,ρ)rN−1v(t,r)dρdr,t∈[0,T], |
then
h∗(t)≤h(t), v(t,r)≤u(t,r) for t∈(0,T],r∈[0,h∗(t)]. |
Remark 2.3. In Lemma 2.2, if r∗(t)≡0, then the conclusions hold without requiring
{v(t,r)≥u(t,r)fort∈[0,T], r∈[0,r∗(t)]={0}inpart(i),v(t,r)≤u(t,r)fort∈[0,T], r∈[0,r∗(t)]={0}inpart(ii). |
Proof. When r∗(t)≡0, Σr∗min=∅, and the conclusion follows directly from the simple proof of Lemma 2.2 in [18] when Lemma 2.1 is used for w over t∈[t2,t1] and r∈[r∗(t),hϵ(t)].
Lemma 2.4 (Behaviour of semi-waves [17]). Let α>0 be a constant. Suppose that f satisfies (f) and the kernel function satisfies
∫∞0J∗(r)rαdr<+∞ for some α≥0, | (2.1) |
and ϕ(x) is a monotone solution of
{d∫0−∞J∗(x−y)ϕ(y)dy−dϕ(x)+cϕ′(x)+f(ϕ(x))=0,x<0,ϕ(−∞)=u∗, ϕ(0)=0 |
for some c>0. Then the following conclusions hold:
(i) If (2.1) holds for some α>0, then
∫−1−∞[u∗−ϕ(x)]|x|α−1dx<∞, |
which implies, by the monotonicity of ϕ(x),
0<u∗−ϕ(x)≤C|x|−αforsomeC>0andallx<0. |
(ii) If (2.1) does not hold for some α>0, then
∫−1−∞[u∗−ϕ(x)]|x|α−1dx=∞. |
In this section, we prove Theorem 1.6. So we always assume that N=3, (J), (f) and (1.12) hold, and there exist some constants 0<C1≤C2 and β>N+1=4 such that
C1|x|β≤J(x)≤C2|x|β for |x|≥1 in R3. | (3.1) |
Theorem 1.6 clearly is a consequence of the following result.
Proposition 3.1. If spreading happens to the solution (u(t,r),h(t)) of (1.3), then for t≫1,
{h(t)−c0t∼−t5−βif β∈(4,5),h(t)−c0t∼−lntif β≥5, | (3.2) |
where c0>0 is given by Proposition 1.2.
By Proposition 1.5, there is C=C(β)>0 such that
{h(t)−c0t≥−Ct5−βif β∈(4,5),h(t)−c0t≥−Clntif β≥5. |
Hence, to prove (3.2), it suffices to obtain the desired upper bound for h(t)−c0t, which will be carried out in subsections 3.1 and 3.2, according to whether d>f′(0) or d≤f′(0). The proof of the latter case is more involved, and is a modification of the proof of the former.
Lemma 3.2. Let the conditions in Proposition 3.1 be satisfied. If d>f′(0), then there is ˜C=˜C(β)>0 such that for large t,
{h(t)−c0t≤−˜Ct5−β,if β∈(4,5),h(t)−c0t≤−˜Clnt,if β≥5. | (3.3) |
Proof. Let α:=min{1,β−4}∈(0,1], and (c0,ϕ0) be the semi-wave pair in Proposition 1.2. Define
ϵ(t):=K1(t+θ)−α, δ(t):=−K2∫t0ϵ(τ)dτ |
and
{ˉh(t):=c0(t+θ)+δ(t), t≥0,¯u(t,r):=(1+ϵ(t))ϕ0(r−ˉh(t))+ω(t,r), t≥0, r≤[0,ˉh(t)], |
where
ω(t,r):=K3ξ(r−¯h(t))ϵ(t), |
with ξ∈C2(R) satisfying
0≤ξ(r)≤1, ξ(r)=1 for |r|<˜ϵ, ξ(r)=0 for |r|>2˜ϵ, | (3.4) |
and the positive constants θ, K1,K2,K3, ˜ϵ are to be determined.
Next we choose suitable θ, K1, K2, K3 and t0>0 such that (ˉu,ˉh) satisfies
{ˉut(t,r)≥d∫ˉh(t)0˜J(r,ρ)ˉu(t,ρ)dρ−dˉu(t,r)+f(ˉu(t,r)),t>0,r∈(ˉh(t)/2,ˉh(t)),ˉh′(t)≥μˉh2(t)∫ˉh(t)0r2ˉu(t,r)∫+∞ˉh(t)˜J(r,ρ)dρdr,t>0,ˉu(t,r)≥u(t+t0,r),ˉu(t,ˉh(t))=0,t>0,r∈[0,ˉh(t)/2],ˉu(0,r)≥u(t0,r), h(t0)≤ˉh(0),r∈[0,h(t0)]. | (3.5) |
If (3.5) is proved, then we can use Lemma 2.2 to obtain
{h(t+t0)≤ˉh(t) for t≥0,u(t+t0,r)≤¯u(t,r) for t≥0, r∈[0,h(t+t0)], |
which yields (3.3).
It remains to show (3.5), which will be carried out in three steps.
Step 1. We varify the second inequality of (3.5).
From (1.10) and (1.11) we see
˜J(r,ρ)≤ρrJ∗(r−ρ) for r,ρ>0. | (3.6) |
A direct computation gives
μˉh2(t)∫ˉh(t)0r2ˉu(t,r)∫+∞ˉh(t)˜J(r,ρ)dρdr= μˉh2∫ˉh0(1+ϵ)r2[ϕ0(r−ˉh)+w(t,r)]∫+∞ˉh˜J(r,ρ)dρdr≤ μ(1+ϵ)ˉh2∫ˉh0r[ϕ0(r−ˉh)+K3ϵ]∫+∞ˉhρJ∗(r−ρ)dρdr= μ(1+ϵ)ˉh2∫0−ˉh(r+ˉh)[ϕ0(r)+K3ϵ]∫+∞0(ρ+ˉh)J∗(r−ρ)dρdr= μ(1+ϵ)[∫0−ˉh∫+∞0[ϕ0(r)+K3ϵ]J∗(r−ρ)dρdr+A] |
with
A:=∫0−ˉh∫+∞0[(1+rˉh)(1+ρˉh)−1][ϕ0(r)+K3ϵ]J∗(r−ρ)dρdr. |
Moreover,
μ∫0−ˉh∫+∞0[ϕ0(r)+K3ϵ]J∗(r−ρ)dρdr= c0+K3ϵμ∫0−∞∫+∞0J∗(r−ρ)dρdr−μ∫−ˉh−∞∫+∞0[ϕ0(r)+K3ϵ]J∗(r−ρ)dρdr= c0+K3ϵμCJ−μ∫−ˉh−∞∫+∞0[ϕ0(r)+K3ϵ]J∗(r−ρ)dρdr, |
where
CJ:=∫0−∞∫+∞0J∗(r−ρ)dρdr=∫∞0ρJ∗(ρ)dρ. | (3.7) |
Here we have used change of integration order to obtain the last identity, and used (3.1) to conclude that CJ is finite.
Claim 1. There exists C3>0 such that for all t≥0,
−∫−ˉh−∞∫+∞0[ϕ0(r)+K3ϵ]J∗(r−ρ)dρdr+A≤−C3[(t+θ)−1+(t+θ)4−β]. |
Since ϕ(r) is non-increasing for r≤0, we have
∫−ˉh−∞∫+∞0[ϕ0(r)+K3ϵ]J∗(r−ρ)dρdr≥ [ϕ0(−ˉh)+K3ϵ]∫−ˉh−∞∫+∞0J∗(r−ρ)dρdr=[ϕ0(−ˉh)+K3ϵ]∫−ˉh−∞∫+∞−rJ∗(ρ)dρdr= [ϕ0(−ˉh)+K3ϵ]∫∞ˉh∫−ˉh−ρJ∗(ρ)drdρ=[ϕ0(−ˉh)+K3ϵ]∫∞ˉh(ρ−ˉh)J∗(ρ)dρ | (3.8) |
and
A=∫0−ˉh∫+∞0(rρˉh2+r+ρˉh)[ϕ0(r)+K3ϵ]J∗(r−ρ)dρdr≤∫0−ˉh∫+∞0rρˉh2[ϕ0(r)+K3ϵ]J∗(r−ρ)dρdr+∫0−ˉh∫ˉh0r+ρˉh[ϕ0(r)+K3ϵ]J∗(r−ρ)dρdr+[ϕ0(−ˉh)+K3ϵ]∫0−ˉh∫+∞ˉhr+ρˉhJ∗(r−ρ)dρdr=: W1(ˉh)+W2(ˉh)ˉh+W3(ˉh). |
Since ˉh(t)≥ˉh(0)=c0θ and ϕ0(−∞)=u∗, for θ≫1 we have ϕ0(−ˉh/2)≥u∗/2. Using this, (3.1) and ˉh(t)≤c0(t+θ), we obtain
W1(ˉh)= ∫0−ˉh∫+∞0rρˉh2[ϕ0(r)+K3ϵ]J∗(r−ρ)dρdr≤∫0−ˉh∫+∞0rρˉh2ϕ0(r)J∗(r−ρ)dρdr= ∫0−ˉh∫+∞−rr(ρ+r)ˉh2ϕ0(r)J∗(ρ)dρdr≤∫−ˉh/2−ˉh∫+∞2ˉhr(ρ+r)ˉh2ϕ0(r)J∗(ρ)dρdr≤ u∗2∫−ˉh/2−ˉh∫+∞2ˉhrρ2ˉh2J∗(ρ)dρdr=−3u∗32∫+∞2ˉhρJ∗(ρ)dρ=−3u∗32∫+∞2ˉhρ∫∞ρηJ(η)dηdρ≤−3u∗32∫+∞2ˉhρ∫∞ρC1η−β+1dηdρ=−3u∗C121+β(β−2)(β−4)ˉh4−β≤−3u∗C1c4−β021+β(β−2)(β−4)(t+θ)4−β. |
To estimate W2(ˉh), we first prove the following claim:
Claim 2. For any constants k,h>0,
B(h):=∫0−h∫h0(r+ρ)[ϕ0(r)+k]J∗(r−ρ)dρdr<0,B′(h)≤0. |
Since J∗ is even, we have
B(h)=∫0−h∫h0(r+ρ)[ϕ0(r)+k]J∗(r−ρ)dρdr=∫h0∫h0(ρ−r)[ϕ0(−r)+k]J∗(ρ+r)dρdr=∫h0∫r0(ρ−r)[ϕ0(−r)+k]J∗(ρ+r)dρdr+∫h0∫hr(ρ−r)[ϕ0(−r)+k]J∗(ρ+r)dρdr=∫h0∫hρ(ρ−r)[ϕ0(−r)+k]J∗(ρ+r)drdρ+∫h0∫hr(ρ−r)[ϕ0(−r)+k]J∗(ρ+r)dρdr=∫h0∫hr(ρ−r)[ϕ0(−r)−ϕ0(−ρ)]J∗(ρ+r)dρdr<0, |
where we have used r→ϕ0(−r) is strictly increasing and J∗(0)>0.
Using the first identity for B(h) above, we obtain
B′(h)=∫h0(ρ−h)[ϕ0(−h)+k]J∗(h+ρ)dρ+∫0−h(r+h)[ϕ0(r)+k]J∗(r−h)dr=∫h0(h−ρ)[ϕ0(−ρ)−ϕ0(−h)]J∗(h+ρ)dρ≤0. |
Claim 2 is thus proved.
Using Claim 2, we have, due to ˉh(t)≥c0θ≫1,
W2(ˉh)≤∫0−1∫10(r+ρ)[ϕ0(r)+K3ϵ]J∗(r−ρ)dρdr=∫0−1∫10(r+ρ)ϕ0(r)J∗(r−ρ)dρdr=:−˜C3<0. |
By change of order of integration and (3.8), we have
W3(ˉh)= [ϕ0(−ˉh)+K3ϵ]∫0−ˉh∫+∞ˉhr+ρˉhJ∗(r−ρ)dρdr= [ϕ0(−ˉh)+K3ϵ]∫0−ˉh∫+∞ˉh−r2r+ρˉhJ∗(ρ)dρdr= [ϕ0(−ˉh)+K3ϵ](∫2ˉhˉh∫0ˉh−ρ+∫∞2ˉh∫0−ˉh)2r+ρˉhJ∗(ρ)drdρ= [ϕ0(−ˉh)+K3ϵ]∫∞ˉh(ρ−ˉh)J∗(ρ)dρ≤ ∫−ˉh−∞∫+∞0[ϕ0(r)+K3ϵ]J∗(r−ρ)dρdr. |
Hence, due to ˉh(t)≤c0(t+θ), we have
−∫−ˉh−∞∫+∞0[ϕ0(r)+K3ϵ]J∗(r−ρ)dρdr+A≤ W1+W2(ˉh)ˉh≤−3u∗C1c4−β021+β(β−2)(β−4)(t+θ)4−β−˜C3c0(t+θ). |
Thus Claim 1 holds with
C3:=min{˜C3c0,3u∗C1c4−β021+β(β−2)(β−4)}. |
We may now use Claim 1, α=min{1,4−β}∈(0,1] and ˉh(t)≤c0(t+θ) to deduce
μˉh2(t)∫ˉh(t)0r2ˉu(t,r)∫+∞ˉh(t)˜J(r,ρ)dρdr≤ (1+ϵ)(c0+μCJK3ϵ−μC3[(t+θ)−1+(t+θ)4−β])≤ c0+K1(c0+2μCJK3)(t+θ)−α−μC3[(t+θ)−1+(t+θ)4−β]≤ c0−K1K2(t+θ)α=h′(t) |
provided that, apart from θ≫1, K1, K2 and K3 are small such that
K1(c0+2μCJK3+K2)≤μC3. | (3.9) |
Step 2. We verify the first inequality of (3.5), namely, for t>0 and r∈(¯h(t)/2,¯h(t)),
ˉut(t,r)≥d∫ˉh(t)0˜J(r,ρ)ˉu(t,ρ)dρ−dˉu(t,x)+f(ˉu(t,r)). | (3.10) |
We first show that for any continuous function ϕ(t,ρ) non-increasing in ρ,
∫ˉh(t)0˜J(r,ρ)ϕ(t,ρ)dρ≤∫ˉh(t)0J∗(r−ρ)ϕ(t,ρ)dρ for r∈[ˉh(t)/2,ˉh(t)]. | (3.11) |
From ˜J(r,ρ)≤ρrJ∗(r−ρ), we deduce, for r∈[¯h(t)/2,¯h(t)],
∫ˉh0˜J(r,ρ)ϕ(t,ρ)dρ≤∫ˉh0ρrJ∗(r−ρ)ϕ(t,ρ)dρ=∫ˉh0J∗(r−ρ)ϕ(t,ρ)dρ+∫ˉh0ρ−rrJ∗(r−ρ)ϕ(t,ρ)dρ=∫ˉh0J∗(r−ρ)ϕ(t,ρ)dρ+1r∫ˉh−r−rρJ∗(ρ)ϕ(t,ρ+r)dρ. |
Since ϕ(t,ρ) is nonincreasing for ρ≤0, and r≥ˉh(t)−r for r∈(¯h(t)/2,¯h(t)), we have
∫ˉh−r−rρJ∗(ρ)ϕ(t,ρ+r)dρ≤∫0−rρJ∗(ρ)ϕ(t,ρ+r)dρ+∫r0ρJ∗(ρ)ϕ(t,ρ+r)dρ=∫r0ρJ∗(ρ)[ϕ(t,r+ρ)−ϕ(t,r−ρ)]dρ≤0, |
which yields (3.11).
Without loss of generality, we may assume that ξ(r) is non-increasing for r∈[0,∞) and so ˉu(t,r) is decreasing in r for r∈[0,ˉh(t)]. Therefore (3.11) holds with ϕ(t,r)=ˉu(t,r).
We are now ready to check (3.10). It is clear that
ˉut(t,r)=−(1+ϵ)[c0+δ′(t)]ϕ′0(r−ˉh(t))+ϵ′(t)ϕ0(r−ˉh(t))+ωt(t,r) |
and
−(1+ϵ)c0ϕ′0(r−ˉh(t))= (1+ϵ)[d∫ˉh(t)−∞J∗(r−ρ)ϕ0(t,ρ)dρ−dϕ0(r−ˉh(t))+f(ϕ0(r−]barh(t)))]= d∫ˉh(t)−∞J∗(r−ρ)[ˉu(t,ρ)−ω]dρ−d[ˉu(t,r)−ω]+(1+ϵ)f(ϕ0(r−ˉh(t)))= d∫ˉh(t)−∞J∗(r−ρ)ˉu(t,ρ)dρ−dˉu(t,r)+d[ω(t,r)−∫ˉh(t)−∞J∗(r−ρ)ω(t,ρ)dρ]+(1+ϵ)f(ϕ0(r−ˉh(t)))≥ d∫ˉh(t)0J∗(r−ρ)ˉu(t,ρ)dρ−dˉu(t,r)+f(ˉu(t,r))+d[ω(t,r)−∫ˉh(t)−∞J∗(r−ρ)ω(t,ρ)dρ]+(1+ϵ)f(ϕ0(r−ˉh(t)))−f(ˉu(t,r)). |
Hence by (3.11) with ϕ=¯u, for t>0 and r∈[ˉh(t)/2,ˉh(t)],
ˉut(t,r)≥ d∫ˉh(t)0˜J(r,ρ)ˉu(t,ρ)dρ−dˉu(t,r)+f(ˉu(t,r))+d[ω(t,r)−∫ˉh(t)−∞J∗(r−ρ)ω(t,r)dρ]+(1+ϵ)f(ϕ0(r−ˉh(t)))−f(ˉu(t,r))−(1+ϵ)δ′(t)ϕ′0(r−ˉh(t))+ϵ′(t)ϕ0(r−h_(t))+ω′(t)=: d∫ˉh(t)0˜J(r,ρ)ˉu(t,ρ)dρ−dˉu(t,r)+f(ˉu(t,r))+B. |
In the following we show that B≥0 if θ≫1 and K1,K2,K3 are suitably chosen.
Claim 3. If ˜ϵ>0 in (3.4) is sufficiently small and θ is sufficiently large, then
d[ω(t,r)−∫ˉh(t)−∞J∗(r−ρ)ω(t,ρ)dρ]+(1+ϵ)f(ϕ0(r−ˉh(t)))−f(¯u(t,r))≥ d−f′(0)2ω(t,r)>0 for r∈[ˉh(t)−˜ϵ,ˉh(t)]. | (3.12) |
We have, for r∈[ˉh(t)−˜ϵ,ˉh(t)],
d[ω(t,r)−∫ˉh(t)−∞J∗(r−ρ)ω(t,ρ)dρ]=K3ϵ(t)[d−d∫0−∞J∗(r−ˉh(t)−ρ)ξ(ρ)dρ]≥ K3ϵ(t)[d−d∫0−2˜ϵJ∗(r−ˉh(t)−ρ)dρ]=K3ϵ(t)[d−d∫ˉh(t)−rˉh(t)−r−2˜ϵJ∗(ρ)dρ]≥ K3ϵ(t)[d−d∫˜ϵ−2˜ϵJ∗(ρ)dρ]≥K3ϵ(t)[d−d−f′(0)4]=[d−d−f′(0)4]ω(t,r), |
provided ˜ϵ∈(0,ϵ1] for some small ϵ1>0 depending on d−f′(0) and J∗.
Moreover, for r∈[ˉh(t)−˜ϵ,ˉh(t)], by (f) we obtain
(1+ϵ)f(ϕ0(r−ˉh(t)))−f(¯u(t,r))≥f((1+ϵ)ϕ0(r−ˉh(t)))−f(¯u(t,r))=f(¯u(t,r)−ω(t,r))−f(¯u(t,r)), |
and
0≤¯u(t,r)≤(1+ϵ)ϕ0(−˜ϵ)+K3ϵ≤2ϕ0(−˜ϵ)+θ−α |
if K1≤1 and K3≤1. So for such r, ¯u(t,r) and ω(t,r) are small when 0<˜ϵ≪1 and θ≫1. Therefore
f(¯u(t,r)−ω(t,r))−f(¯u(t,r))=−ω(t,r)[f′(¯u(t,r))+o(1)]=−ω(t,r)[f′(0)+o(1)]≥−[f′(0)+d−f′(0)4]ω(t,r) |
for r∈[ˉh(t)−˜ϵ,ˉh(t)], provided that ˜ϵ is small and θ is large. Hence, (3.12) holds.
Let us now show B≥0 for r∈[ˉh(t)−˜ϵ,ˉh(t)]. Denote
M:=supr≤0|ϕ′(r)|. |
By (3.12) and ω(t,r)=K3ϵ(t) for r∈[ˉh(t)−˜ϵ,ˉh(t)], we have
B≥ d−f′(0)2ω(t,r)−(1+ϵ)δ′(t)ϕ′0(r−ˉh(t))+ϵ′(t)ϕ0(r−h_(t))+ωt(t,r)≥ ϵ(t)[d−f′(0)2K3−2K2M−αu∗(t+θ)−1−K3α(t+θ)−1]≥ ϵ(t)[d−f′(0)2K3−2K2M−θ−1α(u∗+K3)]≥ 0 |
provided that we first fix K2 and K3 so that (3.9) holds and at the same time
d−f′(0)2K3−2K2M>0, | (3.13) |
and then choose θ sufficiently large.
Next, for fixed small ˜ϵ>0, we estimate B for x∈[ˉh(t)/2,ˉh(t)−˜ϵ].
Claim 4. For any η∈(0,u∗), there exists c1=c1(η)>0 such that
(1+ϵ)f(v)−f((1+ϵ)v)≥c1ϵ for v∈[η,u∗] and 0<ϵ≪1. | (3.14) |
Define
G(v):=f(v)/v. |
Then by (1.12), for v∈[η,u∗] and 0<ϵ≪1, there exists some ˜v∈[η,(1+ϵ)u∗]⊂[η,2u∗] such that
G(v)−G((1+ϵ)v)= G′(˜v)(−ϵv)≥˜c1ϵv, |
where ˜c1=minu∈[η,2u∗][−G′(u)]>0. It follows that
(1+ϵ)f(v)−f((1+ϵ)v)=(1+ϵ)v[G(v)−G((1+ϵ)v)]≥η2˜c1ϵ |
for v∈[η,u∗] and 0<ϵ≪1. This proves Claim 4.
By Claim 4, there exist positive constants Cl and Cf such that, for v=ϕ0(x−ˉh(t))∈[ϕ0(−˜ϵ),u∗],
(1+ϵ)f(v)−f((1+ϵ)v+ω)= (1+ϵ)f(v)−f((1+ϵ)v)+f((1+ϵ)v)−f((1+ϵ)v+ω)≥ Clϵ−CfK3ϵ |
when ϵ=ϵ(t) is small.
We also have
d[ω(t,r)−∫ˉh(t)−∞J∗(r−ρ)ω(t,ρ)dρ]≥−d∫ˉh(t)−∞J∗(r−ρ)ω(t,ρ)dρ≥−dK3ϵ(t), |
and
ωt(t,r)=−ξ′ˉh′K3ϵ(t)+ξK3ϵ′(t)≥−ξ∗K3ϵ(t)−K3α(t+θ)−1ϵ(t)≥−(ξ∗+αθ−1)K3ϵ(t), |
with ξ∗:=c0maxx∈R|ξ′(x)|.
Using these we obtain, for r∈[ˉh(t)/2,ˉh(t)−˜ϵ],
B≥ −dK3ϵ(t)+(1+ϵ)f(ϕ0(r−ˉh(t)))−f(ˉu(t,r))+2Mδ′(t)+ϵ′(t)+ωt(t,r)≥ Clϵ(t)−(Cf+d)K3ϵ(t)−2MK2ϵ(t)−α(t+θ)−1ϵ(t)−(ξ∗+αθ−1)K3ϵ(t)= ϵ(t)[Cl−K3(Cf+d)−2MK2−α(t+θ)−1−(ξ∗+αθ−1)K3]≥ ϵ(t)[Cl−K3(Cf+d)−2MK2−ξ∗K3−αθ−1(1+K3)]≥ 0 |
provided that we choose K2 and K3 small such that
Cl−K3(Cf+Cd)−2MK2−ξ∗K3>0 |
while keeping both (3.9) and (3.13) hold, and then choose θ>0 sufficiently large.
Therefore, (3.10) holds when K2,K3 and θ are chosen as above.
Step 3. We choose t0=t0(θ) such that the last two inequalities of (3.5) hold.
Clearly, for large θ>0 depending on K2,
2c0(t+θ)≥ˉh(t)≥c02(t+θ) for all t≥0. | (3.15) |
For the ODE problem
v′=f(v), v(0)=u∗+ϵ1 |
with small ϵ1>0, from f′(u∗)<0 we see that
u∗<v(t)≤u∗+ϵ1e˜Ft for all t≥0, |
where ˜F=maxu∈[u∗,u∗+ϵ1]f′(u)<0. A simple comparison argument shows that there is t∗>0 such that u(t,r)≤u∗+ϵ1 for t≥t∗ and r∈[0,h(t)]. Using comparison again we obtain
u(t+t∗,r)≤v(t)≤u∗+ϵ1e˜Ft for all t≥0, r∈[0,h(t)]. |
We claim that there is D∗>0 such that
J∗(ρ)≤D∗ρ2−β for ρ≥1. | (3.16) |
Indeed,
J∗(ρ)=ω2∫∞|ρ|ηJ(η)dη≤C2ω2∫∞ρη1−βdη≤C2ω2ρ2−ββ−2. |
Hence (3.16) holds, and it follows that
∫∞0J∗(η)ηα∗dη<∞ for1<α∗<β−3. |
Now we can use Lemma 2.4 to conclude the existence of Cϕ>0 such that
u∗−ϕ0(x)≤Cϕ|x|α∗ for x≤−1. | (3.17) |
In particular, for α∗∈(1,min{β−3,2}) we can use (3.17) to deduce, for t≥0 and r∈[0,ˉh(t)/2],
ˉu(t,r)= (1+ϵ(t))ϕ0(r−ˉh(t))≥(1+ϵ(t))ϕ0(−ˉh(t)/2)≥ (1+K1(t+θ)−1)(u∗−2α∗Cϕˉh(t)−α∗)≥(1+K1(t+θ)−1)[u∗−(4/c0)α∗Cϕ(t+θ)−α∗]= u∗+u∗K1(t+θ)−1−(4/c0)α∗Cϕ(t+θ)−α∗−(4/c0)α∗K1Cϕ(t+θ)−1−α∗≥ u∗+u∗K1(t+θ)−1/2≥u∗+ϵ1e˜F(t+t0−t∗)≥ u(t+t0,r) |
provided that θ≫1 and
u∗K1(t+θ)−1/2≥ϵ1e˜F(t+t0−t∗) for all t≥0. | (3.18) |
We show next that this is possible if t0 is chosen properly. Indeed, by Proposition 1.3, there is C1>0 such that h(t)≤2c0t+C1 for t≥0. It follows that
h(t0)≤c0θ/2<ˉh(0) for t0:=θ/4−C12c0 and θ≫1, |
and (3.18) is satisfied for this choice of t0 if
u∗K1(t+θ)−1/2≥ϵ1e˜F[(t+θ)/4−C1/(2c0)−t∗] for all t≥0, |
which is clearly valid since θ≫1 and ˜F<0.
Now all the remaining inequalities in (3.5) are satisfied and the proof of the lemma is complete.
Lemma 3.3. In Lemma 3.2, if f′(0)≥d, then (3.3) still holds.
Proof. This is a modification of the proof of Lemma 3.2, where in the definition of ˉu, the term ω(t,r) is changed to −ω(t,r), and a new term λ(t) is added; see details below.
As in the proof of Lemma 3.2, let α=min{β−4,1}∈(0,1], ˜ϵ>0 be a small constant and ξ∈C2(R) satisfy
0≤ξ(r)≤1, ξ(r)=1 for |r|<˜ϵ, ξ(r)=0 for |r|>2˜ϵ. |
Define
{ˉh(t):=c0(t+θ)+δ(t), t≥0,¯u(t,r):=(1+ϵ(t))ϕ0(r−ˉh(t)−λ(t))−ω(t,r), t≥0, r≤ˉh(t), |
where
ϵ(t):=K1(t+θ)−α, δ(t):=−K2∫t0ϵ(τ)dτ,ω(t,r):=K3ξ(r−ˉh(t))ϵ(t), λ(t):=K4ϵ(t), |
with the positive constants ˜ϵ, K1,K2,K3,K4 to be determined and θ≫1.
Denote
C˜ϵ:=minr∈[−2˜ϵ,0]|ϕ′0(r)|>0. |
Then for r∈[ˉh(t)−2˜ϵ,ˉh(t)],
ˉu(t,r)≥ ϕ0(−λ(t))−ω(t,r)≥C˜ϵλ(t)−K3ϵ(t)≥ϵ(t)(C˜ϵK4−K3)>0 |
if
K3=C˜ϵK4/2, | (3.19) |
which combined with ξ(r)=0 for |r|≥2˜ϵ implies
¯u(t,r)≥0 for t≥0, r≤ˉh(t). | (3.20) |
Step 1. We verify that for K1,K2 and K4 suitably small,
ˉh′(t)≥μˉh2(t)∫ˉh(t)0r2ˉu(t,r)∫+∞ˉh(t)˜J(r,ρ)dρdr for all t>0. | (3.21) |
By (3.6),
μˉh2(t)∫ˉh(t)0r2ˉu(t,r)∫+∞ˉh(t)˜J(r,ρ)dρdr≤ μ(1+ϵ)ˉh2∫ˉh0rϕ0(r−ˉh−λ)∫+∞ˉhρJ∗(r−ρ)dρdr= μ(1+ϵ)ˉh2∫ˉh0rϕ0(t,r)∫+∞ˉhρJ∗(r−ρ)dρdr+μ(1+ϵ)ˉh2∫ˉh0r[ϕ0(r−ˉh−λ)−ϕ0(t,r)]∫+∞ˉhρJ∗(r−ρ)dρdr=: I+II. |
Let M1:=supx≤0|ϕ′0(x)|. Then
II≤2μM1λˉh2∫ˉh0∫+∞ˉhrρJ∗(r−ρ)dρdr=2μM1λˉh2∫ˉh0∫+∞ˉh−rr(ρ+r)J∗(ρ)dρdr=2μM1λˉh2[∫ˉh0∫ˉhˉh−ρ+∫+∞ˉh∫ˉh0]r(ρ+r)J∗(ρ)drdρ≤2μM1λˉh2[2ˉh2∫ˉh0ρJ∗(ρ)dρ+2ˉh2∫+∞ˉhρJ∗(ρ)dρ]=4μM1CJλ, |
with CJ>0 given by (3.7), which is finite due to (3.16).
By the calculations in Step 1 of Lemma 3.2, we have
I= μˉh2(t)∫ˉh(t)0r2ˉu(t,r)∫+∞ˉh(t)˜J(r,ρ)dρdr≤ c0+K1(c0+2μCJK3)(t+θ)−α−μC3[(t+θ)−1+(t+θ)4−β]. |
Hence, by (3.19), we have
μˉh2(t)∫ˉh(t)0r2ˉu(t,r)∫+∞ˉh(t)˜J(r,ρ)dρdr≤I+II≤ 4μM1CJK4ϵ(t)+c0+K1(c0+2μCJK3)(t+θ)−α−μC3[(t+θ)−1+(t+θ)4−β]= c0+K1(4μM1CJK4+c0+μCJC˜ϵK4)(t+θ)−α−μC3[(t+θ)−1+(t+θ)4−β]≤ c0−K1K2(t+θ)−α=h′(t) |
if K_1 , K_2 and K_4 are small such that
\begin{align} K_1(4\mu M_1 C_JK_4+c_0+\mu C_J C_{\tilde\epsilon} K_4)+K_1K_2\leq \mu C_3. \end{align} | (3.22) |
Step 2. We show that by choosing K_2, K_4 suitably small and \theta sufficiently large, for t > 0 and r\in [\bar h(t)/2, \bar h(t)] ,
\begin{align} \overline u_t(t,r)\geq &\ d \int_{0}^{\bar h(t)} \tilde {J}(r, \rho) \overline u(t,\rho) {{\rm{d}}} \rho -d\, \overline u(t,r)+f( \overline u(t,r)). \end{align} | (3.23) |
Firstly we notice that for \theta\gg 1 and all t > 0 , \overline u_r(t, r) < 0 . Indeed, since \omega(t, r) = 0 for r\not\in [\bar h(t)-2\tilde\epsilon, \bar h(t)-\tilde\epsilon] , and \phi_0'(r) < 0 for r\leq 0 , it suffices to consider r\in [\bar h(t)-2\tilde\epsilon, \bar h(t)-\tilde\epsilon] . For such r ,
u_r(t, r)\leq \phi_0'(r-\bar h(t)-K_4\epsilon(t))-K_3\xi'(r-\bar h(t))\epsilon(t)\leq -C_1(\tilde\epsilon)+C_2(\tilde\epsilon)\theta^{-\alpha} < 0, |
where
C_1(\tilde\epsilon): = \min\limits_{s\in [-3\tilde\epsilon, 0]}|\phi_0'(s)|,\; C_2(\tilde\epsilon): = K_3\|\xi'\|_\infty. |
Hence we can use (3.11) to obtain
\int_{0}^{\bar h(t)} \tilde {J}(r, \rho) \overline u(t, \rho)d\rho\leq \int_{0}^{\bar h(t)} J_*(r-\rho) \overline u(t, \rho)d\rho {\mbox{ for }} r\in [\bar h(t)/2, \bar h(t)], t > 0. |
Using the definition of \overline u , we have
\begin{align*} \overline u_t(t,r) = &-(1+\epsilon)(\bar h'+\lambda')\phi_0'(r-\bar h-\lambda)+\epsilon'\phi_0(r-\bar h-\lambda)-\omega_t\\ = &-(1+\epsilon)\big(c_0+\delta'+\lambda'\big)\phi_0'(r-\bar h-\lambda)+\epsilon'\phi_0(r-\bar h-\lambda)-\omega_t \end{align*} |
and
\begin{align*} & -(1+\epsilon)c_0\phi_0'(r-\bar h-\lambda)\\ = &\ (1+\epsilon) \left[d \int_{- \infty}^{\bar h+\lambda} J_*(r-\rho) \phi_0(\rho-\bar h-\lambda) {{\rm{d}}} \rho -d \phi_0(r-\bar h-\lambda)+f(\phi_0(r-\bar h-\lambda)) \right]\\ \geq &\ (1+\epsilon) \left[d \int_{0}^{\bar h} J_* (r-\rho) \phi_0(\rho-\bar h-\lambda) {{\rm{d}}} \rho -d \phi_0(r-\bar h-\lambda)+f(\phi_0(r-\bar h-\lambda)) \right]\\ = &\ d \int_{0}^{\bar h} J_* (r-\rho) [ \overline u(t,\rho)+\omega] {{\rm{d}}} \rho -d [ \overline u(t,r)+\omega]+(1+\epsilon)f(\phi_0(r-\bar h-\lambda))\\ = &\ \int_{0}^{\bar h} J_*(r-\rho) \overline u(t,\rho) {{\rm{d}}} \rho -d \overline u(t,r)\\ &-d \left[\omega(t,r)-\int_{0}^{\bar h}J_*(r-\rho)\omega(t,\rho) {{\rm{d}}} \rho \right] +(1+\epsilon)f(\phi_0(r-\bar h-\lambda)). \end{align*} |
Hence, for r\in [\bar h(t)/2, \bar h(t)] and t > 0 ,
\begin{align*} \overline u_t(t,r)\geq &\ d \int_{0}^{\bar h(t)} \tilde J(r,\rho) \overline u(t,\rho) {{\rm{d}}} \rho -d \overline u(t,r)+f( \overline u(t,r))+A(t,r) \end{align*} |
with
\begin{align*} A(t,r): = & -d \left[\omega(t,r)-\int_{0}^{\bar h} J_*(r-\rho)\omega(t,\rho) {{\rm{d}}} \rho \right]+(1+\epsilon)f(\phi_0(r-\bar h-\lambda))-f( \overline u(t,r))\\ &-(1+\epsilon)(\delta'+\lambda')\phi_0'(r-\bar h-\lambda)+\epsilon'\phi_0(r-\underline h-\lambda)-\omega_t. \end{align*} |
To show (3.23), it remains to choose suitable K_2, K_4 and \theta such that A(t, r)\geq { 0} for t > 0 and r\in [\bar h(t)/2, \bar h(t)] .
Claim: There exists \tilde J_0 > 0 depending on \tilde\epsilon such that for all small \tilde\epsilon_0\in (0, \tilde{\epsilon}/2) , we have
\begin{equation} \begin{aligned} &-d \left[\omega(t,r)-\int_{0}^{\bar h}{J_*}(r-\rho)\omega(t,\rho) {{\rm{d}}} \rho \right]+(1+\epsilon)f(\phi_0(r-\bar h-\lambda))-f( \overline u(t,r))\\ &\geq \; \tilde J_0\, \omega(t,r) \ {\mbox{ for }} \ r\in [\bar h(t)-\tilde\epsilon_0,\bar h(t)]. \end{aligned} \end{equation} | (3.24) |
Indeed, for r\in [\bar h(t)-\tilde\epsilon_0, \bar h(t)] ,
\begin{align*} &d \left[\omega(t,r)-\int_{0}^{\bar h(t)}{J_*}(r-\rho)\omega(t,\rho) {{\rm{d}}} \rho \right] = K_3\epsilon(t) \left[ d-d \int_{0}^{\bar h(t)}{J}_*(r-\rho) \xi(\rho-\bar h(t)) {{\rm{d}}} \rho \right]\\ \leq&\ K_3\epsilon(t) \left[ d- d \int_{\bar h(t)-\tilde{\epsilon}}^{\bar h(t)}{J}_*(r-\rho) {{\rm{d}}} \rho \right] = K_3\epsilon(t) \left[ d- d \int_{\bar h(t)-\tilde{\epsilon}-r}^{\bar h(t)-r} {J}_*(\rho) {{\rm{d}}} \rho \right]\\ \leq&\ d \omega (t,r) \left[1-\int_{-\tilde{\epsilon}+\tilde\epsilon_0}^{0}{J}_*(\rho ) {{\rm{d}}} \rho \right]\leq d \omega (t,r) \left[1-\int_{-\tilde{\epsilon}/2}^{0}{J}_*(\rho) {{\rm{d}}} \rho \right]. \end{align*} |
On the other hand, for r\in [\bar h(t)-\tilde\epsilon_0, \bar h(t)] , we have
\begin{align*} &(1+\epsilon)f(\phi_0(x-\bar h-\lambda)-f( \overline u) \geq f((1+\epsilon)\phi_0(x-\bar h-\lambda))-f( \overline u)\\ = &\ f( \overline u+\omega)-f( \overline u) = \omega\Big(f'( \overline u)+o(1)\Big) = \Big(f'(0)+o(1)\Big)\omega \end{align*} |
since both \overline u(t, r) and \omega(t, r) are close to { 0} for r\in [\bar h(t)-\tilde\epsilon_0, \bar h(t)] with \tilde\epsilon_0 small and \theta\gg 1 .
Hence, for such r and \tilde\epsilon_0 , since f'(0)\geq d ,
\begin{align*} &-d \left[\omega(t,r)-\int_{0}^{\bar h(t)}{J}_*(r-\rho )\omega(t,\rho) {{\rm{d}}} \rho \right]+(1+\epsilon)f(\phi_0(r-\bar h(t)))-f( \overline u(t,r))\\ \geq &\ d \omega \left[-1+\int_{-\tilde{\epsilon}/2}^{0}{J}_*(\rho ) {{\rm{d}}} \rho \right] + f'(0) \omega +o(1)\omega \\ \geq &\ \tilde J_0\,\omega(t,r), \ \ \ \ \ {\mbox{ with }}\ \ \tilde J_0: = \frac d2 \int_{-\tilde{\epsilon}/2}^{0}{J_*}(\rho) {{\rm{d}}} \rho. \end{align*} |
This proves (3.24).
Clearly for r\in [\bar h(t)-\tilde\epsilon_0, \bar h(t)] ,
\begin{align*} -\omega_t(t,r) = \alpha K_3K_1(t+\theta)^{-\alpha-1} \geq { 0}. \end{align*} |
Denoting M_1: = \sup\limits_{x\leq 0}|\phi_0'(x)| , we obtain, for r\in [\bar h(t)-\tilde\epsilon_0, \bar h(t)] and small \tilde\epsilon_0 ,
\begin{align*} A(t,r)\geq &\ \tilde J_0K_3\epsilon(t)+2(\delta'(t)+\lambda'(t))M_1+\epsilon'(t)u^*\\ = &\ \tilde J_0K_3\epsilon(t)+2\epsilon(t)(-K_2-K_4\alpha(t+\theta)^{-1})M_1 -\alpha (t+\theta)^{-1}\epsilon(t)u^*\\ \geq &\ \epsilon(t)\bigg[ \tilde J_0K_3-2(K_2+K_4\alpha\theta^{-1})M_1 -\alpha\theta^{-1}u^*\bigg]\\ = &\ \epsilon(t)\bigg[ \tilde J_0K_3-2K_2M_1-\theta^{-1}\Big(K_4\alpha M_1 +\alpha u^*\Big)\bigg]\\ \geq&\ { 0} \end{align*} |
provided that K_2 is chosen small so that (3.22) holds and
\begin{equation} \tilde J_0K_3-2K_2M_1 > 0, \end{equation} | (3.25) |
and \theta is chosen sufficiently large.
We next estimate A(t, r) for r\in [\bar h(t)/2, \bar h(t)-\tilde\epsilon_0] . From Claim 4 in the proof of Lemma 3.2, there exist positive constants C_l = C_l(\tilde\epsilon_0) and C_f such that, for v = \phi_0(r-\bar h(t-\lambda(t)))\in [\phi_0(-\tilde{\epsilon}_0), u^*] ,
\begin{align*} &(1+\epsilon)f(v)-f((1+\epsilon)v-\omega)\\ = &\ (1+\epsilon)f(v)-f((1+\epsilon)v)+f((1+\epsilon)v)-f((1+\epsilon)v-\omega)\\ \geq &\ C_l\epsilon -C_f\omega\geq C_l\epsilon -C_fK_3\epsilon \end{align*} |
when \epsilon = \epsilon(t) is small. Hence
\begin{align*} &(1+\epsilon)f(\phi_0(r-\bar h-\lambda))-f(\bar u(t,r))\\ \geq &\ C_l\epsilon -C_fK_3\epsilon \ {\mbox{ for }} \ \ \ r\in [\bar h(t)/2, \bar h(t)-\tilde\epsilon_0], \ 0 < \tilde\epsilon_0\ll 1. \end{align*} |
Clearly,
\begin{align*} -d \left[\omega(t,r)-\int_{- \infty}^{\bar h(t)} {J}(r-\rho)\omega(t,\rho) {{\rm{d}}} \rho \right] \geq\ -dK_3\epsilon(t), \end{align*} |
and
\begin{align*} \omega_t(t,r) = -K_3\xi'\bar h'(t)\epsilon(t) + K_3\xi \epsilon'(t) \leq \xi_*K_3\epsilon(t) \end{align*} |
with \xi_*: = c_0\max_{x\in {\mathbb R}}|\xi'(r)| .
We thus obtain, for r\in [\bar h(t)/2, \bar h(t)-\tilde\epsilon_0] and 0 < \tilde\epsilon_0\ll 1 ,
\begin{align*} A(t,r)\geq &\ -K_3\epsilon(t)d +(1+\epsilon)f(\phi_0(r-\bar h))-f( \overline u)+2M_1 (\delta'+\lambda') +\epsilon' -\omega_t\\ \geq &\ C_l\epsilon (t) -K_3\epsilon(t) (d +C_f +\xi_* )+2M_1 (-K_2\epsilon(t)+K_4\epsilon'(t)) +\epsilon'(t) \\ \geq &\ \epsilon(t)\bigg[C_l-K_3(d+C_f +\xi_* )-2M_1(K_2+K_4\alpha(t+\theta)^{-1}) -\alpha(t+\theta)^{-1} \bigg]\\ \geq &\ \epsilon(t)\bigg[C_l-K_3\Big(d+C_f +\xi_* \Big)-2M_1K_2 -\theta^{-1}\alpha \Big(2M_1K_4 +1\Big)\bigg]\\ \geq &\ { 0} \end{align*} |
if we choose K_2 and K_4 small so that (3.22) and (3.25) hold and at the same time, recalling (3.19),
C_l-K_4\Big(d+C_f+\xi_*\Big)C_{\tilde\epsilon}/2-2M_1K_2 > 0, |
and then choose \theta sufficiently large. Hence, (3.23) is satisfied if K_2 and K_4 are chosen small as above, and \theta is sufficiently large.
Step 3. We show that (3.3) holds.
As in the proof of Lemma 3.2, we can choose sufficient large \theta and t_0 such that
\begin{align*} &\bar h(0)\geq 2h(t_0),\\ &\bar u(t,r)\geq u(t+t_0,r)\ {\mbox{ for }} \ \ r\in [0,\bar h(t)/2],\ t\geq 0. \end{align*} |
It follows that
\bar u(0, r)\geq u(t_0,r) {\mbox{ for }} r\in [0, h(t_0)]. |
From (3.20), we have
\begin{align*} \overline u(t,\bar h(t))\geq { 0} \ {\mbox{ for }} \ t\geq 0. \end{align*} |
These inequalities together with (3.21) and (3.23) allow us to use the comparison principle to conclude that
\begin{align*} h(t+t_0)\leq \bar h(t),\ u(t+t_0,r)\leq \overline u(t,r) \ {\mbox{ for }}\ t\geq 0,\ r\in [ 0, h(t+t_0)], \end{align*} |
which implies (3.3). The proof of the lemma is now complete.
The research of both authors was supported by the Australian Research Council.
The authors declare no conflict of interest.
[1] |
M. Alfaro, J. Coville, Propagation phenomena in monostable integro-differential equations: Acceleration or not?, J. Differ. Equations, 263 (2017), 5727–5758. https://doi.org/10.1016/j.jde.2017.06.035 doi: 10.1016/j.jde.2017.06.035
![]() |
[2] | F. Andreu-Vaillo, J. M. Mazón, J. D. Rossi, J. Toledo-Melero, Nonlocal diffusion problems, Providence, Rhode Island: AMS, 2010. https://doi.org/10.1090/surv/165 |
[3] |
P. W. Bates, G. Zhao, Existence, uniqueness and stability of the stationary solution to a nonlocal evolution equation arising in population dispersal, J. Math. Anal. Appl., 332 (2007), 428–440. https://doi.org/10.1016/j.jmaa.2006.09.007 doi: 10.1016/j.jmaa.2006.09.007
![]() |
[4] |
H. Berestycki, J. Coville, H. Vo, On the definition and the properties of the principal eigenvalue of some nonlocal operators, J. Funct. Anal., 271 (2016), 2701–2751. https://doi.org/10.1016/j.jfa.2016.05.017 doi: 10.1016/j.jfa.2016.05.017
![]() |
[5] |
E. Bouin, J. Garnier, C. Henderson, F. Patout, Thin front limit of an integro-differential Fisher-KPP equation with fat-tailed kernels, SIAM J. Math. Anal., 50 (2018), 3365–3394. https://doi.org/10.1137/17M1132501 doi: 10.1137/17M1132501
![]() |
[6] |
X. Cabré, J.-M. Roquejoffre, The influence of fractional diffusion in Fisher-KPP equations, Commun. Math. Phys., 320 (2013), 679–722. https://doi.org/10.1007/s00220-013-1682-5 doi: 10.1007/s00220-013-1682-5
![]() |
[7] |
J. Cao, Y. Du, F. Li, W. T. Li, The dynamics of a Fisher-KPP nonlocal diffusion model with free boundaries, J. Funct. Anal., 277 (2019), 2772–2814. https://doi.org/10.1016/j.jfa.2019.02.013 doi: 10.1016/j.jfa.2019.02.013
![]() |
[8] |
C. Cortázar, F. Quirós, N. Wolanski, A nonlocal diffusion problem with a sharp free boundary, Interface. Free Bound., 21 (2019), 441–462. https://doi.org/10.4171/ifb/430 doi: 10.4171/ifb/430
![]() |
[9] |
J. Coville, L. Dupaigne, On a non-local equation arising in population dynamics, Proc. Roy. Soc. Edinb. A, 137 (2007), 727–755. https://doi.org/10.1017/S0308210504000721 doi: 10.1017/S0308210504000721
![]() |
[10] |
J. Coville, J. Davila, S. Martinez, Pulsating fronts for nonlocal dispersion and KPP nonlinearity, Ann. Inst. Henri Poincaré (C) Non Linear Anal., 30 (2013), 179–223. https://doi.org/10.1016/j.anihpc.2012.07.005 doi: 10.1016/j.anihpc.2012.07.005
![]() |
[11] |
F. del Teso, J. Endal, J. L. Vazquez, The one-phase fractional Stefan problem, Math. Mod. Meth. Appl. Sci., 31 (2021), 83–131. https://doi.org/10.1142/S0218202521500032 doi: 10.1142/S0218202521500032
![]() |
[12] |
Y. Du, Z. Guo, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary Ⅱ, J. Differ. Equations, 250 (2011), 4336–4366. https://doi.org/10.1016/j.jde.2011.02.011 doi: 10.1016/j.jde.2011.02.011
![]() |
[13] |
Y. Du, Z. Guo, The Stefan problem for the Fisher-KPP equation, J. Differ. Equations, 253 (2012), 996–1035. https://doi.org/10.1016/j.jde.2012.04.014 doi: 10.1016/j.jde.2012.04.014
![]() |
[14] |
Y. Du, F. Li, M. Zhou, Semi-wave and spreading speed of the nonlocal Fisher-KPP equation with free boundaries, J. Math. Pure Appl., 154 (2021), 30–66. https://doi.org/10.1016/j.matpur.2021.08.008 doi: 10.1016/j.matpur.2021.08.008
![]() |
[15] |
Y. Du, Z. Lin, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal., 42 (2010), 377–405. https://doi.org/10.1137/090771089 doi: 10.1137/090771089
![]() |
[16] |
Y. Du, H. Matsuzawa, M. Zhou, Spreading speed and profile for nonlinear Stefan problems in high space dimensions, J. Math. Pure. Appl., 103 (2015), 741–787. https://doi.org/10.1016/j.matpur.2014.07.008 doi: 10.1016/j.matpur.2014.07.008
![]() |
[17] | Y. Du, W. Ni, Rate of propagation for the Fisher-KPP equation with nonlocal diffusion and free boundaries, 2021, preprint. Available from: http://turing.une.edu.au/ydu/papers/SpreadingRate-July2021.pdf |
[18] | Y. Du, W. Ni, The high dimensional Fisher-KPP nonlocal diffusion equation with free boundary and radial symmetry, Part 1, SIAM J. Math. Anal., in press. |
[19] | Y. Du, W. Ni, The high dimensional Fisher-KPP nonlocal diffusion equation with free boundary and radial symmetry: sharp estimates, 2022, preprint. Available from: http://turing.une.edu.au/ydu/papers/dn-highD-2-March2022.pdf |
[20] |
J. Fang, G. Faye, Monotone traveling waves for delayed neural field equations, Math. Mod. Meth. Appl. Sci., 26 (2016), 1919–1954. https://doi.org/10.1142/S0218202516500482 doi: 10.1142/S0218202516500482
![]() |
[21] |
D. Finkelshtein, P. Tkachov, Accelerated nonlocal nonsymmetric dispersion for monostable equations on the real line, Appl. Anal., 98 (2019), 756–780. https://doi.org/10.1080/00036811.2017.1400537 doi: 10.1080/00036811.2017.1400537
![]() |
[22] |
R. A. Fisher, The wave of advance of advantageous genes, Annals of Eugenics, 7 (1937), 355–369. https://doi.org/10.1111/j.1469-1809.1937.tb02153.x doi: 10.1111/j.1469-1809.1937.tb02153.x
![]() |
[23] |
J. Garnier, Accelerating solutions in integro-differential equations, SIAM J. Math. Anal., 43 (2011), 1955–1974. https://doi.org/10.1137/10080693X doi: 10.1137/10080693X
![]() |
[24] | A. N. Kolmogorov, I. G. Petrovski, N. S. Piskunov, A study of the diffusion equation with increase in the amount of substance, and its application to a biological problem, Byul. Moskovskogo Gos. Univ., 1 (1937), 1–25. |
[25] |
F. Li, J. Coville, X. Wang, On eigenvalue problems arising from nonlocal diffusion models, Discrete Contin. Dyn. Syst., 37 (2017), 879–903. https://doi.org/10.3934/dcds.2017036 doi: 10.3934/dcds.2017036
![]() |
[26] |
X. Liang, T. Zhou, Spreading speeds of nonlocal KPP equations in almost periodic media, J. Funct. Anal., 279 (2020), 108723. https://doi.org/10.1016/j.jfa.2020.108723 doi: 10.1016/j.jfa.2020.108723
![]() |
[27] |
W. Shen, A. Zhang, Spreading speeds for monostable equations with nonlocal dispersal in space periodic habitats, J. Differ. Equations, 249 (2010), 747–795. https://doi.org/10.1016/j.jde.2010.04.012 doi: 10.1016/j.jde.2010.04.012
![]() |
[28] |
P. E. Souganidis, A. Tarfulea, Front propagation for integro-differential KPP reaction-diffusion equations in periodic media, Nonlinear Differ. Equ. Appl., 26 (2019), 29. https://doi.org/10.1007/s00030-019-0573-7 doi: 10.1007/s00030-019-0573-7
![]() |
[29] |
H. F. Weinberger, Long-time behavior of a class of biological models, SIAM J. Math. Anal. 13 (1982), 353–396. https://doi.org/10.1137/0513028 doi: 10.1137/0513028
![]() |
[30] |
H. F. Weinberger, X. Q. Zhao, An extension of the formula for spreading speeds, Math. Biosci. Eng., 7 (2010), 187–194. https://doi.org/10.3934/mbe.2010.7.187 doi: 10.3934/mbe.2010.7.187
![]() |
[31] |
H. Yagisita, Existence and nonexistence of traveling waves for a nonlocal monostable equation, Publ. Res. Inst. Math. Sci., 45 (2009), 925–953. https://doi.org/10.2977/prims/1260476648 doi: 10.2977/prims/1260476648
![]() |
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