A nonlocal diffusion model with free boundaries in spatial heterogeneous environment

Accepted Manuscript A Nonlocal Diffusion Model with Free Boundaries in Spatial Heterogeneous Environment

Jia-Feng Cao, Wan-Tong Li, Meng Zhao

PII: DOI: Reference:

S0022-247X(16)30840-X http://dx.doi.org/10.1016/j.jmaa.2016.12.044 YJMAA 20984

To appear in:

Journal of Mathematical Analysis and Applications

Received date:

2 August 2016

Please cite this article in press as: J.-F. Cao et al., A Nonlocal Diffusion Model with Free Boundaries in Spatial Heterogeneous Environment, J. Math. Anal. Appl. (2017), http://dx.doi.org/10.1016/j.jmaa.2016.12.044

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A Nonlocal Diffusion Model with Free Boundaries in Spatial Heterogeneous Environment Jia-Feng Cao, Wan-Tong Li∗ and Meng Zhao School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu, 730000, People’s Republic of China December 16, 2016

Abstract This paper is concerned with the nonlocal diffusive model with double free boundaries in spatial heterogeneous environment, where the spatial heterogeneity is described by the sign indefinite coefficients. Such a model can be used to illustrate the spreading or vanishing of a new or invasive species. Due to the lack of comparison principle in the nonlocal reaction-diffusion equation, many classical methods cannot be used directly to this nonlocal problem. This motivates us to find new techniques. We first establish the spreading-vanishing dichotomy as well as some criteria that ensure the species spreading or vanishing by principal eigenvalues of associated scalar elliptic eigenvalue problems. And then we determine the spreading speed when spreading occurs. Keywords: Nonlocal diffusion; Free boundary; Spatial heterogeneity; Sign indefinite; Spreading-vanishing dichotomy; Spreading speed AMS Subject Classification (2000): 35K57, 35R20, 92D25

1

Introduction

We are interested in the effect of spatial heterogeneity on the long-term viability of some species, governed by the following nonlocal diffusive free boundary model ⎧ ⎪ ut = dΔu + u(a(x) − b(x)u − c(x)(φ ∗ u)), t > 0, g(t) < x < h(t), ⎪ ⎪ ⎪ ⎪ ⎨ u(t, h(t)) = 0, h (t) = −μux (t, h(t)), t > 0, (1.1)  ⎪ t > 0, ⎪ u(t, g(t)) = 0, g (t) = −μux (t, g(t)), ⎪ ⎪ ⎪ ⎩ − h0 ≤ x ≤ h 0 , u(0, x) = u0 (x), h(0) = −g(0) = h0 , ∗

Corresponding author ([email protected]).

1

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Cao, Li and Zhao

where x = g(t) and x = h(t) are free boundaries to be determined together with u; h0 , d and μ are given positive constants; u0 satisfies u0 (x) ∈ C 1 ([−h0 , h0 ]), u0 (−h0 ) = u0 (h0 ) = 0 and u0 (x) > 0 for x ∈ (−h0 , h0 ), (1.2) where u0 is the density of the species in the very early stage of its introduction. For u(t, x) satisfying u(t, x) = 0 for x ≤ g(t) or x ≥ h(t), we define the convolution (φ ∗ u)(t, x) =  R φ(x − y)u(t, y)dy, where the kernel φ(·) satisfies (K) φ(x) is continuous and nonnegative, and φ ∈ L1 (R) with



R φ(x)dx

= 1.

The functions a(x), b(x) and c(x) are of class C ν0 (R) ∩ L∞ (R) for some ν0 ∈ (0, 1), and satisfy (H) a(x) > 0 on a set of positive measure, b(x) and c(x) are positive on R. Moreover, there exist positive constants b, b, c and c such that b = inf b(x) ≤ sup b(x) = b and c = inf c(x) ≤ sup c(x) = c. x∈R

x∈R

x∈R

x∈R

Problem (1.1) models the interactions of one theoretical species which is allowed to move throughout a growing habitat domain [g(t), h(t)] via a “random walk” process characterised by the diffusive term dΔ, where d denotes the diffusive rate. The local rate of change in the species density is described by the density dependent term a(x) − b(x)u − c(x)(φ ∗ u). In this term, a(x) represents the intrinsic growth rate at location x in the absence of crowding or limitation on the availability of resources. The sign of a(x) is indefinite, and in the natural world, it is evident that a(x) will be positive on favorable habitats and negative on unfavorable ones. The intra-specific competition at the point x depends not simply on the density at x but on the neighbouring points as well, since the species individuals may move to find resources. And hence, the term b(x)u denotes the competition among species for local space, while c(x)(φ ∗ u) measures the competition for resources in the neighborhood of species individuals. We further note that −c(x)(φ ∗ u) represents a disadvantage in global species levels being too high because of the consume of resources. The reason that this is a global term is that the individuals are moving and then the force of intra-species competition depends on species levels in a neighbourhood of the original position, that is, on a spatial average weighted according to distance from the original position, please see [1] for a detailed illustration. When φ ∗ u is a temporal convolution, the equation in problem (1.1) on bounded smooth domain Ω ⊂ RN has been studied by Wu [27], where the global attractivity of the associated steady-state solution is demonstrated by using the approach of ω-limit. In

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the case that a(x), b(x) and c(x) are positive constants, the equation in (1.1) was studied by Deng et al. [7], and by Deng et al. [8] with spatial-temporal convolution. In [7, 8], the authors proposed a comparison principle and then showed the existence and uniqueness of the solution to the model involved by constructing monotone sequences. It was shown that the unique positive steady-state solution is global attractive. It is observed that for more than three decades, the effect of the spatial heterogeneity of the environment resources on the total size of a single new invasion species or the coexistence of multiple interacting species has attracted much attention to both mathematicians and ecologists. Spatial heterogeneity of the environment not only seems to be crucial in creating large amount of patterns, it also brings about interesting mathematical questions. There are various forms of spatial heterogeneity, and many of them lead to models where the coefficients vary in space. Reaction-diffusion models of the form ut − dΔu = u(a(x) − cu) in Ω × (0, ∞)

(1.3)

has been studied extensively, see [4, 5, 17] for instance, where a(x) is the same as that in condition (H), c > 0 is a constant describing the limiting effects of crowding. It was shown that there will be a critical value d∗ > 0 such that for all d ∈ (0, d∗ ), the model (1.3) possesses a unique stable positive steady state (which implies persistence for the species), while all solutions to (1.3) decay to 0 as t → ∞ in the case that d ≥ d∗ . The critical value 1 d∗ = λ+ (a(x)) , where λ+ 1 (a(x)) is the principal eigenvalue of problem 1

−Δφ = λa(x)φ in Ω and φ = 0 on ∂Ω.

(1.4)

Concerning with the balance between competitive strength and diffusive rate that affected by the spatial heterogeneity of resources, Hutson et al. [15] considered the following reaction-diffusion system  ut = d1 Δu + u [1 + γa(x) − u − bv] in Ω × (0, ∞), (1.5) vt = d2 Δv + v [1 + γa(x) − cu − v] in Ω × (0, ∞)  subject to the zero Neumann boundary condition. With the assumption that Ω a(x)dx = 0 (to make sure that they account for the effects of the heterogeneity rather than the total carrying capacity of the environment), they investigated the stability of the equilibrium and the existence of the coexistence solutions, and in particular discussed the implications of their results for the principle of competitive exclusions. Lou et al. [18] investigated the loops and branches of coexistence states in a Lotka-Volterra competition diffusion model with spatial heterogeneous reaction term, and obtained the bifurcation diagram of the steady state. We refer to Section 5.2 in [19] for linear eigenvalue problems with

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indefinite weight (coefficient), which is concerned with the question of determining the optimal spatial arrangement of favorable and unfavorable regions for species to survive. Note that for free boundary problems, it turns out that spatial heterogeneity also performs an important role in the predictions of spreading (persistence) or vanishing (extinction) for the species. For example, Wang [25] considered the following free boundary problem in spatial heterogeneous environment as follows ⎧ ⎪ u = dΔu + u(a(x) − u), t > 0, 0 < x < h(t), ⎪ ⎨ t  (1.6) B[u](t, 0) = u(t, h(t)) = 0, h (t) = −μux (t, h(t)), t > 0, ⎪ ⎪ ⎩ u(0, x) = u0 (x), h(0) = h0 , 0 ≤ x ≤ h0 , where B[u] = αu − βux , α, β ≥ 0 are constants with α + β = 1, the initial function u0 (x) satisfies u0 ∈ C 2 ([0, h0 ]), u0 > 0 in (0, h0 ) and B[u0 ](0) = u0 (h0 ) = 0. It was shown in [25] that if a(x) ∈ C([0, ∞)) ∩ L∞ ([0, ∞)) and is positive somewhere in (0, ∞), problem (1.6) admits a unique positive solution (u, h) defined for all t > 0 with u(x, t) > 0 and h (t) > 0 for all x ∈ (0, h(t)) and t > 0. Particularly, the author showed the uniform estimates of u(t, ·) C 1 [0,h(t)] for t ≥ 1 and h (t) C ν2 ([n+1,n+3]) for n ≥ 0 regardless of the size of h∞ := limt→∞ h(t). Further, it was proved that if there are two positive constants m1 and m2 such that m1 ≤ lim inf x→∞ a(x) ≤ lim supx→∞ a(x) ≤ m2 holds and spreading happens, then [k0 (μ, m1 ) + o(1)]t ≤ h(t) ≤ [k0 (μ, m2 ) + o(1)]t as t → ∞ for positive constants k0 (μ, m1 ) and k0 (μ, m2 ), which is the estimate of the spreading speed of the species. We see that Wang [26] extended the above results to the case of time-periodic environment, and we refer to Zhou et al. [28] to free boundary problems with sign indefinite coefficients in spatial heterogeneous environment. This paper is devoted to examine the positive solution of (1.1) in the spatial heterogeneous case, a situation that more closely reflects the variation of the natural environment resources. In Section 2, we are concerned with some basic results that including the global existence and uniqueness of positive solutions to (1.1), as well as some different comparison principles over different suitable parabolic regions since the nonlocal equation in (1.1) does not admit the general comparison principle. Some of these results will be frequently used in the coming sections and also have their independent interest. Section 3 focuses on the analysis of the principal eigenvalue λ+ 1 of (3.1). Due to the sign indefinite coefficient a(x) and its continuity, we address our conclusion by looking at 1 as the maximum of the reciprocal of the form of the general variational formulation. λ+ 1 It is worth pointing out that the approach to determine the eigenvalues for problems with sign-changing coefficients was introduced by Manes et al. [20] for the case of Dirichlet boundary conditions.

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In Section 4, we exhibit the long term behaviors of positive solutions (u, g, h) to (1.1) and give some sufficient conditions that make the species spreading or vanishing. Finally, we give some estimates to the spreading speed when spreading happens in Section 5 to finish this paper.

2

Positive solutions to (1.1)

In this section, we establish the global existence and uniqueness of the positive solution to (1.1), as well as some properties that the solution involved. Theorem 2.1 Assume that (H) and (K) hold. Then for any given u0 (x) satisfying (1.2) and any ν ∈ (0, 1), problem (1.1) admits a unique global positive solution 1+ν ,1+ν 2

ν

ν

(D∞ ) × C 1+ 2 (0, ∞) × C 1+ 2 (0, ∞), (2.1)  where D∞ = (t, x) ∈ R2 : t ∈ [0, ∞), x ∈ [g(t), h(t)] . In addition, there exists a positive constant C1 = C1 ( a, b, c, u0 ∞ , u0 C 1 ([−h0 ,h0 ]) ) such that (u, g, h) ∈ C

0 < u(t, x) ≤ C1 and 0 < h (t), −g  (t) ≤ μC1 for t > 0 and g(t) < x < h(t).

(2.2)

Moreover, if h∞ := limt→∞ h(t) < ∞ and −g∞ := − limt→∞ g(t) < ∞, then there exists some positive constant C2 = C2 (μ, a, b, c ∞ , h∞ , g∞ ) such that

g  (t), h (t) C ν2 ([1,∞) ≤ C2 and u(t, ·) C 1 ([g(t),h(t)]) ≤ C2 (∀t ≥ 1).

(2.3)

Proof. With the assumption that a(x), b(x) and c(x) are of class C ν0 (R) ∩ L∞ (R), we get (2.1) by the methods that used in [10]. At the same time, there is u(t, x) > 0, ux (t, h(t)) < 0 and ux (t, g(t)) > 0 for all t > 0 and g(t) < x < h(t) due to the strong maximum principle to the equation of u, and hence there holds h (t), −g  (t) > 0 for t > 0 by the Stefan conditions h (t) = −μux (t, h(t)) and g  (t) = −μux (t, g(t)). In addition, it follows from the general comparison principle that

 |a(x)| u(t, x) ≤ C1 := max u0 ∞ , max for all t > 0. x∈R b(x) + c(x) The proof for 0 < h (t), −g  (t) ≤ μC1 is similar to that of [10]. Now, we prove (2.3), the method used here is motivated by Theorem 2.1 in [26]. In order to do this, we first transfer the free boundaries x = g(t) and x = h(t) into fixed lines s = −1 and s = 1 respectively, that is, by letting (t, x) → (t, s) with s =

2x h(t) + g(t) − . h(t) − g(t) h(t) − g(t)

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It is obvious that

2 ∂s = := F (t), ∂x h(t) − g(t)

∂2s =0 ∂x2

and

∂s h (t) − g  (t) h (t) + g  (t) = −s − := E(t, s). ∂t h(t) − g(t) h(t) − g(t) 

= w(t, s), then direct calculation shows that free If we set u(t, x) = w t, s(h−g)+(h+g) 2 boundary problem (1.1) becomes following ⎧

 ˜ ˜ ⎪ w − dF w + Ew = w a ˜ − bw − c ˜ ( φ ∗ w) , t > 0, − 1 < s < 1, ⎪ t ss s ⎪ ⎪ ⎪ √ ⎪ ⎨ w(t, 1) = 0, h (t) = −μ F ws (t, 1), t > 0, (2.4) √ ⎪  ⎪ (t) = −μ F w (t, −1), t > 0, w(t, −1) = 0, g ⎪ s ⎪ ⎪ ⎪ ⎩ w(0, s) = u (h s), h(0) = −g(0) = h , − 1 ≤ s ≤ 1, 0 0 0



 ˜ = φ s(h−g)+(h+g) and k˜ = k(t, ˜ s) = k s(h−g)+(h+g) where F = F (t), E = E(t, s), φ(s) 2 2 may denote one of the functions a(x), b(x) or c(x). It is easy to see that problem (2.4) is an initial-boundary value problem with fixed boundary. For any integer n ≥ 0, let wn (t, s) = w(t + n, s), then (2.4) deduces to 

⎧ n n n n n n n ˜bn wn − c˜n (φ˜ ∗ wn ) , 0 < t ≤ 3, − 1 < s < 1, ⎪ − dF w + E w = w − w a ˜ ss s ⎪ t ⎪ ⎪ ⎨ n n w (t, 1) = w (t, −1) = 0, 0 < t ≤ 3,   ⎪ ⎪ ⎪ s(h(n) − g(n)) + (h(n) + g(n)) ⎪ ⎩ wn (0, s) = u n, , − 1 ≤ s ≤ 1, 2 (2.5) ˜ + n, s). It follows with F n (t) = F (t + n), E n (t, s) = E(t + n, s) and k˜n (t, s) = k(t from (2.2) that wn , F n and E n are bounded uniformly on n. Applying the interior Lp estimate (see [26]) and the embedding theorem, there exists a positive constant C2 independent of n such that wn 1+ν ,1+ν ≤ C2 for all n ≥ 0, which indicates that w

C

1+ν C 2 ,1+ν (In )

2

([1,3]×[−1,1])

≤ C2 with In = [n + 1, n + 3] × [−1, 1]. This fact combined with

2 ws (t, 1), h(t) − g(t) 2 g  (t) = −μux (t, g(t)), ux (t, g(t)) = ws (t, −1) h(t) − g(t) h (t) = −μux (t, h(t)), ux (t, h(t)) =

and 0 < h (t), −g  (t) ≤ μC1 shows that g  (t), h (t) C ν2 ([n+1,n+3]) ≤ C2 . Since rectangles In overlap and C2 is independent of n, we see that w C 0,1 ([1,∞)×[−1,1]) ≤ C2 . √ √ Using ux (t, h(t)) = F ws (t, 1) and ux (t, g(t)) = F ws (t, −1) again, we obtain that

u(t, ·) C 1 ([g(t),h(t)]) ≤ C2 for all t ≥ 1. 

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Note that Theorem 2.1 indicates that h(t) and −g(t) are monotone increasing in time t, then h∞ := limt→∞ h(t) ∈ (0, +∞] and g∞ := limt→∞ g(t) ∈ [−∞, 0) are well defined. In what follows, we discuss the comparison principles over some suitable parabolic regions, which have been proved to be essential tools for the analysis of the asymptotic behavior of (u, g, h) obtained in Theorem 2.1. Following are two comparison principles for the free boundary problem in a extension scalar equation. The proof is almost identical to that of Lemma 3.5 in [10] and we omit the details. ¯ ∈ C 1,2 (D∗ ) × C 1 ([0, +∞)) × C 1 ([0, +∞)) with D∗ = Lemma 2.2 Assume that (¯ u, g¯, h)  ¯ satisfying (t, x) ∈ R2 : t > 0, g¯(t) ≤ x ≤ h(t) ⎧ ¯ ⎪ u+u ¯(a(x) − b(x)¯ u), t > 0, g¯(t) < x < h(t), u ¯ ≥ dΔ¯ ⎪ ⎨ t ¯ u ¯(t, g¯(t)) = u ¯(t, h(t)) = 0, t > 0, ⎪ ⎪ ⎩ ¯ ¯ ¯ h (t) ≥ −μ¯ ux (t, h(t)), g¯ (t) ≤ −μ¯ ux (t, g¯(t)), t > 0, g¯(t) < x < h(t)

(2.6)

¯ ¯(0, x) ≥ u0 (x) for all x ∈ [−h0 , h0 ]. Then the unique positive with h(0), −¯ g (0) ≥ h0 and u ¯ for all t > 0 and x ∈ [g(t), h(t)]. solution (u, g, h) of (1.1) satisfies u ≤ u ¯, g ≥ g¯ and h ≤ h

Lemma 2.3 For fixed positive constant k∗ , let (u, g, h) be the unique positive solution to ⎧ ⎪ u = dΔu − k ∗ u, t > 0, g(t) < x < h(t), ⎪ ⎪ t ⎪ ⎪ ⎨ u(t, g(t)) = 0, g  (t) = −μux (t, g(t)), t > 0, (2.7) ⎪ u(t, h(t)) = 0, h (t) = −μux (t, h(t)), t > 0, ⎪ ⎪ ⎪ ⎪ ⎩ u(0, x) = u0 (x), h(0) = −g(0) = h0 , − h0 ≤ x ≤ h0 .  ¯ ¯ ∈ C 1,2 (D∗ ) × , define (¯ u, g¯, h) Meanwhile, for D∗ = (t, x) ∈ R2 : t > 0, g¯(t) ≤ x ≤ h(t) C 1 ([0, +∞)) × C 1 ([0, +∞)) satisfying ⎧ ¯ ⎪ u ¯ ≥ dΔ¯ u − k∗ u ¯, t > 0, g¯(t) < x < h(t), ⎪ ⎨ t ¯ (2.8) u ¯(t, g¯(t)) = u ¯(t, h(t)) = 0, t > 0, ⎪ ⎪ ⎩ ¯  ¯ ¯ ux (t, h(t)), g¯ (t) ≤ −μ¯ ux (t, g¯(t)), t > 0, g¯(t) < x < h(t) h (t) ≥ −μ¯ ¯ ¯(0, x) ≥ u0 (x) for all x ∈ [−h0 , h0 ]. Then there hold with h(0), −¯ g (0) ≥ h0 and u ¯ for all t > 0 and x ∈ [g(t), h(t)]. u≤u ¯, g ≥ g¯ and h ≤ h We can define (u, g, h) by reversing all the inequalities in (2.8) and get an analogue conclusion.

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Below, we show a comparison principle to prove the global existence and uniqueness of solutions to a spatial heterogeneous problem with the lack of an order-preserving property that is caused by the nonlocal nature of the nonlinear term. Lemma 2.4 Assume that (H) and (K) hold. For T ∈ (0, ∞), let u, u ∈ C 1,2 (DT ) ∩ CB (ΓT ) satisfy ⎧ ⎪ u − dΔu ≥ u(a(x) − b(x)u − c(x)φ ∗ u), (t, x) ∈ DT , ⎪ ⎨ t (2.9) ut − dΔu ≤ u(a(x) − b(x)u − c(x)φ ∗ u), (t, x) ∈ DT , ⎪ ⎪ ⎩ u(0, x) ≥ u0 (x) ≥ u(0, x), x ∈ R,   where DT = (t, x) ∈ R2 : t ∈ (0, T ), x ∈ R , ΓT = (t, x) ∈ R2 : t ∈ [0, T ), x ∈ R and CB (ΓT ) denotes the space of all bounded continuous functions in ΓT . Then u ≥ u for all (t, x) ∈ ΓT . Proof. The idea here is motivated by Theorem 2.2 in [7]. Define u ˆ = u − u, we need to obtain that u ˆ ≥ 0. Note that u ˆ satisfies u ≥ u [a(x) − b(x)u − c(x)φ ∗ u] − u [a(x) − b(x)u − c(x)φ ∗ u] u ˆt − dΔˆ =u ˆ [a(x) − b(x)(u + u) − c(x)(φ ∗ u)] + c(x)u(φ ∗ u ˆ) ˆ), = K(t, x)ˆ u + c(x)u(φ ∗ u where K(t, x) = a(x) − b(x)(u + u) − c(x)(φ ∗ u). Since u, u ∈ C 1,2 (DT ) ∩ CB (ΓT ), and a(x), b(x) and c(x) are of class C ν0 (R) ∩ L∞ (R) for some ν0 ∈ (0, 1), then we can choose some constant ς ∗ such that ς ∗ ≥ max(t,x)∈DT K(t, x), further, let ς ∗ = a(x) L∞ +2b+c > ∗ 0, where  represents the upper bound of u and u, and define w ˆ = e−ς t u ˆ. Hence, w ˆ satisfies  w ˆt − dΔw ˆ + (ς ∗ − K(t, x))w ˆ ≥ c(x)u(φ ∗ w) ˆ in DT , w(0, ˆ x) ≥ 0 on R. If we can conclude that w ˆ ≥ 0, then u ˆ ≥ 0 follows. To state this, we suppose to the contrary that there exists some point (t0 , x0 ) ∈ DT0 (T0 will be chose later) such that w(t ˆ 0 , x0 ) < 0, then w ˆinf = inf (t,x)∈DT0 w(t, ˆ x) ≤ w(t ˆ 0 , x0 ) < 0, which indicates that for any ∗ ∗ ∗ ∗ > 0, we can find (t , x ) ∈ DT0 such that w(t ˆ , x∗ ) ≤ w ˆinf + ∗ . If we choose ∗ = − 12 w ˆinf , 1 ∗ ∗ ˆinf . Below, we propose an auxiliary function vˆ defined as then there is w(t ˆ , x ) ≤ 2w vˆ =

w ˆ , 1 + x2 + ηt

where, η > 0 is a constant to be determined later. Then lim|x|→∞ vˆ(t, x) = 0 and  vt − dΔˆ v + (ς ∗ − K(t, x))ˆ v ] − 4dˆ vx x + (η − 2d)ˆ v ≥ c(x)u(φ ∗ w) ˆ in DT , (1 + x2 + ηt) [ˆ vˆ(0, x) ≥ 0 on R.

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In addition, vˆ attains its negative minimum vˆmin at (t1 , x1 ) ∈ DT0 , and there hold w(t ˆ ∗ ,x∗ ) v |(t1 ,x1 ) ≥ 0. Moreover, there is vˆmin ≤ 1+(x vˆt |(t1 ,x1 ) ≤ 0, vˆx |(t1 ,x1 ) = 0 and Δˆ ∗ )2 +ηt∗ ≤ w ˆinf 2(1+(x∗ )2 +ηt∗ )

< 0. Therefore, we can get

vmin + (η − 2d)ˆ vmin ≥ c(x)u(t1 , x1 )w ˆinf , (1 + x21 + ηt1 )(ς ∗ − K(t1 , x1 ))ˆ ˆinf . Further, with the negativity of vˆmin , we which implies (η − 2d)ˆ vmin ≥ c(x)u(t1 , x1 )w arrive at   η − 2d ≤ c 1 + (x∗ )2 + ηt∗ .     1 and taking η is large such that (1−cT0 )η ≥ c 1 + (x∗ )2 + By letting T0 = min T, c 2d, then there is a contradiction. Thus, the assumption that there exists (t0 , x0 ) ∈ DT0 such that w(t ˆ 0 , x0 ) < 0 is false, and u ≥ u for all (t, x) ∈ ΓT follows.  We see that u and u can be called a pair of nonnegative upper and lower solutions to  ut − dΔu = u(a(x) − b(x)u − c(x)φ ∗ u), (t, x) ∈ DT , (2.10) u(0, x) = u0 (x), x∈R respectively, a problem similar to (2.10) with x ∈ Ω and temporal convolution has been studied by Wu et al. [27]. Now we are in a position to prove the global existence and uniqueness of the solution to the following initial-value problem  ut − dΔu = u(a(x) − b(x)u − c(x)(φ ∗ u)), t > 0, x ∈ R, (2.11) x ∈ R, u(0, x) = u0 (x), which is for later use and has its own interest. We arrive at our conclusions by combining the following three theorems. Theorem 2.5 Assume that (K) and (H) hold. Then if (2.11) admits a solution, then it must be unique. Proof. Suppose on the contrary that u1 , u2 ∈ C 1,2 (DT ) are two distinct bounded solutions to (2.11) in DT , we get our conclusion by deriving a contradiction.  Letting Φ(t, x) with R Φ(t, x)dx = 1 be the fundamental solution to the heat equation, then ui (i = 1, 2) can be formulated as   t Φ(t, x − y)u0 (y)dy + Φ(t − s, x − y)f (ui (s, y))dyds, ui (t, x) = R

0

R

  where f (ui (s, y)) = ui (s, y) a(y) − b(y)ui (s, y) − c(y) R φ(y − z)ui (s, z)dz . Define u = u1 − u2 = 0, then u satisfies  t Φ(t − s, x − y) {a(y)u − b(y)(u1 + u2 )u − c(y)[u1 (φ ∗ u) − u(φ ∗ u2 )]} dyds. u(t, x) = 0

R



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Cao, Li and Zhao

Combining



R φ(x)dx



= 1, u ∈ C 1,2 (DT ) and assumption (H) yields that

 

Φ(t − s, ·) L1 (R) a(·) L∞ (R) + 2M b + 2M c u(s, ·) L∞ (R) ds 0    t = a(·) L∞ (R) + 2M b + 2M c

u(s, ·) L∞ (R) ds,

u(t, ·) L∞ (R) ≤

t

0

where M > C1 is a positive constant. Then it follows from the Gronwall inequality that 

u(t, ·) L∞ (R) = 0 for all t ∈ (0, T ), a contradiction. Thus, our conclusion follows. Theorem 2.6 Assume that (K) and (H) hold. Then for u, u proposed in Lemma 2.4, problem (2.11) admits a unique solution u(t, x) in DT , and u(t, x) ≤ u(t, x) ≤ u(t, x) for any (t, x) ∈ ΓT . Proof. We want to use the monotone iterative technique, induction and Lemma 2.4 ˜ > 0 denote the bound for (comparison principle) to deduce the desired results. Let M u and u for all (t, x) ∈ ΓT , with the help of (H), we can choose L∗ > 0 large such that ˜. L∗ ≥ a(x) L∞ + 2bM  ∞ Putting v 0 = u and w0 = u, we are going to construct two sequences v k k=0 and  k ∞ w k=0 by iteratively solving the following problems for k = 1, 2, · · · 

vtk − dΔv k = v k−1 (a(x) − b(x)v k−1 ) − v k c(x)(φ ∗ wk−1 ) − L∗ (v k − v k−1 ) in DT , v k (0, x) = u0 (x) on R

and 

wtk − dΔwk = wk−1 (a(x) − b(x)wk−1 ) − wk c(x)(φ ∗ v k−1 ) − L∗ (wk − wk−1 ) in DT , wk (0, x) = u0 (x) on R.

Firstly, we prove that v 0 ≤ v 1 ≤ w1 ≤ w0 in DT . By letting v˜ = v 0 − v 1 , we get  v ≤ −˜ v [c(x)(φ ∗ w0 ) + L∗ ], (t, x) ∈ DT , v˜t − dΔ˜ v˜(0, x) ≤ 0,

x ∈ R.

It follows from the comparison principle that v˜(t, x) ≤ 0 for (t, x) ∈ ΓT , which in turn implies that v 0 ≤ v 1 for (t, x) ∈ ΓT . By using the similar manner, we can arrive at w1 ≤ w0 for (t, x) ∈ ΓT . Furthermore, taking u ˜ = v 1 − w1 and then u ˜ satisfies u = −c(x)(φ ∗ (w0 − v 0 ))˜ u − L∗ u ˜ + (v 0 − w0 )[L∗ − b(x)(v 0 + w0 ) + a(x)] ≤ −L∗ u ˜ u ˜t − dΔ˜ ˜(0, x) ≤ 0 for x ∈ R. Using the comparison principle once again, for all (t, x) ∈ ΓT and u we obtain that u ˜(t, x) ≤ 0, that is, v 1 ≤ w1 for (t, x) ∈ ΓT .

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Below, we will state that v 1 and w1 are a pair of lower and upper solutions to (2.11) ˜ , then by in ΓT , respectively. Since v 0 ≤ v 1 ≤ w1 ≤ w0 in ΓT and L∗ ≥ a(x) L∞ + 2bM  k ∞  k ∞ 1 1 substituting v and w into equations satisfied by sequences v k=0 and w k=0 yields vt1 − dΔv 1 = v 0 (a(x) − b(x)v 0 ) − v 1 c(x)(φ ∗ w0 ) − L∗ (v 1 − v 0 ) ≤ v 0 (a(x) − b(x)v 0 ) − v 1 c(x)(φ ∗ w1 ) − L∗ (v 1 − v 0 ) = v 1 [a(x) − b(x)v 1 − c(x)(φ ∗ w1 )] + (v 0 − v 1 )[L∗ − b(x)(v 0 + v 1 ) + a(x)] ≤ v 1 [a(x) − b(x)v 1 − c(x)(φ ∗ w1 )] and wt1 − dΔw1 = w0 (a(x) − b(x)w0 ) − w1 c(x)(φ ∗ v 0 ) − L∗ (w1 − w0 ) ≥ w0 (a(x) − b(x)w0 ) − w1 c(x)(φ ∗ v 1 ) − L∗ (w1 − w0 ) = w1 [a(x) − b(x)w1 − c(x)(φ ∗ v 1 )] + (w0 − w1 )[L∗ − b(x)(w0 + w1 ) + a(x)] ≥ w1 [a(x) − b(x)w1 − c(x)(φ ∗ v 1 )] for all (t, x) ∈ ΓT . In addition, there holds v 1 (0, x) = w1 (0, x) = u0 (x) for all x ∈ R. Then it follows from Lemma 2.4 that v 1 and w1 are a pair of lower and upper solutions to (2.11) in ΓT respectively. Next, we suppose that v k and wk are a pair of lower and upper solutions to (2.11) in ΓT for k > 1. Repeating the same arguments as that among v 1 and w1 deduces that v k ≤ v k+1 ≤ wk+1 ≤ wk , and also v k+1 and wk+1 are a pair of lower and upper solutions to problem (2.11) in ΓT respectively. Thus, by induction, we can obtain two monotone  ∞  ∞ sequences, v k k=0 is monotone increasing and wk k=0 is monotone decreasing such that for k = 0, 1, 2, · · · and (t, x) ∈ DT , that is, v 0 ≤ v 1 ≤ · · · ≤ v k ≤ wk ≤ · · · ≤ w1 ≤ w0 . Then there exist two functions v and w with v ≤ w such that v k → v and wk → w pointwise on DT , and v and w also are a pair of lower and upper solutions to (2.11) in DT . Meanwhile, we find that as k → ∞, there is  vt − dΔv = v(a(x) − b(x)v − c(x)(φ ∗ w)) in DT , v(0, x) = u0 (x) on R and



wt − dΔw = w(a(x) − b(x)w − c(x)(φ ∗ v)) in DT , w(0, x) = u0 (x) on R.

Hence, v and w can be used as a pair of upper and lower solutions to problem (2.11) in DT , and then there is v = w in DT . By Theorem 2.5, the existence of the unique solution of (2.11) in DT denoted by u(t, x) (u = v = w) follows and u(t, x) ≤ u(t, x) ≤ u(t, x) for any (t, x) ∈ ΓT . 

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Cao, Li and Zhao

Theorem 2.7 The unique solution u(t, x) of problem (2.11) obtained in Theorem 2.6 is defined for all t > 0.   Proof. Choosing constant K > 0 large such that K > max u0 ∞ , maxx∈R |a(x)| b(x) , then one can verify that u = 0 and u = K are a pair of lower and upper solutions to (2.11) in DT respectively. It follows from Theorem 2.6 that (2.11) admits a unique solution u(t, x) in DT , and 0 ≤ u(t, x) ≤ K for all (t, x) ∈ DT . It is noticed that K is independent of T , then the solution u(t, x) is defined for all t > 0. This completes the proof. 

3

Some eigenvalue problems

In this section, we focus on an eigenvalue problem and study the property of the corresponding principal eigenvalue, which play an important role in determining the spreading or vanishing of the species in a heterogeneous environment. Let us consider the following eigenvalue problem −dΔϕ = λa(x)ϕ in Ω and ϕ = 0 on ∂Ω,

(3.1)

where Ω ⊂ Rn (n ≥ 1) is a bounded smooth domain. If the sets Ω+ = {x ∈ Ω : a(x) > 0} and Ω− = {x ∈ Ω : a(x) < 0} both have positive measure, it follows from Cantrell et at. [4] that (3.1) has a doubly infinite sequence of eigenvalues − − + + + · · · ≤ λ− 3 ≤ λ2 ≤ λ1 < 0 < λ1 ≤ λ2 ≤ λ3 ≤ · · ·

with variational characterisations



   a(x)ϕ2 (x)dx a(x)ϕ2 (x)dx 1 1 Ω Ω   and , = sup inf = inf sup Hn Hn d Ω |∇ϕ(x)|2 dx d Ω |∇ϕ(x)|2 dx λ+ λ− n n    ∂ϕ 2 where Ω |∇ϕ(x)|2 dx = ni=1 Ω ∂x dx and Hn varies over all n-dimensional subspaces i 1 of H0 (Ω). For 1-dimensional space, choose Ω = (−l, l), where l > 0 is a given constant, and we are mainly interested in l  2 1 −l a(x)ϕ (x)dx . (3.2) = sup inf H01 (−l,l) l λ+ d |∇ϕ(x)|2 dx 1 −l

Since a(x) is positive on an open set, then we have λ+ 1 > 0. Eigenvalue problems with indefinite weights as (3.1) have been widely studied, see [3, 14] for example. It is obvious + + that λ+ 1 depends continuously on d, l and a(x), then we write λ1 as λ1 (d, l, a(x)). If we + + fix l and a(x), and investigate the property of λ+ 1 in d, then we write λ1 (d, l, a(x)) = λ1 (d) for brevity. Similarly, we write λ+ 1 (l) for fixed d and a(x) and varying l. It follows from + Cantrell et al. [6] that λ1 (d, l, a(x)) enjoys following properties.

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13

Theorem 3.1 λ+ 1 (d, l, a(x)) is strictly decreasing in l and a(x), and strictly increasing in d. Furthermore, there is + lim λ+ 1 (l) = lim λ1 (d) = +∞ and

l→0+

lim λ+ 1 (l) = 0.

d→+∞

l→+∞

Proof. We only prove limd→+∞ λ+ 1 (d) = +∞ for fixed l and a(x), the left conclusions can ∗ be found in the monograph [6]. Now letting λ+ 1 (d, l, a ) denote the principal eigenvalue of −dΔϕ = λa∗ ϕ in Ω and ϕ = 0 on ∂Ω

(3.3)

 +  ∗ ∗ ∗ with

2 a = supx∈Ω a(x) > 0, we see that (3.3) admits a solution pair λ1 (d, l, a ), ϕ = + dπ π ∗ , cos 2l x . Combining the fact that λ+ 1 (d, l, a ) ≤ λ1 (d, l, a(x)) immediately deduces 4l2 a∗ that limd→+∞ λ+ 1 (d) = +∞.

 l∗

such that λ+ 1 (l) > 1 1 [5] that d∗ = λ+ (a(x)) 1

l∗ (d, a(x))

Observe that for fixed d and a(x), there is a unique = + ∗ ∗ for l < l∗ , λ+ 1 (l) < 1 for l > l and λ1 (l ) = 1. It was shown in (with fixed d and l) is regarded as a critical value, and whether the unique positive steady state solution is global attractive or not depends on the relationship between d∗ and 1. Further, λ+ 1 (l) enjoys following property. Theorem 3.2 For fixed d and a(x), there exists t∗ > 0 such that λ+ 1 (l) > 1 for l < + + ∗ ∗ ∗ ∗ ∗ h(t ) − g(t ), λ1 (l) = 1 for l = h(t ) − g(t ) and λ1 (l) < 1 for l > h(t ) − g(t∗ ), we denote h∗ = h(t∗ ) − g(t∗ ) for brevity.

4

Asymptotic behaviors of (u, g, h)

In this section, we will derive some sufficient conditions that govern the long time behaviors of (u, g, h), and then the spreading-vanishing dichotomy of the involved species. Theorem 4.1 Assume that (H) and (K) hold. Then for any u0 satisfying (1.2), the following initial-boundary value problem ⎧ ⎪ u = dΔu + u(a(x) − b(x)u − c(x)(φ ∗ u)), t > 0, x ∈ Ω, ⎪ ⎨ t (4.1) u(t, x) = 0, t > 0, x ∈ ∂Ω, ⎪ ⎪ ⎩ x∈Ω u(0, x) = u0 (x), has a unique positive solution u(t, x) and u(t, x; u0 ) depends continuously on initial function u0 (x), where Ω is a bounded smooth domain in R.

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Cao, Li and Zhao

Proof. Let X = C(Ω) be a Banach space equipped with the supreme norm · ∞ . In X, let A be a closed linear operator with dense domain D(A) given by  Au = −Δu and D(A) = u ∈ ∩1≤p<∞ W 2,p (Ω) : Au ∈ X and u = 0 on ∂Ω . It is easy to see that −A generates an analytic semigroup e−At on X. For u0 ∈ X, choose T > 0 to be small. Denote by C the Banach space C([0, T ] × Ω) equipped with the supreme norm · ∞ . Let L be the operator, which maps C into itself, given by  t −At (Lu)(t, x) = u0 e + e−A(t−s) u(s, x)[a(x) − b(x)u − c(x)(φ ∗ u)(s, x)]ds. 0

We are now in a position to prove that L has a unique fixed point u ∈ C such that Lu = u. Define U = {u ∈ C : u − u0 ∞ ≤ 1}, then for any u, v ∈ U with u = v and t ∈ [0, T ], there is  t    −A(t−s) 

Lu − Lv =  e [u (a − bu − cφ ∗ u) − v (a − bv − cφ ∗ v)] (s, x)ds  0  t  t

a(u − v) ds + M ∗

b(u − v)(u + v) ds ≤ M∗ 0 0  t  t + M∗

c(u − v)(φ ∗ u) ds + M ∗

cvφ ∗ (u − v) ds 0

0

≤ M1 T u − v ∞ ,   where M ∗ = max0≤t≤T e−At  and M1 = M ∗ [ a + (2 b + c )( u0 ∞ + 1)]. Meanwhile, we have  t   −At    −A(t−s)    − I u0 + [u (a − bu − cφ ∗ u)] (s, x) ds

Lu − u0 = e e 0  −At     − I u0 + M ∗ T u ∞ [ a + ( b + c ) u ∞ ] ≤ e  −At   − I u0  + M2 T ≤ e   with M2 = M ∗ ( u0 ∞ + 1) [ a + ( b + c )( u0 ∞ + 1)]. For small T , e−At − I u0 → 0 in X. Then it follows that L is contractive in U . Thus, L admits a unique fixed point in U , which is denoted by u such that  t −At u(t, x) = u0 e + e−A(t−s) u(s, x)[a(x) − b(x)u − c(x)(φ ∗ u)(s, x)]ds. 0

Below, we will prove that u(t, x) obtained above is the classical solution of (4.1). Let w(t, x) be the solution of ⎧ ⎪ w = dΔw + u(a(x) − b(x)u − c(x)(φ ∗ u)), t ∈ [0, T ], x ∈ Ω, ⎪ ⎨ t (4.2) w(t, x) = 0, t ∈ [0, T ], x ∈ ∂Ω, ⎪ ⎪ ⎩ x ∈ Ω. w(0, x) = u0 (x),

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15

Since Lu = u ∈ C, then for all u0 ∈ X, w(t, x) is locally H¨older continuous in [0, T ] (see Theorem 4.3.1 in [21]), which in turn indicates that the map G : [0, T ] × Ω → [0, T ] × Ω, (t, x) → u(a(x) − b(x)u − c(x)(φ ∗ u)) is locally H¨ older continuous. Hence, u = w is the unique classical solution to (4.2) and then u is the solution to (4.1). In addition, by Theorem 4.5.3 in [16], there is u ∈ C 1,2 ([0, T ]×Ω) if u0 ∈ C 2 (Ω). It follows from the comparison principle for parabolic differential equations that u(t, x) ≥ 0 for (t, x) ∈ [0, T ] × Ω if u0 (x) ≥ 0 for x ∈ Ω. Moreover, there is u(t, x) > 0 if u0 = 0. On the other hand, again by the comparison principle, it can be shown that u(t, x) is uniformly bounded with the following estimate (also see Theorem 2.1)

|a(x)| u(t, x) ≤ max u0 ∞ , max x∈Ω b(x)

 in [0, T ] × Ω. 

Therefore, we see that u(t, x) exists globally. This completes the proof.

Theorem 4.1 shows the existence and uniqueness of the positive solution of problem (4.1). Following result is a summary of [5, Theorem A.1], which states some sufficient conditions for ensuring the global attractive of the steady-state of (4.1). Theorem 4.2 Suppose that f (x, u) is Lipschitz in x ∈ Ω and continuously differentiable in u with ∂f ∂u < 0 for u > 0, f (x, u) ≤ 0 for all x ∈ Ω and u ≥ k, where k is a positive constant, and f (x0 , 0) > 0 for some x0 ∈ Ω. Consider the eigenvalue problem −Δϕ = λf (x, 0)ϕ in Ω and Bϕ = 0 on ∂Ω,

(4.3)

where Bϕ = ϕ or Bϕ = ∂ϕ (n is the outer normal derivative). In the case that Bϕ = ∂n  assume in addition that Ω f (x, 0)dx < 0. Then the following problem ut = dΔu + f (x, u)u in (0, ∞) × Ω and Bu = 0 on (0, ∞) × ∂Ω

∂ϕ ∂n ,

(4.4)

has a unique positive steady-state u which is a global attractor for nonnegative nontrivial solutions if 0 < d < λ1 (f 1(x,0)) . In addition, there is no positive steady-state for (4.4) and all nonnegative solutions to (4.4) decay to 0 as t → ∞ if d ≥ λ1 (f 1(x,0)) . A rather complete analysis of (4.3) or (4.4) suitable to our present needs is given in [4]. The remained section is devoted to the proof of the spreading-vanishing dichotomy described in the introduction. The spreading-vanishing dichotomy is a consequence of the following two theorems.

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Theorem 4.3 Assume that (H) and (K) hold. Let (u, g, h) be the unique positive solution to (1.1) with h∞ − g∞ < ∞, then limt→+∞ u(t, ·) C([g(t),h(t)]) = 0. Furthermore, if inf x∈[g∞ ,h∞ ] {b(x)M (x) − c(x)(φ ∗ M (x))} ≥ 0, where M (x) is some positive function, then there hold h∞ − g∞ ≤ h∗ . Proof. Firstly, we prove limt→+∞ u(t, ·) C([g(t),h(t)]) = 0 by deriving some contradictions. Suppose on the contrary that lim supt→+∞ u(t, ·) C([g(t),h(t)]) = ε > 0. Then there exists a sequence (tk , xk ) in (0, +∞) × (g(t), h(t)) such that u(tk , xk ) ≥ 2ε for all k ∈ N and tk → ∞ as k → ∞. Meanwhile, since −∞ < g∞ < g(t) < xk < h(t) < h∞ < ∞, we then have that a subsequence of {xk } converges to x0 ∈ (g∞ , h∞ ). Without loss of generality, we assume xk → x0 as k → ∞. Define uk (t, x) = u(tk + t, x) for (t, x) ∈ (−tk , ∞) × (g(tk + t), h(tk + t)). It follows from the parabolic regularity theory that {uk } has a subsequence {uki } such ˜ as i → ∞, where u ˜ satisfies that uki → u u+u ˜(a(x) − b(x)˜ u − c(x)φ ∗ u ˜) for (t, x) ∈ R × (g∞ , h∞ ). u ˜t = dΔ˜

(4.5)

˜ > 0 in R × (g∞ , h∞ ) by the strong maximum theorem. Note that u ˜(0, x0 ) ≥ 2ε , therefore u ˆu ˆ ˜t −dΔ˜ u ≥ −M ˜ at Letting M = a(x)−b(x)−c(x)(φ∗˜ u) L∞ (R) , then using Hopf lemma to u (0, h∞ ) and (0, g∞ ) yields that u ˜x (0, h∞ ) ≤ −σ0 < 0 and u ˜x (0, g∞ ) ≥ σ0 > 0 respectively for some σ0 > 0. On the other hand, for some positive constant Cˆ ≥ C2 with C2 defined in Theorem 2.1, we already have g  (t), h (t) C ν2 ([1,∞)) + u(t, ·) C 1 ([g(t),h(t)]) ≤ Cˆ (∀t ≥ 1), which agrees with the fact that h∞ − g∞ < ∞ implies that h (t) → 0, g  (t) → 0 as t → ∞. We then get ux (tk , h(tk )) → 0 and ux (tk , g(tk )) → 0 as tk → ∞ by Stefan conditions. Moreover, it is not hard to notice that the fact u(t, ·) C 1 ([g(t),h(t)]) ≤ C2 indicates that ux (tk , h(tk )) = (uk )x (0, h(tk )) → u ˜x (0, h∞ ) and ux (tk , g(tk )) = (uk )x (0, g(tk )) → u ˜x (0, g∞ ) as k → ∞, which contradict to u ˜x (0, h∞ ) ≤ −σ0 < 0 and u ˜x (0, g∞ ) ≥ σ0 > 0. Hence, there holds limt→+∞ u(t, ·) C([g(t),h(t)]) = 0 if h∞ − g∞ < ∞. Below, we will prove h∞ − g∞ < h∗ . Again, we obtain our desired result by deriving a contradiction. Suppose that h∞ − g∞ > h∗ , then there is some T˜ > 0 and > 0 small such that h∞ − g∞ > h(t) − g(t) > h∞ − − (g∞ + ) > h∗ for t ≥ T˜. Owing to Theorem + ∗ 3.2, we have λ+ 1 (h∞ − g∞ ) < λ1 (h ) = 1. Letting w(t, x) be the solution of the following ⎧ ⎪ w − dΔw = w(a(x) − b(x)w), t > 0, g∞ < x < h∞ , ⎪ ⎨ t (4.6) w(t, g∞ ) = w(t, h∞ ) = 0, t > 0, ⎪ ⎪ ⎩ w(0, x) = u0 (x), g ∞ < x < h∞ .

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The existence of w(t, x) can be ensured by Theorem 4.1. Then it follows from Theorem 4.2 that w(t, x) → W (x) uniformly in x, where W (x) denotes the unique positive steady state of problem (4.6). And hence, by the comparison principle theorem, for any τ ∗ > 0, there exists some T ∗ such that for all t ≥ T ∗ and x ∈ (g(t), h(t)) (h∞ − g∞ > h(t) − g(t)), there holds u(t, x) ≤ w(t, x) < W (x) + τ ∗ , which in turn indicates that  ut − dΔu = u(a(x) − b(x)u − c(x) φ(x − y)u(t, y)dy) R ≥ u(a(x) − b(x)u − c(x) φ(x − y)(W (y) + τ ∗ )dy). R

Let us now consider the following problem ⎧ ⎪ w − dΔw = w(a(x) − b(x)w − c(x)(φ ∗ (W (x) + τ ∗ ))), t ≥ T ∗ , g∞ + < x < h∞ − , ⎪ ⎨ t w(t, g∞ + ) = w(t, h∞ − ) = 0, t ≥ T ∗, ⎪ ⎪ ⎩ g∞ + < x < h∞ − . w(T ∗ , x) = u(T ∗ , x), (4.7) ∗ There holds u(t, x) ≥ w (t, x) for t ≥ T and x ∈ (g∞ + , h∞ − ) by the comparison principle again, where w (t, x) represents the solution of problem (4.7). Choosing ν > 0 to be a sufficiently small number, for small τ ∗ , we find that −νdΔW ≤ νW (a(x) − νbW − c(x)(φ ∗ (W (x) + τ ∗ ))) provided that inf x∈[g∞ ,h∞ ] {b(x)W (x) − c(x)(φ ∗ W (x))} ≥ 0. This assumption means that the nonlocal interaction is dominated by the local interaction, we find that if b(x), c(x) are positive constants, and φ ∗ u denotes the temporal convolution, then we arrive at b > c, which has been proposed in many references, see [23] for instance. Now νW is a positive lower solution to problem (4.7). Thus, there is u(t, x) ≥ w (t, x) ≥ νW (x) > 0 for all t ≥ T ∗ and x ∈ [g(t), h(t)], which is equivalent to lim inf t→∞ minx∈[g(t),h(t)] u(t, x) > 0. Following the first step, then we can obtain that h∞ = −g∞ = ∞, a contradiction. Then the desired result follows.  Remark 4.4 Note that if we suppose further that the kernel function φ(x) = 1r Φ( xr ), where Φ(x) ∈ C 1 is symmetric and supported on [−1, 1], and u ∈ C 3 ([g(t), h(t)]), then the Taylor’s formula implies that  1 x−y φ∗u−u= Φ( )[u(y) − u(x)]dy r r R 1 x−y uxx = Φ( )[ux (y − x) + (y − x)2 + o((y − x)2 )]dy r 2 R r

18

Cao, Li and Zhao r2 = uxx 2

 R

Φ(z)z 2 dz + o(r2 ) → 0

as r → 0. Then in the above proof, we just need impose the assumption that b − c ≥ 0. Theorem 4.3 also implies that if 2h0 ≥ h∗ or lim inf t→+∞ u(·, t) C([g(t),h(t)]) > 0, then h∞ − g∞ > h∗ and h∞ = −g∞ = ∞, that is spreading happens. Theorem 4.5 Assume that (K) and (H) hold. If (u, g, h) is the unique positive solution to (1.1) with h∞ = −g∞ = ∞, then lim inf t→+∞ u(t, ·) C([g(t),h(t)]) > 0. Further, if the following equation −dΔu = u(a(x) − b(x)u − c(x)(φ ∗ u)) in R

(4.8)

admits a positive solution U (x) and it is unique, then limt→+∞ u(t, x) = U (x) uniformly in any bounded subset of R. Proof. The first part of the theorem is obtained directly from the proof of Theorem 4.3. Below, we prove the second part. Note that (4.8) is the corresponding steady-state problem of (2.11), and the existence and uniqueness of the positive solutions for (2.11) is ensured by Theorems 2.5-2.7. since h∞ = −g∞ = ∞, we can find some t∗ > t∗ such that h(t∗ ) − g(t∗ ) > h∗ . If we choose L ≥ L∗ := h(t∗ ) − g(t∗ ), then 0 < λ+ 1 (L) < 1. Let us now consider the following initialboundary value problem ⎧ ⎪ u − dΔu = u(a(x) − b(x)u − c(x)(φ ∗ u)), t ≥ t∗ , x ∈ [−L, L], ⎪ ⎨ t (4.9) u(t, −L) = u(t, L) = 0, t ≥ t∗ , ⎪ ⎪ ⎩ x ∈ [−L, L]. u(t∗ , x) > 0, It then follows from Theorem 4.1 that problem (4.9) admits a unique positive solution uL (t, x). Since 0 < λ+ 1 (L) < 1, by Theorem 4.2, we find that (4.9) has a unique steady state which is denoted by UL (x) such that uL (t, x) → UL (x) uniformly for all x ∈ [−L, L] as t → ∞, and UL (x) satisfies following −dΔu = u(a(x) − b(x)u − c(x)(φ ∗ u)) in (−L, L) and u(−L) = u(L) = 0. Choosing sequence {Ln }∞ n=1 such that L1 ≤ L2 ≤ · · · ≤ Ln and Ln → ∞ as n → ∞, we ∞ arrive at sequences {uLn }∞ n=1 and {ULn }n=1 satisfying ⎧ ⎪ (u ) − dΔuLn = uLn (a(x) − b(x)uLn − c(x)(φ ∗ uLn )), t ≥ t∗ , x ∈ [−Ln , Ln ], ⎪ ⎨ Ln t uLn (t, −Ln ) = uLn (t, Ln ) = 0, t ≥ t∗ , ⎪ ⎪ ⎩ x ∈ [−Ln , Ln ] uLn (t∗ , x) > 0,

Free boundary problems

19

and −dΔULn = ULn (a(x)−b(x)ULn −c(x)(φ∗ULn )) in (−Ln , Ln ), ULn (−Ln ) = ULn (Ln ) = 0. Moreover, uLn (t, x) → ULn (x) uniformly for all x ∈ [−Ln , Ln ] as t → ∞. If we let n → ∞, then by classical elliptic regularity theory and a diagonal procedure, it follows that ULn (x) converges to U (x) uniformly on any compact bounded subset of R, where U > 0 satisfying −dΔU = U (a(x) − b(x)U − c(x)(φ ∗ U )) for x ∈ R.

(4.10)

Furthermore, if U (x) is the unique positive solution of (4.10), then the solution u(t, x) of 

ut − dΔu = u(a(x) − b(x)u − c(x)(φ ∗ u)), t ≥ t∗ , x ∈ R, x∈R

u(t∗ , x) > 0,

satisfies limt→∞ u(t, x) = U (x) uniformly in any compact subset of R. Due to our assumptions, the result follows.  Remark 4.6 The uniqueness of the positive solution U (x) of problem (4.8) (resp. 4.10) is not a easy task, it may rely on the properties of a(x) and u(t, x) at infinity. Just as Theorem 7.12 in [9], in order to prove the uniqueness of the positive solution to following logistic type elliptic equation −Δu = u(a(x) − b(x)uq−1 ) for x ∈ RN ,

(4.11)

where q is a constant great than 1, a(x) and b(x) are continuous functions with b(x) positive on RN , the author added assumption: there exist γ ∗ > −2 and τ ∗ ∈ (−∞, +∞) and positive numbers α1 , α2 , β1 and β2 such that α1 = lim inf |x|→∞

a(x) a(x) a(x) a(x) and β2 = lim sup τ ∗ . ∗ , α2 = lim sup ∗ , β1 = lim inf ∗ γ γ τ |x| |x|→∞ |x| |x|→∞ |x| |x|→∞ |x| q−1

q−1

(x) (x) u u α1 α2 After proving that lim inf |x|→∞ |x| γ ∗ −τ ∗ ≥ β2 and lim sup|x|→∞ |x|γ ∗ −τ ∗ ≤ β1 , the uniqueness of the positive solution of (4.11) followed by a Safonov type iteration technique. Also, all of those procedures are mainly based on the comparison principle. We leave this for further study, and get limt→∞ u(t, x) = U (x) uniformly in any compact subset of R by supposing the uniqueness.

Combining Theorems 4.3 and 4.5, we obtain the following spreading-vanishing dichotomy for (1.1).

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Theorem 4.7 Let (u, g, h) be the unique positive solution of free boundary problem (1.1). Then the following alternative holds: Either (i) spreading: h∞ = −g∞ = ∞ and lim inf t→+∞ u(t, ·) C([g(t),h(t)]) > 0; or (ii) vanishing: h∞ − g∞ ≤ h∗ and limt→+∞ u(x, t) C([g(t),h(t)]) = 0. Following are some criteria governing the alternatives in the spreading-vanishing dichotomy in the case that 2h0 < h∗ . Theorem 4.8 Assume that (K) and (H) hold. Let (u, g, h) be the unique positive solution to (1.1) with 2h0 < h∗ . Then h∞ − g∞ < ∞ if there exists μ > 0 such that μ ≤ μ. Proof. We are going to construct a suitable upper solution to (1.1) and then apply the + ∗ comparison principle. Since 2h0 < h∗ , then we have λ+ 1 (h0 ) > 1 = λ1 (h ), and there exists an eigenfunction ϕ(x) corresponding to λ+ 1 (h0 ) such that 0 < ϕ(x) ≤ 1 and −dΔϕ = λ+ 1 a(x)ϕ in (−h0 , h0 ) and ϕ(−h0 ) = ϕ(h0 ) = 0. At the same time, there hold ϕ (x) < 0 for 0 < x ≤ h0 and ϕ (x) > 0 for −h0 ≤ x < 0. As in Du et al. [10], define δ ∗ σ(t) = h0 (1 + δ − e−γ t ), ϑ(t) = −σ(t) for t ≥ 0, 2  xh0 ∗ u ¯(t, x) = M ∗ e−γ t ϕ for t ≥ 0 and ϑ(t) ≤ x ≤ σ(t), σ(t)   in which δ, γ ∗ , M ∗ are positive constants that determined later. Note that h0 1 + 2δ ≤ ∗ σ(t) ≤ h0 (1 + δ) and σ  (t) = h20 δ γ ∗ e−γ t . Then, direct calculation yields ut − dΔu − a(x)u + b(x)u2     xh0 σ  (t)  dh20  xh0 ∗ −γ ∗ t ∗ − a(x)ϕ + b(x)uϕ −γ ϕ − = M e ϕ − 2 ϕ σ(t)2 σ (t) σ(t)     h2 xh0 − a(x) + b(x)u ≥ u −γ ∗ + 2 0 λ+ 1a σ (t) σ(t) and

   h20 δ −γ ∗ t 2 xh0 + − a(x) = 2 λ1 a − (1 + δ − e ) a(x) . σ (t) σ(t) 2 

xh0 → a(x) as δ → 0, then by λ+ It is evident that σ(t) → h0 and a σ(t) 1 (h0 ) > 1, there is h20 + λ a σ 2 (t) 1



ζ :=

xh0 σ(t)



min t>0, x∈[ϑ(t),θ(t)]

λ+ 1a



xh0 σ(t)



δ ∗ − (1 + δ − e−γ t )2 a(x) > 0 2

Free boundary problems

21

ζ directly yields that ut − dΔu − for sufficiently small δ > 0. Hence, choosing γ ∗ = 1+δ a(x)u + b(x)u2 ≥ 0. On the other hand, there is u(t, σ(t)) = u(t, −σ(t)) = 0, and     x h0 x ∗ ∗ =M ϕ u(0, x) = M ϕ ≥ u0 ∞ ≥ u0 (x) σ(0) 1 + 2δ

provided that M ∗ > 0 is sufficiently large. Furthermore, there holds σ  (t) =

 h0  h0 δ ∗ −γ ∗ t ∗ |ϕ (h0 )| ∗ ≥ μM ∗ e−γ t ≥ μM ∗ e−γ t γ e |ϕ (h0 )| = −μux (t, σ(t)) 2 σ(t) 1 + 2δ

provided that

  h0 δγ ∗ 1 + 2δ μ≤ := μ. 2M ∗ |ϕ (h0 )| Therefore, for our choice of δ, γ ∗ , M ∗ and μ, the triplet (u, ϑ(t), σ(t)) satisfies ⎧ ⎪ u ¯t ≥ dΔ¯ u+u ¯(a(x) − b(x)¯ u), t > 0, ϑ(t) < x < σ(t), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ¯(t, σ(t)) = u ¯(t, ϑ(t)) = 0, t > 0, ⎨u ux (t, σ(t)), ϑ (t) ≤ −μ¯ ux (t, ϑ(t)), t > 0, ϑ(t) < x < σ(t), σ  (t) ≥ −μ¯ ⎪ ⎪ ⎪   ⎪ ⎪ δ ⎪ ⎪ > h0 , u ¯(0, x) ≥ u0 (x), ϑ(0) ≤ x ≤ σ(0). ⎩ σ(0) = −ϑ(0) = h0 1 + 2

Thus, (¯ u(x, t), ϑ(t), σ(t)) is an upper solution to (1.1), and it follows from Lemma 2.2 that h(t) ≤ σ(t) and g(t) ≥ ϑ(t). An immediate result is h∞ − g∞ ≤ 2 limt→∞ σ(t) = 2h0 (1 + δ) < ∞. Then our result follows.  Theorem 4.9 Assume that (K) and (H) hold. Let (u, g, h) be the unique positive solution to (1.1) with 2h0 < h∗ . Then h∞ = −g∞ = ∞ provided that there exists μ > 0 depending on (u0 , h0 ) such that μ ≥ μ. Proof. The proof here is motivated by Lemma 3.6 in [22] (see also Lemma 3.2 in [24]). Combining (2.2) with assumption (H) yields that there exists some positive constant δ ∗ such that u(a(x) − b(x)u − c(x)(φ ∗ u)) ≥ −δ ∗ u for u ∈ [0, C1 ], + ∗ where C1 is defined in Theorem 2.1. Recalling Theorem 3.2, we have λ+ 1 (h0 ) > 1 = λ1 (h ) due to 2h0 < h∗ , which in turn deduces that h0 ≤ h∗ . Now, consider the following auxiliary free boundary problem ⎧ ⎪ t > 0, r1 (t) < x < r2 (t) wt − dΔw = −δ ∗ w, ⎪ ⎪ ⎪ ⎪ ⎨ w(t, r1 (t)) = 0, r (t) = −μwx (t, r1 (t)), t > 0, 1 (4.12) ⎪ w(t, r2 (t)) = 0, r2 (t) = −μwx (t, r2 (t)), t > 0, ⎪ ⎪ ⎪ ⎪ ⎩ w(0, x) = u0 (x), r2 (0) = −r1 (0) = h0 , − h0 ≤ x ≤ h0 .

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Just as the proof of the existence and uniqueness of the global positive solutions of problem (1.1), it follows that problem (4.12) admits a unique positive solution (w, r1 (t), r2 (t)) defined for all t > 0. In addition, w is bounded above by some positive constant, and r2 (t), −r1 (t) > 0 for all t > 0. In what follows, we rewrite (uμ , g μ (t), hμ (t)) and (wμ , r1μ (t), r2μ (t)) rather than (u, g, h) and (w, r1 (t), r2 (t)) to emphasize the strong dependence of the solutions on the expanding ability μ. Taking advantage of Lemma 2.3, we can obtain that uμ (t, x) ≥ wμ (t, x), g μ (t) ≤ r1μ (t) and hμ (t) ≥ r2μ (t) for t ≥ 0 and x ∈ [r1μ (t), r2μ (t)]. Below, we claim that for all large μ and t > 0, there hold r2μ (t) ≥ 2h∗ and r1μ (t) ≤ −2h∗ . In order to prove this, we define smooth functions r1 (t) and r2 (t) such that h0 , r2 (t), −r1 (t) > 0 and r2 (2) = −r1 (2) = 2h∗ . 2 Then, by the standard theory for parabolic equations, the following initial-boundary value problem ⎧ ⎪ wt − dΔw = −δ ∗ w, t > 0, r1 (t) < x < r2 (t), ⎪ ⎪ ⎨ w(t, r1 (t)) = w(t, r2 (t)) = 0, t > 0, (4.13) ⎪ ⎪ h h ⎪ 0 0 ⎩ w(0, x) = u (x), − ≤x≤ , 0 2 2

 

 satisfies u0 − h20 = has a unique positive solution w(t, x), where u0 (x) ∈ C 2 − h20 , h20

 

u0 h20 = 0 and u0 (x) ≥ u0 (x) > 0 for x ∈ − h20 , h20 . Due to the Hopf boundary lemma, we have wx (t, r2 (t)), −wx (t, r1 (t)) < 0 for all t > 0. Owing to our choice of r1 (t) and r2 (t), we can find some positive constant μ, such that r2 (0) = −r1 (0) =

r2 (t) ≤ −μwx (t, r2 (t)) and r1 (t) ≥ −μwx (t, r1 (t)) for μ ≥ μ and 0 ≤ t ≤ 2. Moreover, it follows from Lemma 2.3 that w(t, x) ≤ wμ (t, x), r1 (t) ≥ r1μ (t) and r2 (t) ≤ r2μ (t) for (t, x) ∈ [0, 2] × [r1 (t), r2 (t)] μ μ h0 since r2 (0)  = −r1 (0) = 2 < r2 (0) = −r1 (0) = h0 and u0 (x) ≤ u0 (x) for all x ∈

− h20 , h20 . Thus, we can arrive at hμ (t) ≥ r2 (t) and g μ (t) ≤ r1 (t) for t ∈ [0, 2]. Particularly, there is

lim hμ (t) > hμ (2) ≥ r2 (2) = 2h∗ and

t→+∞

lim g μ (t) < g μ (2) ≤ r1 (2) = −2h∗ ,

t→+∞

which is equivalent to h∞ − g∞ > 4h∗ . It follows from Theorem 4.3 that h∞ = −g∞ = ∞. This completes the proof.  Now, we give a sharp criteria governing the alternatives in the spreading-vanishing dichotomy in the case that 2h0 < h∗ .

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23

Theorem 4.10 Assume that (K) and (H) hold. Let (u, g, h) be the unique positive solution to (1.1) with 2h0 < h∗ . Then there exists a positive constant μ∗ which depends on u0 (x) such that h∞ = −g∞ = ∞ if μ > μ∗ , and h∞ − g∞ < ∞ if μ ≤ μ∗ . Proof. Motivated by [10, Theorem 3.9], define Σ = {μ > 0 : h∞ − g∞ < ∞}. Then it follows from Theorem 4.8 that (0, μ] ⊂ Σ. Meanwhile, there is Σ ∩ [μ, ∞) = ∅ by Theorem 4.9. By letting μ∗ := sup Σ, we have μ ≤ μ∗ ≤ μ and spreading happens, that is, h∞ = −g∞ = ∞ if μ > μ∗ . If we can prove that Σ = (0, μ∗ ], then our conclusion follows. To emphasize the dependence of the solution (u, g, h) of problem (1.1) on μ, we rewrite (u, g, h) as (uμ , gμ , hμ ). Assume on the contrary that h∞ = −g∞ = ∞ if μ = μ∗ . Then there exists T ∗ > 0 such that hμ (T ∗ ) − gμ (T ∗ ) > h∗ . By the continuous dependence of (uμ , gμ , hμ ) on μ, there is a small τ > 0 such that hμ (T ∗ ) − gμ (T ∗ ) > h∗ for all μ ∈ [μ∗ − τ, μ∗ + τ ]. Therefore, for such a μ, there holds limt→∞ (hμ − gμ )(t) > hμ (T ∗ ) − gμ (T ∗ ) > h∗ , which indicates that [μ∗ − τ, μ∗ + τ ] ∩ Σ = ∅ and u∗ ≤ u∗ − τ , a contradiction, which in turn implies that μ∗ ∈ Σ. For μ ∈ (0, μ∗ ), one can verify that (uμ∗ , gμ∗ , hμ∗ ) is an upper solution to (1.1). It follows from Lemma 2.2 that hμ (t) − gμ (t) ≤ hμ∗ (t) − gμ∗ (t) for all t > 0, and hence limt→∞ (hμ (t) − gμ (t)) ≤ limt→∞ (hμ∗ (t) − gμ∗ (t)) < ∞. Then we arrive at Σ = (0, μ∗ ]. This completes the proof. 

5

Spreading-speed

This section is concerned with the estimate of the spreading speed of the expanding fronts h(t) and g(t) for large time when spreading occurs. Observe that the spreading speed in the evolution of the expanding front h(t), governed by the following free boundary diffusive logistic model in a radially symmetric environment ⎧ ⎪ ut = dΔu + u(α(r) − β(r)u), t > 0, 0 < r < h(t), ⎪ ⎪ ⎪ ⎪ ⎨ ur (t, 0) = u(t, h(t)) = 0, t > 0, (5.1)  ⎪ (t) = −μu (t, h(t)), t > 0, h ⎪ r ⎪ ⎪ ⎪ ⎩ u(0, r) = u0 (r), h(0) = h0 , 0 ≤ r ≤ h0 has been established by Du et al. [13], in which α(r) and β(r) are both positive and satisfy: there exist positive L-periodic functions a(r) and b(r) in C ν (R) such that limr→+∞ (|α(r)− a(r) + |β(r) − b(r)| = 0, i.e., the spatial environment is assumed to be asymptotically periodic at infinity in the radical direction. Problem (5.1) was considered earlier in

24

Cao, Li and Zhao

Du et al. [11], it was shown that for some positive constants k∗ = k∗ (α∞ , β ∞ ) and h(t) ∗ k ∗ = k ∗ (α∞ , β∞ ), there holds k∗ ≤ lim inf t→+∞ h(t) t ≤ lim supt→+∞ t ≤ k when spread∞ ing occurs, where α∞ = lim inf r→∞ α(r), α = lim supr→∞ α(r), β∞ = lim inf r→∞ β(r) and β ∞ = lim supr→∞ β(r). Moreover, it follows that limt→+∞ h(t) t = k exists if both limr→∞ α(r) and limr→∞ β(r) exist. For indefinite coefficient diffusive logistic model with free boundary, the spreading speed of the expanding front h(t), governed by the following model in time-periodic environment as ⎧ ⎪ ut = dΔu + u(a(t, x) − b(t, x)u), t > 0, 0 < x < h(t), ⎪ ⎪ ⎪ ⎪ ⎨ B[u](t, 0) = u(t, h(t)) = 0, t > 0, (5.2)  ⎪ t > 0, h (t) = −μux (t, h(t)), ⎪ ⎪ ⎪ ⎪ ⎩ u(0, x) = u0 (x), h(0) = h0 , 0 ≤ x ≤ h0 was shown in [26] with the assumption that the intrinsic growth rate a(t, x) would be very ∞ negative in the sense that both |{a(t, x) > 0}|  |{a(t, x) < 0}| and 0 a(t, x)dx = −∞ hold for any t > 0, the boundary condition B[u] is the same as that in Introduction. In [26], the author obtained the estimate of the spreading speed by supposing that a∞ (t) = lim inf a(t, x), a∞ (t) = lim sup a(t, x), x→∞

x→∞

b∞ (t) = lim inf b(t, x), x→∞

b∞ (t) = lim sup b(t, x), x→∞

ν

where a∞ (t), a∞ (t), b∞ (t) and b∞ (t) ∈ C 2 ([0, T ]) are T -periodic positive constants. For our model (1.1), following summarized result in [2] will be used. √ Lemma 5.1 For any given constants a > 0, b > 0, d > 0 and k ∈ [0, 2 ad), the problem −dU  + kU  = aU − bU 2 in [0, +∞), U (0) = 0

(5.3)

admits a unique positive solution Uk = Ua,b,d,k and it satisfies U (x) → ab as x → ∞. Moreover, Uk (x) > 0 for x ≥ 0, Uk 1 (0) > Uk 2 (0), Uk1 (x) > Uk2 (x) for x > 0 and √ k1 < k2 , and for each μ > 0, there exists a unique k0 = k0 (μ, a, b, d) ∈ (0, 2 ad) such that μUk 0 (0) = k0 . In what follows, we assume that there exist positive constants a and a such that a ≤ lim inf a(x) ≤ lim sup a(x) ≤ a. x→∞

(5.4)

x→∞

Now we are in a position to establish the upper bound of the spreading speed when spreading happens.

Free boundary problems

25

Theorem 5.2 Let (u, g, h) be the solution of (1.1) with h∞ = −g∞ = +∞, then there g(t) hold lim supt→+∞ h(t) t , lim supt→+∞ − t ≤ k0 = k0 (μ, a, b, d). Proof. It follows from Theorem 2.1 in [10] that the ⎧ ⎪ ut = dΔu + u(a − bu), ⎪ ⎪ ⎪ ⎪ ⎨ u(t, g(t)) = 0, g  (t) = −μux (t, g(t)), ⎪ u(t, h(t)) = 0, h (t) = −μux (t, h(t)), ⎪ ⎪ ⎪ ⎪ ⎩ u(0, x) = u0 (x), h(0) = −g(0) = h0 ,

following free boundary problem t > 0, g(t) < x < h(t), t > 0,

(5.5)

t > 0, − h0 ≤ x ≤ h0

admits a unique positive solution (u, g, h) and there exists a unique k0 = k0 (μ, a, b, d) ∈ √ (0, 2 ad) such that μuk0 (0) = k0 . Recalling Lemma 2.2 that (u, g, h) is an upper solution to (1.1) and then an immediate result is h(t) ≤ h(t) and g(t) ≥ g(t). Thus, lim sup t→+∞

h(t) , t

follows. This completes the proof.

lim sup − t→+∞

g(t) ≤ k0 = k0 (μ, a, b, d) t 

Remark 5.3 It is worth pointing out that different method should be proposed to determine the lower bound of the spreading speed if spreading happens, since we have no idea about the sign of a − cC1 . If we can assume that a − cC1 ≥ 0, then the lower bound of the spreading speed can be established, and there is lim inf t→+∞ h(t) t , g(t) lim inf t→+∞ − t ≥ k0 = k0 (μ, a − cC1 , b, 1) by the same method used in Theorem 5.12 in [10].

Acknowledgments The second author was supported by NSF of China (11671180) and the Fundamental Research Funds for the Central Universities (lzujbky-2016-ct12).

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