The diffusive logistic equation with a free boundary and sign-changing coefficient
aa r X i v : . [ m a t h . A P ] J un The diffusive logistic equation with a free boundaryand sign-changing coefficient Mingxin Wang Natural Science Research Center, Harbin Institute of Technology, Harbin 150080, PR China
Abstract.
This short paper concerns a diffusive logistic equation with a free boundaryand sign-changing coefficient, which is formulated to study the spread of an invasivespecies, where the free boundary represents the expanding front. A spreading-vanishingdichotomy is derived, namely the species either successfully spreads to the right-half-spaceas time t → ∞ and survives (persists) in the new environment, or it fails to establishand will extinct in the long run. The sharp criteria for spreading and vanishing is alsoobtained. When spreading happens, we estimate the asymptotic spreading speed of thefree boundary. Keywords:
Diffusive logistic equation; sign-changing coefficient; Free boundary; Spreading-vanishing; Sharp criteria.
AMS subject classifications (2000) : 35K57, 35K61, 35R35, 92D25.
Understanding the nature of establishment and spread of invasive species is a central problem ininvasion ecology. A lot of mathematicians have made efforts to develop various invasion modelsand investigated them from a viewpoint of mathematical ecology, refer to [2]-[4], [7]-[18], [22] and[24]-[28] for example. Most theoretical approaches are based on or start with single-species models.In consideration of the environmental heterogeneity, the following problem u t − d ∆ u = u ( m ( x ) − u ) , t > , x ∈ Ω ,B [ u ] = 0 , t ≥ , x ∈ ∂ Ω ,u (0 , x ) = u ( x ) , x ∈ Ωis a typical one to describe the spread of invasive species and has received an astonishing amountof attention, see, for example [2, 21] and the references therein. In this model, u ( t, x ) representsthe population density; constant d > m ( x )accounts for the local growth rate (intrinsic growth rate) of the population and is positive on favorablehabitats and negative on unfavorable ones; Ω is a bounded domain of R N ; the boundary operator B [ u ] = αu + β ∂u∂ν , α and β are non-negative functions and α + β > ν is the outward unit normalvector of the boundary ∂ Ω. The corresponding systems with heterogeneous environment have alsobeen studied extensively, please refer to [3, 4, 18, 21] and the references cited therein.To realize the spreading mechanism of an invading species (how fast spreads into new terri-tory, and what factors influence the successful spread), Du and Lin [11] proposed the following free This work was supported by NSFC Grant 11371113 E-mail: [email protected]; Tel: 86-15145101503; Fax: 86-451-86402528
Mingxin Wang boundary problem of the diffusive logistic equation u t − du xx = u ( a − bu ) , t > , < x < h ( t ) ,u x ( t,
0) = 0 , u ( t, h ( t )) = 0 , t ≥ ,h ′ ( t ) = − µu x ( t, h ( t )) , t ≥ ,h (0) = h , u (0 , x ) = u ( x ) , ≤ x ≤ h , (1.1)where x = h ( t ) is the moving boundary to be determined; a, b, d, h and µ are given positive constants, h denotes the size of initial habitat, µ is the ratio of expanding speed of the free boundary andpopulation gradient at expanding front, it can also be considered as the “moving parameter”; u isa given positive initial function. They have derived various interesting results.Since then, this kind of problems describing the spread by free boundary have been studiedintensively. For example, when the boundary condition u x = 0 at x = 0 in (1.1) is replaced by u = 0,such free boundary problem was studied by Kaneko & Yamada [17]. Du & Guo [7, 8], Du, Guo &Peng [9] and Du & Liang [10] considered the higher space dimensions, heterogeneous environmentand time-periodic environment case, where the heterogeneous environment coefficients were requiredto have positive lower and upper bounds. Peng & Zhao [22] studied the seasonal succession case.Instead of u ( a − bu ) by a general function f ( u ), this problem has been investigated by Du & Lou[13] and Du, Matsuzawa & Zhou [14]. The diffusive competition system with a free boundary hasbeen studied by Guo & Wu [15], Du & Lin [12] and Wang & Zhao [26]. The diffusive prey-predatormodel with free boundaries has been studied by Wang & Zhao [24, 25, 27].Recently, Zhou and Xiao [28] studied the following diffusive logistic model with a free boundaryin the heterogeneous environment: u t − du xx = u ( m ( x ) − u ) , t > , < x < h ( t ) ,u x ( t,
0) = 0 , u ( t, h ( t )) = 0 , t ≥ ,h ′ ( t ) = − µu x ( t, h ( t )) , t ≥ ,h (0) = h , u (0 , x ) = u ( x ) , ≤ x ≤ h , where the initial function u ∈ C ([0 , h ]), u ′ (0) = u ( h ) = 0, u ′ ( h ) < u > , h ). Inthe strong heterogeneous environment, i.e, (H1) m ∈ C ([0 , ∞ )) ∩ L ∞ ([0 , ∞ )) and m changes sign in (0 , h ),Zhou and Xiao took d and µ as variable parameters and derived some sufficient conditions for speciesspreading (resp. vanishing); While in the weak heterogeneous environment, i.e., (H2) m ∈ C ([0 , ∞ )) and 0 < m ≤ m ( x ) ≤ m < ∞ for all x ≥ < d ≤ d ∗ with some d ∗ .Motivated by the above works, in this paper we consider the following problem u t − du xx = u ( m ( x ) − u ) , t > , < x < h ( t ) ,B [ u ]( t,
0) = 0 , u ( t, h ( t )) = 0 , t ≥ ,h ′ ( t ) = − µu x ( t, h ( t )) , t ≥ ,h (0) = h , u (0 , x ) = u ( x ) , ≤ x ≤ h , (1.2)ogistic equation with a free boundary and sign-changing coefficient 3where, B [ u ] = αu − βu x , α , β ≥ α + β = 1; the initial function u ( x ) satisfies • u ∈ C ([0 , h ]) , u > , h ), B [ u ](0) = u ( h ) = 0.Throughout this paper, we suppose that the function m ( x ) satisfies (A) m ∈ C ([0 , ∞ )) ∩ L ∞ ([0 , ∞ )) and m ( x ) is positive somewhere in (0 , ∞ ).Actually, if m ( x ) ≤ , ∞ ), the problem (1.2) may not have the biological background.The objective of this paper is to study the dynamics of (1.2) under weaker assumptions onthe heterogeneous environment function m ( x ). In Section 2, we shall give the global existence,uniqueness, regularity and estimate of ( u, h ). Especially, the uniform estimates of k u ( t, · ) k C [0 , h ( t )] for t ≥ k h ′ k C ν/ ([ n +1 ,n +3]) for n ≥ h ∞ , whichis different from the previous works. Section 3 is devoted to the sharp criteria for spreading andvanishing. We shall use the pairs ( h , µ ) and ( d, µ ), respectively, as varying parameters to describethe sharp criteria. In Section 4, we study the long time behavior of u for spreading case. Tothis aim, in this section we first discuss the existence and uniqueness of the positive solution to acorresponding stationary problem. As a consequence of the results obtained in Sections 3 and 4, aspreading-vanishing dichotomy is obtained. In Section 5 we estimate the asymptotic spreading speedof the free boundary when spreading occurs. The last section is a brief discussion.We remark that for the higher dimensional and radially symmetric case of (1.2), the methods ofthis paper are still valid and the corresponding results can be retained. Besides, the present shortpaper can be regarded as the simplify, improvement and generalization of [28] in some sense. ( u, h ) In this section, we give the existence, uniqueness, regularity and estimate of solution.
Theorem 2.1
Problem (1.2) has a unique global solution ( u, h ) , and for some ν ∈ (0 , , u ∈ C ν , ν ( D ∞ ) , h ∈ C ν (0 , ∞ ) , (2.1) where D ∞ = { ( t, x ) : t ∈ (0 , ∞ ) , x ∈ [0 , h ( t )] } . Furthermore, there exist positive constants M = M ( k m, u k ∞ ) and C = C ( µ, k m, u k ∞ ) , such that < u ( t, x ) ≤ M, < h ′ ( t ) ≤ µM, ∀ t > , < x < h ( t ) , (2.2) k h ′ k C ν/ ([ n +1 ,n +3]) ≤ C, ∀ n ≥ , k u ( t, · ) k C ([0 , h ( t )]) ≤ C, ∀ t ≥ . (2.3) Proof.
Noting that the function m is bounded, and applying the methods used in [1, 11] withsome modifications, we can prove that (1.2) has a unique global solution ( u, h ), and satisfies (2.1)and the first estimate of (2.2). The details are omitted here. Because of the condition (A) , theregularity of ( u, h ) can not be promoted.Now we prove h ′ ( t ) >
0. Firstly, as u > < x < h ( t ) and u = 0 at x = h ( t ), we see that u x ( t, h ( t )) ≤ h ′ ( t ) ≥
0. Since we only know h ∈ C ν ([0 , ∞ )), it can not be guaranteed thatthe domain D ∞ has an interior sphere property at the right boundary x = h ( t ). Hence, the Hopfboundary lemma cannot be used directly to get h ′ ( t ) >
0. To solve this, we use a transformation tostraighten the free boundary x = h ( t ). Define y = x/h ( t ) and w ( t, y ) = u ( t, x ). A series of detailedcalculation yield w t − dζ ( t ) w yy − ξ ( t, y ) w y = w [ m ( h ( t ) y ) − w ] , t > , < y < , ( αw − βh ( t ) w y )( t,
0) = 0 , w ( t,
1) = 0 , t ≥ ,w (0 , y ) = u ( h y ) , ≤ y ≤ , Mingxin Wang where ζ ( t ) = h − ( t ), ξ ( t, y ) = yh ′ ( t ) /h ( t ). This is an initial and boundary value problem with fixedboundary. Since w > t > < y <
1, by the Hopf boundary lemma, we have w y ( t, < t >
0. This combines with the relation u x = h − ( t ) w y derives that u x ( t, h ( t )) <
0, and so h ′ ( t ) > t >
0. The proof of h ′ ( t ) ≤ µM is similarly to that in [11].Now we prove (2.3). For the integer n ≥
0, let w n ( t, y ) = w ( t + n, y ), then we have w nt − dζ ( t + n ) w nyy − ξ ( t + n, y ) w ny = w n [ m ( h ( t + n ) y ) − w n ] , t > , < y < , ( αw n − βh ( t + n ) w ny )( t,
0) = 0 , w n ( t,
1) = 0 , t ≥ ,w n (0 , y ) = u ( n, h ( n ) y ) , ≤ y ≤ . Noticing (2.2), apply the interior L p estimate (see [20, Theorems 7.15 and 7.20]) and embeddingtheorem, we can find a constant C > n such that k w n k C ν , ν ([1 , × [0 , ≤ C for all n ≥
0. This implies k w k C ν , ν ( E n ) ≤ C , where E n = [ n + 1 , n + 3] × [0 , h ′ ( t ) = − µu x ( t, h ( t )), u x ( t, h ( t )) = h − ( t ) w y ( t,
1) and 0 < h ′ ( t ) ≤ µM , allows us toget the first estimate of (2.3). Since these rectangles E n overlap and C is independent of n , one has k w k C , ([1 , ∞ ) × [0 , ≤ C . Using u x = h − ( t ) w y again, the second estimate of (2.3) is obtained.It follows from Theorem 2.1 that h ( t ) is monotonic increasing. Therefore, there exists h ∞ ∈ (0 , ∞ ]such that lim t →∞ h ( t ) = h ∞ . We first prove that if h ∞ < ∞ then lim t →∞ max ≤ x ≤ h ( t ) u ( t, x ) = 0. This conclusion will help us toestablish the sharp criteria for spreading and vanishing. Lemma 3.1
Let d, µ and B be as above, c ∈ R . Assume that s ∈ C ([0 , ∞ )) , w ∈ C ν , ν ([0 , ∞ ) × [0 , s ( t )]) and satisfy s ( t ) > , w ( t, x ) > for t ≥ and < x < s ( t ) . We further suppose that lim t →∞ s ( t ) < ∞ , lim t →∞ s ′ ( t ) = 0 and there exists a constant C > such that k w ( t, · ) k C [0 , s ( t )] ≤ C for t > . If ( w, s ) satisfies w t − dw xx ≥ cw, t > , < x < s ( t ) ,B [ w ] = 0 , t ≥ , x = 0 ,w = 0 , s ′ ( t ) ≥ − µw x , t ≥ , x = s ( t ) , then lim t →∞ max ≤ x ≤ s ( t ) w ( t, x ) = 0 . Proof . When α = 0 or β = 0, this is exactly [24, Proposition 3.1]. When α > β >
0, thatproof is still valid. The details are omitted here.Applying (2.3) and Lemma 3.1, we have the following theorem.
Theorem 3.1
Let ( u, h ) be the solution of (1.2) . If h ∞ < ∞ , then lim t →∞ max ≤ x ≤ h ( t ) u ( t, x ) = 0 .This shows that if the species cannot spread successfully, it will extinct in the long run. For any given ℓ >
0, let λ ( ℓ ; d, m ) be the first eigenvalue of ( − dφ ′′ − m ( x ) φ = λφ, < x < ℓ,B [ φ ](0) = 0 , φ ( ℓ ) = 0 . (3.1)Remember the boundary condition φ ( ℓ ) = 0 and m ( x ) is bounded, the following conclusions are wellknown (see, for example, [3, 21, 23]).ogistic equation with a free boundary and sign-changing coefficient 5 Proposition 3.1 (i) λ ( ℓ ; d, m ) is continuous in d, m and ℓ ; (ii) λ ( ℓ ; d, m ) is strictly increasing in d , strictly decreasing in m and ℓ ; (iii) lim d →∞ λ ( ℓ ; d, m ) = lim ℓ → + λ ( ℓ ; d, m ) = ∞ , lim d → + λ ( ℓ ; d, m ) = − max [0 ,ℓ ] m ( x ) . Lemma 3.2 If h ∞ < ∞ , then λ ( h ∞ ; d, m ) ≥ . Proof . We assume λ ( h ∞ ; d, m ) < λ ( ℓ ; d, m ) in ℓ and h ( t ) → h ∞ , there exists τ ≫ λ ( h ( τ ); d, m ) <
0. Let w be the solution of w t − dw xx = w ( m ( x ) − w ) , t ≥ τ, < x < h ( τ ) ,B [ w ]( t,
0) = w ( t, h ( τ )) = 0 , t ≥ τ,w ( τ, x ) = u ( τ, x ) , ≤ x ≤ h ( τ ) . Then u ≥ w in [ τ, ∞ ) × [0 , h ( τ )]. As λ ( h ( τ ); d, m ) <
0, we have lim t →∞ w ( t, x ) = z ( x ) uniformly on[0 , h ( τ )], where z is the unique positive solution of ( − dz ′′ = z ( m ( x ) − z ) , < x < h ( τ ) ,B [ z ](0) = z ( h ( τ )) = 0 . Hence, lim inf t →∞ u ( t, x ) ≥ z ( x ) > , h ( T )). This contradicts Theorem 3.1.The following lemma is the analogue of [11, Lemma 3.5] and the proof will be omitted. Lemma 3.3 ( Comparison principle ) Let ¯ h ∈ C ([0 , ∞ )) and ¯ h > in [0 , ∞ ) , ¯ u ∈ C , ( O ) ∩ C , ( O ) ,with O = { ( t, x ) : t > , < x < ¯ h ( t ) } . Assume that (¯ u, ¯ h ) satisfies ¯ u t − d ¯ u xx ≥ ¯ u ( m ( x ) − ¯ u ) , t > , < x < ¯ h ( t ) ,B [¯ u ]( t, ≥ , ¯ u ( t, ¯ h ( t )) = 0 , t ≥ , ¯ h ′ ( t ) ≥ − µ ¯ u x ( t, ¯ h ( t )) , t ≥ . If ¯ h (0) ≥ h , ¯ u (0 , x ) ≥ in [0 , ¯ h (0)] , and ¯ u (0 , x ) ≥ u ( x ) in [0 , h ] . Then the solution ( u, h ) of (1.2) satisfies h ( t ) ≤ ¯ h ( t ) in [0 , ∞ ) , and u ≤ ¯ u in D , where D = { ( t, x ) : t ≥ , ≤ x ≤ h ( t ) } . Lemma 3.4 If λ ( h ; d, m ) > , then there exists µ > , depending on d, h , m ( x ) and u ( x ) , suchthat h ∞ < ∞ provided µ ≤ µ . By Lemma , λ ( h ∞ ; d, m ) ≥ for µ ≤ µ . Proof . The idea comes from [11, 15, 24], but the proof given here is more simple. Let φ be thecorresponding positive eigenfunction to λ := λ ( h ; d, m ). Noting that φ ′ ( h ) < φ (0) > β >
0, and φ ′ (0) > β = 0, it is easy to see that there exists k > xφ ′ ( x ) ≤ kφ ( x ) , ∀ ≤ x ≤ h . (3.2)Let 0 < δ, σ < K > s ( t ) = 1 + 2 δ − δ e − σt , v ( t, x ) = K e − σt φ ( x/s ( t )) , t ≥ , ≤ x ≤ h s ( t ) . Firstly, for any given 0 < ε ≪
1, since m ( x ) is uniformly continuous in [0 , h ], it is easy to seethat there exists 0 < δ ( ε ) ≪ < δ ≤ δ ( ε ) and 0 < σ < (cid:12)(cid:12) s − ( t ) m ( x/s ( t )) − m ( x ) (cid:12)(cid:12) ≤ ε, ∀ t > , ≤ x ≤ h s ( t ) . (3.3) Mingxin Wang
Denote y = x/s ( t ). Owing to (3.2), (3.3) and λ >
0, the direct calculation yields, v t − dv xx − v ( m ( x ) − v ) = v (cid:18) − σ + m ( y ) s ( t ) − m ( x ) − yφ ′ ( y ) φ ( y ) σδs ( t ) e − σt + λ s ( t ) (cid:19) + v ≥ v ( − σ − ε − kσ + λ / > , ∀ t > , < x < h s ( t ) (3.4)provided 0 < σ, ε ≪
1. Evidently, v ( t, h s ( t )) = K e − σt φ ( h ) = 0. If either α = 0 or β = 0,then B [ v ]( t,
0) = 0. If α, β >
0, then αφ (0) = βφ ′ (0) and φ ′ (0) >
0. Therefore, B [ v ]( t,
0) = βK e − σt φ ′ (0)[1 − /s ( t )] > s ( t ) >
1. In a word, B [ v ]( t, ≥ , v ( t, h s ( t )) = 0 , ∀ t ≥ . (3.5)Fix 0 < σ, ε ≪ < δ ≤ δ ( ε ). Thanks to the regularities of u ( x ) and φ ( x ), we can choose a K ≫ u ( x ) ≤ Kφ ( x/ (1 + δ )) = v (0 , x ) , ∀ ≤ x ≤ h . (3.6)Thanks to h s ′ ( t ) = h σδ e − σt and v x ( t, h s ( t )) = s ( t ) K e − σt φ ′ ( h ), there exists µ > h s ′ ( t ) ≥ − µv x ( t, h s ( t )) , ∀ < µ ≤ µ , t ≥ . (3.7)Remember (3.4)-(3.7). Applying Lemma 3.3 to ( u, h ) and ( v, h s ), it yields that h ( t ) ≤ h s ( t )for all t ≥
0. Hence h ∞ ≤ h s ( ∞ ) = h (1 + 2 δ ) for all 0 < µ ≤ µ .Instead of K by η , from the proof of Lemma 3.4 we see that the following lemma holds. Lemma 3.5 If λ ( h ; d, m ) > , then there exist δ, η > , such that h ∞ < ∞ provided u ( x ) ≤ ηφ ( x/ (1 + δ )) in [0 , h ] . The following lemma is the analogue of [26, Lemma 3.2] and the proof will be omitted.
Lemma 3.6
Let
C > be a constant. For any given constants ¯ h , H > , and any function ¯ u ∈ C ([0 , ¯ h ]) satisfying B [¯ u ](0) = ¯ u (¯ h ) = 0 and ¯ u > in (0 , ¯ h ) , there exists µ > such thatwhen µ ≥ µ and (¯ u, ¯ h ) satisfies ¯ u t − d ¯ u xx ≥ − C ¯ u, t > , < x < ¯ h ( t ) ,B [¯ u ]( t,
0) = 0 = ¯ u ( t, ¯ h ( t )) , t ≥ , ¯ h ′ ( t ) = − µ ¯ u x ( t, ¯ h ( t )) , t ≥ , ¯ h (0) = ¯ h , ¯ u (0 , x ) = ¯ u ( x ) , ≤ x ≤ ¯ h , we must have lim t →∞ ¯ h ( t ) > H . To establish the sharp criteria, we define two sets. For any given d , let P d = { ℓ > λ ( ℓ ; d, m ) =0 } . By the monotonicity of λ ( ℓ ; d, m ) in ℓ , the set P d contains at most one element. For any given ℓ , we define P ℓ = { d > λ ( ℓ ; d, m ) = 0 } . Similarly, it contains at most one element. Remark 3.1
For the fixed d > , due to lim ℓ → + λ ( ℓ ; d, m ) = ∞ and lim ℓ →∞ λ ( ℓ ; d, m ) := λ ∞ ( d, m ) exists, we have that P d = ∅ is equivalent to λ ∞ ( d, m ) < . As a consequence, if m satisfies one of the following assumptions: ogistic equation with a free boundary and sign-changing coefficient 7 (A1) There exist a constant ρ > and y n > x n > such that y n − x n → ∞ as n → ∞ and m ( x ) ≥ ρ in [ x n , y n ] ; (A2) There exist three constants ρ > , k > , − < γ ≤ and x n satisfying x n → ∞ as n → ∞ , such that m ( x ) ≥ ρx γ in [ x n , kx n ] .Then λ ∞ ( d, m ) < , and so P d = ∅ for all d > . In fact, when the condition (A1) holds, we use the following expression of λ ( ℓ ; d, m ): λ ( ℓ ; d, m ) = inf φ ∈ H ((0 ,ℓ )) dφ (0) φ ′ (0) + d R ℓ ( φ ′ ( x )) d x − R ℓ m ( x ) φ ( x )d x R ℓ φ ( x )d x . Take a function φ n with φ n ( x ) = 0 in [0 , x n ], φ n ( x ) = x − x n in [ x n , x n +1], φ n ( x ) = 1 in [ x n +1 , y n − φ n ( x ) = y n − x in [ y n − , y n ]. Then φ n ∈ H ((0 , y n )), φ n (0) = 0, and Z y n ( φ ′ n ( x )) d x = 2 , Z y n m ( x ) φ n ( x )d x > ρ ( y n − x n − , Z y n φ n ( x )d x < y n − x n . Hence, for any fixed d >
0, we have λ ∞ ( d, m ) < λ ( y n ; d, m ) ≤ d − ρ ( y n − x n − y n − x n → − ρ < n → ∞ . When the condition (A2) holds, we use the idea of [5, Lemma 3.1] to derive our conclusion. Let λ ( n ) be the principal eigenvalue of − dψ ′′ = λψ, x n < x < kx n ; ψ ( x n ) = ψ ( kx n ) = 0 , and ψ ( x ) be the corresponding positive eigenfunction. Through a simple rescaling ψ ( x ) = Ψ( x/x n ) :=Ψ( y ), we see that Ψ( y ) satisfies − d Ψ ′′ ( y ) = x n λ ( n )Ψ( y ) , < y < k ; Ψ(1) = Ψ( k ) = 0 . Since Ψ >
0, we have λ ∗ = x n λ ( n ), where λ ∗ is the principal eigenvalue of − dφ ′′ = λφ, < x < k ; φ (1) = φ ( k ) = 0 . Make the zero extension of ψ to [0 , x n ), then ψ (0) = 0 and Z kx n (cid:2) d ( ψ ′ ) − m ( x ) ψ (cid:3) d x = Z kx n x n (cid:2) d ( ψ ′ ) − m ( x ) ψ (cid:3) d x = Z kx n x n (cid:2) λ ( n ) ψ − m ( x ) ψ (cid:3) d x ≤ Z kx n x n (cid:0) x − n λ ∗ − ρk γ x γn (cid:1) ψ d x = x − n Z kx n x n (cid:0) λ ∗ − ρk γ x γn (cid:1) ψ d x < n ≫ x n → ∞ and 2 + γ >
0. This implies λ ( kx n ; d, m ) < n ≫
1, and then λ ∞ ( d, m ) < (A1) and (A2) seem to be “weaker” because m ( x ) may be “very negative” inthe sense that both |{ m ( x ) > }| ≪ |{ m ( x ) < }| and R ∞ m ( x )d x = −∞ are allowed. Remark 3.2
For each fixed ℓ > , as lim d →∞ λ ( ℓ ; d, m ) = ∞ , lim d → + λ ( ℓ ; d, m ) = − max [0 ,ℓ ] m ( x ) ,we see that P ℓ = ∅ is equivalent to max [0 ,ℓ ] m ( x ) > . By the condition (A) , we have max [0 ,ℓ ] m ( x ) > for each suitable large ℓ . So, P ℓ = ∅ for such ℓ . Mingxin Wang
Now we fix d , and consider h and µ as varying parameters to depict the sharp criteria forspreading and vanishing. Assume that P d = ∅ and let h ∗ = h ∗ ( d ) ∈ P d , i.e., λ ( h ∗ ; d, m ) = 0.Recalling the estimate (2.2), as the consequence of Lemmas 3.2, 3.4 and 3.6, we have Corollary 3.1 (i) If h ∞ < ∞ , then h ∞ ≤ h ∗ . Hence, h ≥ h ∗ implies h ∞ = ∞ for all µ > ; (ii) When h < h ∗ . There exist µ , µ > , such that h ∞ ≤ h ∗ for µ ≤ µ , h ∞ = ∞ for µ ≥ µ . Finally, we give the sharp criteria for spreading and vanishing.
Theorem 3.2 (i) If h ≥ h ∗ = h ∗ ( d ) , then h ∞ = ∞ for all µ > ; (ii) If h < h ∗ , then there exist µ ∗ > , depending on d , m ( x ) , u ( x ) and h , such that h ∞ = ∞ for µ > µ ∗ , while h ∞ ≤ h ∗ for µ ≤ µ ∗ . Proof . Noticing Corollary 3.1, by use of Lemma 3.3 and the continuity method, we can proveTheorem 3.2. Please refer to the proof of [11, Theorem 3.9] for details.When h is fixed, d and µ are regarded as the varying parameters, we have the following sharpcriteria for spreading and vanishing. Theorem 3.3
Assume that max [0 ,h ] m ( x ) > , and let d ∗ = d ∗ ( h ) ∈ P h ( see Remark . (i) If d ≤ d ∗ , then h ∞ = ∞ for all µ > ; (ii) If d > d ∗ and P d = ∅ , then there exists µ ∗ > , depending on d , m , u and h , such that h ∞ = ∞ when µ > µ ∗ , h ∞ < ∞ when µ ≤ µ ∗ . Remark 3.3
If one of (A1) and (A2) holds, then P d = ∅ for any d > see Remark . Proof of Theorem 3.3 . (i) When d < d ∗ , we have λ ( h ; d, m ) < λ ( h ; d ∗ , m ) = 0. So, P d = ∅ and h > h ∗ ( d ). When d = d ∗ , we have λ ( h ; d, m ) = 0 and h = h ∗ ( d ). By Theorem 3.2(i), h ∞ = ∞ for all µ > d > d ∗ , we have λ ( h ; d, m ) > λ ( h ; d ∗ , m ) = 0. By Lemma 3.4, there exists µ > h ∞ < ∞ for µ ≤ µ . On the other hand, as P d = ∅ , there exists H ≫ λ ( H ; d, m ) <
0. In view of Lemma 3.6, there exists µ > h ∞ > H provided µ ≥ µ ,which implies λ ( h ∞ ; d, m ) < λ ( H ; d, m ) <
0. Hence, h ∞ = ∞ for µ ≥ µ by Lemma 3.2. Theremaining proof is the same as that of [11, Theorem 3.9].When α = 0 and the condition (H2) holds, Theorem 3.3 has been given by [28, Theorem 5.2]. u for the spreading case: h ∞ = ∞ For the vanishing case: h ∞ < ∞ , we have known lim t →∞ max ≤ x ≤ h ( t ) u ( t, x ) = 0 (cf. Theorem 3.1).In this section we study the long time behavior of u for the spreading case: h ∞ = ∞ . To this aim,we first study the existence and uniqueness of positive solution to the stationary problem: ( − du ′′ = u ( m ( x ) − u ) , < x < ∞ ,B [ u ](0) = 0 . (4.1)The following lemma is a special case of [19, Proposition 2.2].ogistic equation with a free boundary and sign-changing coefficient 9 Lemma 4.1 ( Comparison principle ) Let ℓ > , u , u ∈ C ([0 , ℓ )) be positive functions in (0 , ℓ ) andsatisfy in the sense of distributions that − du ′′ − m ( x ) u + u ≥ ≥ − du ′′ − m ( x ) u + u and B [ u ](0) ≥ ≥ B [ u ](0) , lim sup x → ℓ ( u − u ) ≤ . Then u ≥ u in (0 , ℓ ) . Theorem 4.1
Assume that there exist constants − < γ ≤ and m , m > , such that m = lim inf x →∞ m ( x ) x γ , m = lim sup x →∞ m ( x ) x γ . (4.2) Then (4 . has a unique positive solution ˆ u and m ≤ lim inf x →∞ ˆ u ( x ) x γ , lim sup x →∞ ˆ u ( x ) x γ ≤ m . (4.3) Proof . The existence of positive solution to (4.1) can be proved as that of [6, Lemma 7.16]. Infact, for any large ℓ >
0, in the same way as that of [19], we can prove that the problem ( − du ′′ = u ( m ( x ) − u ) , < x < ℓ,B [ u ](0) = 0 , u ( ℓ ) = ∞ has a unique positive solution u ℓ (when β = 0, this conclusion is exactly [6, Theorem 6.15]). Followingthe proof of [6, Lemma 7.16] step by step (using Lemma 4.1 instead of lemma 5.6 there), we canprove that (4.1) has at least one positive solution.The uniqueness of positive solution to (4.1) and the conclusion (4.3) can be proved by the similarway to that of [6, Theorem 7.12] with suitable modifications. We omit the details here. Actually,proofs of the uniqueness and (4.3) only rely on the properties of m and u at infinity, have nothingto do with the condition of u at x = 0.It is easy to see that if the condition (4.2) holds, then the assumption (A2) must be true.Therefore, P d = ∅ by Remark 3.1. Lemma 4.2
Assume that (4 . holds. Let h ∗ = h ∗ ( d ) satisfy λ ( h ∗ ; d, m ) = 0 . For ℓ > h ∗ , whichimplies λ := λ ( ℓ ; d, m ) < , let u ℓ ( x ) be the unique positive solution of ( − du ′′ = u ( m ( x ) − u ) , < x < ℓ,B [ u ](0) = 0 , u ( ℓ ) = 0 . (4.4) Then lim ℓ →∞ u ℓ ( x ) = ˆ u ( x ) uniformly in [0 , L ] for any L > . Proof . Let φ be the positive eigenfunction of (3.1) corresponding to λ . Since λ <
0, it iseasy to verify that εφ and sup x ≥ m ( x ) are the ordered lower and upper solutions to (4.4) provided0 < ε ≪
1. So, the problem (4.4) has at least one positive solution. The uniqueness of positivesolution to (4.4) is followed by Lemma 4.1.By Lemma 4.1, u ℓ ≤ ˆ u in [0 , ℓ ], and u ℓ is increasing in ℓ . Utilizing the regularity theory andcompactness argument, it follows that there exists a positive function u , such that u ℓ → u in C ([0 , ∞ )) as ℓ → ∞ , and u solves (4.1). By the uniqueness, u = ˆ u .Finally, we give the main result of this section.0 Mingxin Wang
Theorem 4.2
Let (4 . hold. If h ∞ = ∞ , then lim t →∞ u ( t, x ) = ˆ u ( x ) in C loc ([0 , ∞ )) . Proof . Choose
K > K ˆ u ≥ u in [0 , h ]. Then ϕ := K ˆ u satisfies ϕ t − dϕ xx >ϕ ( m ( x ) − ϕ ). Let w be the solution of w t − dw xx = w ( m ( x ) − w ) , t > , < x < ∞ ,B [ w ]( t,
0) = 0 , t > ,w (0 , x ) = K ˆ u ( x ) , x ≥ . Then u ≤ w , and w is monotone decreasing in t . Because ˆ u is the unique positive solution of (4.1),by the standard method we can prove that lim t →∞ w ( t, x ) = ˆ u ( x ) uniformly in [0 , L ] for any L > h ∞ = ∞ , it follows that lim sup t →∞ u ( t, x ) ≤ ˆ u ( x ) uniformly in [0 , L ].Let h ∗ = h ∗ ( d ) be such that λ ( h ∗ ; d, m ) = 0. When ℓ > h ∗ , we have λ := λ ( ℓ ; d, m ) <
0. As h ∞ = ∞ , there exists T ≫ h ( t ) > ℓ for all t ≥ T . Let φ be the positive eigenfunction of(3.1) corresponding to λ . Choose 0 < σ ≪ u ( T, x ) ≥ σφ ( x ) in [0 , ℓ ] and σφ is a lowersolution of (4.4). Let u ℓ be the unique solution of u t − du xx = u ( m ( x ) − u ) , t ≥ T, < x < ℓ,B [ u ]( t,
0) = 0 , u ( t, ℓ ) = 0 , t ≥ T,u ( T, x ) = σφ ( x ) , x ∈ [0 , ℓ ] . Then u ≥ u ℓ in [ T, ∞ ) × [0 , ℓ ], and u ℓ is increasing in t . So, lim t →∞ u ℓ ( t, x ) = u ℓ ( x ) uniformly in[0 , ℓ ] since u ℓ is the unique positive solution of (4.4). Hence, lim inf t →∞ u ( t, x ) ≥ u ℓ ( x ) uniformly in[0 , ℓ ]. By Lemma 4.2, lim inf t →∞ u ( t, x ) ≥ ˆ u ( x ) uniformly in [0 , L ] for any L > α = 0, Theorem 4.2 has been obtained by [28] under one of thefollowing assumptions:(i) the condition (H2) holds (see [28, Lemma 5.2]);(ii) the function m ∈ C ([0 , ∞ )), is positive somewhere in (0 , h ) and satisfies (4 .
2) with γ = 0.The diffusion rate d satisfies 0 < d ≤ d ∗ for some d ∗ > (H2) implies (4 .
2) with γ = 0.Combining Theorems 3.1, 3.2, 3.3 and 4.2, we have the following two theorems concerningspreading-vanishing dichotomy and sharp criteria for spreading and vanishing. Theorem 4.3
Let (4.2) hold, d > be fixed and h ∗ = h ∗ ( d ) satisfy λ ( h ∗ ; d, m ) = 0 . Then either (i) Spreading: h ∞ = ∞ and lim t →∞ u ( t, x ) = ˆ u ( x ) uniformly in [0 , L ] for any L > ; or (ii) Vanishing: h ∞ ≤ h ∗ and lim t →∞ max ≤ x ≤ h ( t ) u ( t, x ) = 0 , where ˆ u ( x ) is the unique positivesolution of (4.1) .Moreover, (iii) If h ≥ h ∗ , then h ∞ = ∞ for all µ > ; (iv) If h < h ∗ , then there exist µ ∗ > , depending on d , m , u and h , such that h ∞ = ∞ for µ > µ ∗ , while h ∞ ≤ h ∗ for µ ≤ µ ∗ . Theorem 4.4
Assume that (4.2) holds, h > is fixed and max [0 ,h ] m ( x ) > . Let d ∗ = d ∗ ( h ) ∈ P h Then either (i)
Spreading: h ∞ = ∞ and lim t →∞ u ( t, x ) = ˆ u ( x ) uniformly in [0 , L ] for any L > ; or (ii) Vanishing: h ∞ < ∞ and lim t →∞ max ≤ x ≤ h ( t ) u ( t, x ) = 0 .Moreover, ogistic equation with a free boundary and sign-changing coefficient 11(iii) If d ≤ d ∗ , then h ∞ = ∞ for all µ > ; (iv) If d > d ∗ , then there exist µ ∗ > , depending on d , m , u and h , such that h ∞ = ∞ for µ > µ ∗ , while h ∞ < ∞ for µ ≤ µ ∗ . In this section, we shall estimate the asymptotic spreading speed of the free boundary h ( t ) whenspreading occurs. Throughout this section, we assume that (4 .
2) holds with γ = 0, which implies P d = ∅ for all d > Proposition 5.1 ([11, Proposition 4.1])
Let d and c be given positive constants. Then for any k ≥ , the problem ( − dw ′′ + kw ′ = w ( c − w ) , < x < ∞ ,w (0) = 0 , w ( ∞ ) = c has a unique positive solution w k ( x ) . Moreover, for each µ > , there exists a unique k = k ( µ, c ) > such that µw ′ k (0) = k . Theorem 5.1
When h ∞ = ∞ , we have ( no other restrictions on d, h , m and u ) k ( µ, m ) ≤ lim inf t →∞ h ( t ) t , lim sup t →∞ h ( t ) t ≤ k ( µ, m ) . (5.1) Proof . The proof is similar to those of [11, Theorem 4.2], [7, Theorem 3.6] and [28, Theorem 6.1].Here we give the sketch for completeness and readers’ convenience.For any given 0 < ε ≪
1, by (4.2) and (4.3) with γ = 0, there exists ℓ = ℓ ( ε ) ≫ m − ε < m ( x ) < m + ε, m − ε < ˆ u ( x ) < m + ε, ∀ x ≥ ℓ. Take advantage of h ∞ = ∞ and Theorem 4.2, there exists T = T ( ℓ ) ≫ h ( T ) > ℓ, m − ε < u ( t + T, ℓ ) < m + 2 ε, ∀ t > . Follow the proof of [7, Theorem 3.6] or [28, Theorem 6.1] step by step, we can get (5.1). The detailsare omitted here.When α = 0, Theorem 5.1 has been given in [28] for the case that 0 < d ≤ d ∗ with some d ∗ > From the above discussions we have seen that λ ∞ ( d, m ) := lim ℓ →∞ λ ( ℓ ; d, m ) < d and m , and is independent of the moving parameter µ and initial value u ( x ). It seems that λ ∞ ( d, m ) is determined by d and R ∞ m ( x )d x .The main conclusions of this paper can be briefly summarized as follows:(I) If one of the following holds:(i) d is suitable small ( h and m ( x ) are fixed, m ( x ) is positive somewhere in (0 , h )),(ii) m ( x ) is suitable “larger” in the sense of “distribution” ( h and d are fixed),2 Mingxin Wang (iii) h is suitable “larger” ( d and m ( x ) are fixed, m ( x ) satisfies either (A1) or (A2) ),then the species will successfully spread and survive in the new environment (maintain a positivedensity distribution), regardless of initial population size and value of the moving parameter.(II) When the above situations are not appeared, we can control the moving parameter µ andfind a critical value µ ∗ such that the species will spread successfully when µ > µ ∗ , the species failsto establish and will extinct in the long run when µ ≤ µ ∗ . The better way to reduce the movingparameter might be by controlling the surrounding environment.These theoretical results may be helpful in the prediction and prevention of biological invasions. References [1] X.F. Chen & A. Friedman,
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