Appl. Math. J. Chinese Univ. 2013, 28(3): 290-302

A reaction-diﬀusion model with nonlinearity driven diﬀusion MA Man-jun1

HU Jia-jia1

ZHANG Jun-jie2

TAO Ji-cheng1

Abstract. In this paper, we deal with the model with a very general growth law and an M driven diﬀusion

∂u(t, x) u(t, x) = DΔ( ) + μ(t, x)f (u(t, x), M (t, x)). ∂t M (t, x)

For the general case of time dependent functions M and μ, the existence and uniqueness for positive solution is obtained. If M and μ are T0 -periodic functions in t, then there is an attractive positive periodic solution. Furthermore, if M and μ are time-independent, then the non-constant stationary solution M (x) is globally stable. Thus, we can easily formulate the conditions deriving the above behaviors for speciﬁc population models with the logistic growth law, Gilpin-Ayala growth law and Gompertz growth law, respectively. We answer an open problem proposed by L. Korobenko and E. Braverman in [Can. Appl. Math. Quart. 17(2009) 85-104].

§1

Introduction

The reaction-diﬀusion equation has been widely used to model spatial propagation or spreading of biological species. The original and simplest reaction-diﬀusion equation ut = Δu + f (u), x ∈ RN

(1.1)

has been continually improved in order to model real situation better. From a mass of the biological experiments it is known that the diﬀusion term or the reaction term greatly aﬀects population density and their dynamical behaviors. Therefore, the alternative types of diﬀusion and reaction were considered in many publications, for example, see [1-11, 14-15]. Various diﬀusion types, in particular nonlinear, were considered in population dynamics, see [1,4-8,12,14] and references therein. Some general approach to the equation with a nonlinear diﬀusion term ∇ · (D(u)∇(u)) was developed in [12]. In [4], the authors investigated the existence of positive solutions, the existence and the global attractivity of positive periodic Received: 2011-11-15. MR Subject Classiﬁcation: 35K55, 35K57, 35K45, 35K50. Keywords: general form of growth law, nonlinearity-driven diﬀusion, periodic solution, global attractivity, rate of convergence. Digital Object Identiﬁer(DOI): 10.1007/s11766-013-2966-4. Supported by the National Natural Science Foundation of China (11271342).

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solutions for a Logistic model with a carrying capacity driven diﬀusion as follows: u(t, x) ∂u(t, x) u(t, x) = DΔ + μ(t, x)u(t, x) 1 − . (1.2) ∂t M (t, x) M (t, x) The type of diﬀusion in (1.2) was ﬁrst considered in [1] motivated by the choice of optimal harvesting strategies. The detailed biological signiﬁcance of (1.2) can be found in [4] and its references. Compared with (1.2), Korobenko and Braverman [4] introduced a more general form of population model with nonlinearity-driven diﬀusion as follows u(t, x) ∂u(t, x) = DΔ + μ(t, x)f (u(t, x), M (t, x)), t > 0, x ∈ Ω. (1.3) ∂t M (t, x) Researching the dynamics of (1.3) was presented as an open problem in [4]. In this paper, we aim to investigate (1.3) with the Neumann boundary condition u(t,x) ∂ M(t,x) = 0, t > 0, x ∈ ∂Ω (1.4) ∂n and the initial condition u(0, x) = u0 (x) ≥ 0, x ∈ Ω, (1.5) where the constant D > 0 is a diﬀusive coeﬃcient, Ω is an open nonempty bounded domain of RN (usually, N = 1, 2 or 3) with ∂Ω ∈ C 1+α , 0 < α < 1, n is the exterior normal to the boundary ∂Ω, boundary condition (1.4) means that the quantity u(t, x)/M (t, x) is closed and there is no ﬂux through the isolated boundaries or that immigration to the domain is compensated by emigration. Throughout this paper, we set QT = (0, T ] × Ω, ∂QT = (0, T ] × ∂Ω for any T > 0, ¯ = [0, ∞) × Ω. ¯ Q = (0, ∞) × Ω, ∂Q = (0, ∞) × ∂Ω, Q ¯ ¯ C(QT ) denotes the set of functions which are continuous in (t, x) ∈ QT and C 1,2 (QT ) denotes the set of functions which are once continuously diﬀerentiable in t ∈ (0, T ] and twice continuously diﬀerentiable in x ∈ Ω. Furthermore, M (t, x) is H¨ older continuous in x, continuously ¯ μ(t, x) is continuous in diﬀerentiable periodic in t and strictly positive for any (t, x) ∈ Q; ¯ f (u, M ) is locally Lipschitz (t, x) ∈ Q, periodic in t and strictly positive for any (t, x) ∈ Q; continuous with respect to u uniformly for (t, x) ∈ QT . The present paper is organized as follows. In the next section we give some deﬁnitions and lemmas. Main results and their proofs are presented in Section 3. As applications, we directly conclude some related consequences for the model with Gompertz growth law in Section 4.

§2

Preliminaries

This section is devoted to some deﬁnitions and important lemmas which plays vitally important role in the derivation of the main results in this paper. These lemmas can easily be obtained from the basic theories of partial diﬀerential equations and the related references. Therefore, we omit the proofs.

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We ﬁrst consider the following equation ∂u(t, x) u(t, x) = DΔ( ) + μ(t, x)f (u(t, x), M (t, x)), (t, x) ∈ QT , (2.1) ∂t M (t, x) with boundary condition (1.4) and initial condition (1.5). We set u(t, x) . (2.2) v(t, x) = M (t, x) ¯ T , v(t, x) is well deﬁned. Then (2.1), Because M (t, x) is positive and bounded from above in Q (1.4) and (1.5) can be rewritten as ⎧ ∂v μ(t,x) D ⎪ ∂t − M Δv = M(t,x) f (M (t, x)v(t, x), M (t, x)) − ϕ(t, x)v(t, x), (t, x) ∈ QT , ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ∂v(t,x) = 0, (t, x) ∈ ∂QT , ∂n ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ u0 (x) v(0, x) = v0 (x) = M(0,x) , x ∈ Ω, where ϕ(t, x) =

∂M(t,x) 1 . M(t,x) ∂t

(2.3)

It is easy to see that the operator

D Δ M (t, x) is uniformly elliptic, with H¨ older continuous coeﬃcients in QT . L :=

(2.4)

¯ T ) ∩ C 1,2 (QT ) is called an upper solution of (2.3) Definition 2.1. A function v¯(t, x) ∈ C(Q if it satisﬁes the following inequalities: ⎧ ∂¯v μ(t,x) D v ≥ M(t,x) f (M (t, x)¯ v (t, x), M (t, x)) − ϕ(t, x)¯ v (t, x), (t, x) ∈ QT , ⎪ ⎪ ∂t − M Δ¯ ⎪ ⎪ ⎪ ⎪ ⎨ ∂¯ v (t,x) (2.5) ≥ 0, (t, x) ∈ ∂QT , ∂n ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ v¯(0, x) ≥ v0 (x), x ∈ Ω. ¯ T ) ∩ C 1,2 (QT ) is called a lower solution of (2.3) if it satisﬁes the A function v(t, x) ∈ C(Q ¯ following inequalities: ⎧ ∂v μ(t,x) D ≤ M(t,x) f (M (t, x)v(t, x), M (t, x)) − ϕv(t, x), (t, x) ∈ QT , ⎪ ¯ − M Δv ⎪ ∂t ⎪ ¯ ¯ ¯ ⎪ ⎪ ⎪ ⎨ ∂ v(t,x) (2.6) ¯∂n ≤ 0, (t, x) ∈ ∂QT , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ v(0, x) ≤ v0 (x), x ∈ Ω. ¯ The pair of functions v¯, v is said to be ordered if v¯ ≥ v in QT . We set v, v¯ ≡ {u ∈ ¯ ¯ ¯ ¯ C(QT ) : v ≤ u ≤ v¯). Then, in the sector v, v¯ there exist some bounded functions c ≡ c(t, x) ¯ ¯ ¯ ¯ and c¯ ≡ c¯(t, x) such that the function μ(t, x) f (M (t, x)v(t, x), M (t, x)) − ϕ(t, x)v(t, x) F (t, x, v) = M (t, x) satisﬁes −c(v1 − v2 ) ≤ F (t, x, v1 ) − F (t, x, v2 ) ≤ c¯(v1 − v2 ), v ≤ v2 ≤ v1 ≤ v¯, (t, x) ∈ QT . ¯ ¯

(2.7)

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Then function G(t, x, v) = c · v + F (t, x, v) is monotonically nondecreasing in v for v ≤ v ≤ v¯ ¯ ¯ and (t, x) ∈ QT . Now we are going to construct an upper and a lower sequences for (2.3) which will converge to an unique solution of problem (2.3). Definition 2.2. Consider a sequence {v (k) }∞ k=0 deﬁned by the following iteration process ⎧ (k) ∂v (t,x) D ⎪ − M(t,x) Δv (k) (t, x) + cv (k) (t, x) = G(t, x, v (k−1) ), (t, x) ∈ QT , ⎪ ⎪ ∂t ⎪ ¯ ⎪ ⎪ ⎪ ⎨ ∂v (k) (t,x) (2.8) = 0, (t, x) ∈ ∂QT , ∂n ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ v (k) (0, x) = v (x), x ∈ Ω. 0

Denote the sequence with the initial iteration v (0) = v by {v(k) }∞ k=0 (the lower sequence), ¯ ¯ (0) (k) ∞ v }k=0 (the upper sequence). and the sequence with v = v¯ by {¯ In view of the above assumptions and deﬁnitions, the three lemmas below can be immediately derived from [13]. v (k) } of (2.3) introduced in Definition 2.2 are well Lemma 2.1. The two sequences {v(k) } and {¯ ¯ defined and v(k) , v¯(k) are in C α (QT ), 0 < α ≤ 1, for each k. ¯ Lemma 2.2. If v, v¯ are ordered lower and upper solutions of (2.3) introduced in Definition ¯ 2.1, then the sequences {v(k) } and {¯ v (k) } converge monotonically to a unique solution v of (2.3) ¯ and v ≤ v(k) ≤ v ≤ v¯(k) ≤ v¯. ¯ ¯

(2.9)

To prove the positivity of the solution we will need the strong maximum principle for parabolic equation (2.3). ¯ T ) ∩ C 1,2 (QT ) be such that Lemma 2.3. Let v(t, x) ∈ C(Q ∂v(t, x) − Lv(t, x) ≥ 0, (t, x) ∈ QT ∂t where L is defined by (2.4). If v(t, x) attains a minimum value m0 at some point in QT , then v(t, x) ≡ m0 throughout QT . If v(t, x) attains a minimum at some point (t0 , x0 ) on ∂QT , then ∂v/∂n < 0 at (t0 , x0 ) whenever v(t, x) is not a constant. In order to apply the global attractivity result in [16] to the following problem: ⎧ ∂v μ(t,x) D ⎪ ⎨ ∂t − M Δv = M(t,x) f (M (t, x)v(t, x), M (t, x)) − ϕ(t, x)v(t, x), (t, x) ∈ Q, ⎪ ⎩

(2.10) ∂v(t,x) ∂n

= 0, (t, x) ∈ ∂Q, with the initial condition v(0, x) = v0 (x), x ∈ Ω, we need further supposed that

(2.11)

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(T1 ) both M (t, x) and μ(t, x) are periodic bounded functions with the same period T0 for (t, x) ∈ Q; (C1 ) f (u, M ) is continuous in M for (t, x) ∈ Q; For a given pair of constant upper and lower solutions u ˜ and u ˆ, the function f (u, M ) is continuously diﬀerentiable in u for u ∈ ˆ u, u ˜ . We have the following result: Lemma 2.4. Let v˜ ≥ vˆ be a pair of constant upper and lower solution of (2.10). Then there exists a pair of T0 -periodic solutions v¯ and v of (2.10) with vˆ ≤ v ≤ v¯ ≤ v˜. Moreover, for ¯ ¯ v , v˜ in Ω, the corresponding solution v(x, t) of any initial function v0 (x) satisfying v0 (x) ∈ ˆ (2.10)-(2.11) satisfies ¯ v ≤ lim inf v ≤ lim sup v ≤ v¯, f or any x ∈ Ω. t→∞ ¯ t→∞ Furthermore, if v = v¯ ≡ v ∗ , then v ∗ is the unique T0 -periodic solution in ˆ v , v˜ which satisfies ¯ ∗ ¯ lim |v(t, x) − v (t, x)| = 0, f or any x ∈ Ω. t→∞

Compared with Lemma 2.4, the proof of a more general result can be found in [16, Theorem 3.2].

§3

Main results

In this section we will formulate the conditions for the existence and uniqueness of positive solution and an attracting positive periodic solution as well as the global attractivity of the stationary solution.

3.1

Existence and uniqueness of positive solution

We are going to study the existence and uniqueness of positive solution for problem (2.1) with (1.4) and (1.5). To proceed, we assume that (F1 ) there exists an unique function vˇ(t, x) satisfying F (t, x, vˇ) = 0; moreover, vˇ(t, x) can be described as vˇ(t, x) = H(μ(t, x), ϕ(t, x), M (t, x)), for (t, x) ∈ QT ,

(3.1)

where F (t, x, v) is deﬁned in Section 2. By (2.2), we have u ˇ(t, x) = M (t, x)ˇ v (t, x). (F2 ) For ∀ (t, x) ∈ QT , s −→

f (s,M) s

is bounded and decreasing in s > 0.

Examples of functions f (u, M ) satisfying (F1 ) and (F2 ) are functions f (u, M ) = u(1−u/M ), f (u, M ) = u(t, x) ln(M/u) and f (u, M ) = u(μ(x) − (u/M )θ ), just to name a few. Now we state the main result of this subsection.

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Theorem 3.1. Under the assumptions (F1 ) and (F2 ), if let u0 (x) ∈ C(Ω), u0 (x) ≥ 0 in Ω and there exists Ω1 ⊂ Ω such that u0 (x) > 0 for x ∈ Ω1 , then there exists a unique positive solution u(t, x) to problem (2.1) with (1.4) and (1.5). Proof. By substitution (2.2), seeking a unique positive solution u(t, x) to problem (2.1) with (1.4) and (1.5) is equivalent to seeking a unique positive solution v(t, x) to problem (2.3). Therefore, according to Lemma 2.2, in order to show the existence of the unique solution of (2.3) we only need to construct an ordered pair of upper and lower solutions of (2.3). We take v¯ = max{sup(v(0, x)), x∈Ω

sup

H(μ(t, x), ϕ(t, x), M (t, x))} > 0,

(t,x)∈QT

where H(μ(t, x), ϕ(t, x), M (t, x)) deﬁned in (3.1). Therefore, from (F2 ) it follows that D μ(t, x) ∂¯ v − Δ¯ v = 0, f (M (t, x)¯ v , M (t, x)) − ϕ(t, x)¯ v ≤ 0, (3.2) ∂t M (t, x) M (t, x) ∂¯ v = 0 and v¯ ≥ v0 (x), So the ﬁrst inequality of (2.5) in Deﬁnition 2.1 is satisﬁed. Furthermore, ∂n

therefore v¯ is an upper solution of (2.3) by Deﬁnition 2.1. The function v ≡ 0 is obviously a lower ¯ solution. By Lemma 2.2, we immediately draw a conclusion that there exists a unique solution of problem (2.3) satisfying v(t, x) ∈ v, v¯ . Making the inverse substitution we have a unique ¯ solution to (2.1) with (1.4) and (1.5) which is u(t, x) = M (t, x)v(t, x), 0 ≤ u(t, x) ≤ M (t, x)¯ v.

Let us now turn to the proof of the positivity of the solution for any nonnegative initial function v0 (x). We consider u(t, x) ωt e , ψ(t, x) = v(t, x)eωt = M (t, x) v,M(t,x))| where ω = sup(t,x)∈QT |ϕ(t, x)|+sup(t,x)∈QT μ(t,x)|f (M(t,x)¯ , then by substituting v(t, x) = M(t,x)¯ v −ωt into (2.3), we get ψ(t, x)e ⎧ ∂ψ μ D ωt ⎪ ≥ 0, (t, x) ∈ QT , ⎪ ∂t − M Δψ = [ M f (M v, M ) − ϕv + ωv]e ⎪ ⎪ ⎪ ⎪ ⎨ ∂ψ (3.3) ∂n = 0, (t, x) ∈ ∂QT , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ψ(0, x) = v(0, x) ≥ 0, x ∈ Ω.

Since v(t, x) ≥ v = 0, ψ(t, x) is nonnegative. Next, we will prove that it is impossible that there ¯ ¯ T such that ψ(t, x) = 0. Suppose to the contrary. Assume ψ(t, x) attains a exists a point in Q zero value at some point (t0 , x0 ). There are two cases to discuss. Case i: If (t0 , x0 ) ∈ QT , then Lemma 2.3 leads to ψ(t, x) ≡ 0 in QT . However, according ¯ T ) since v(t, x) is a to the assumption of Theorem 3.1, ψ(0, x) > 0 in Ω1 and ψ(t, x) ∈ C(Q solution of (2.3). So we have ψ(t, x) > 0 in Ω1 for some t > 0, thus ψ(t, x) is not identically equal to zero in QT . This contradiction indicates ψ(t, x) > 0 in QT . Case ii: If (t0 , x0 ) ∈ ∂QT , then by Lemma 2.3 we have ∂ψ(t, x)/∂n|(t0 ,x0 ) < 0 which contradicts (3.3). Thus, we also obtain ψ(t, x) > 0 in ∂QT . Therefore, ψ(t, x) is positive. the proof of Theorem 3.1 is complete.

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Existence of an attracting positive periodic solution

In this section, we consider the existence of an attracting positive periodic solution for (1.3)-(1.5). Through using the same substitution v(t, x) = u(t, x)/M (t, x), problem (1.3)-(1.5) is equivalent to (2.10)-(2.11). We ﬁrst set ¯ ¯ gmin = inf{g(t, x), (t, x) ∈ Q}, gmax = sup{g(t, x), (t, x) ∈ Q} for any bounded continuous periodic function g(t, x). In addition, we suppose that ¯ (F3 ) The function H(μ, ϕ, M ) > 0 for (t, x) ∈ Q. Denote vˆ = Hmin , v˜ = Hmax

(3.4)

and β = sup

sup f1 (M ε, M ),

¯ ε∈ˆ v ,˜ v (t,x)∈Q

(3.5)

where f1 (·, ·) stands for the partial derivation of f with respect to the ﬁrst variable. The main result of this section is stated as follows: Theorem 3.2. Under assumptions (T1 ), (C1 ) and (F1 ) − (F3 ), there exists a unique periodic v , v˜ provided that solution v ∗ (t, x) of (2.10) in the interval ˆ 1 ϕmin − βμmax > 0. (3.6) 2 Moreover, for any initial function v0 (x) satisfying v0 (x) ∈ ˆ v , v˜ in Ω, the corresponding solution of (2.10) − (2.11) satisfies ¯ v(t, x) → v ∗ (t, x) as t → +∞ f or x ∈ Ω. Proof. By Lemma 2.4, ﬁrstly it is necessary to ﬁnd a pair of constant upper and lower solutions of (2.10). From (3.1), (3.4) and (F2 ) , it follows that D f (M vˇ, M ) μ ∂˜ v − Δ˜ v = 0 = v˜ μ −ϕ ≥ f (M v˜, M ) − ϕ˜ v, ∂t M M vˇ M ∂ˆ v D f (M vˇ, M ) μ − Δˆ v = 0 = vˆ μ −ϕ ≤ f (M vˆ, M )) − ϕˆ v. ∂t M M vˇ M The above two inequalities indicate that v˜, vˆ are the constant upper and lower solutions of (2.10)-(2.11) with the initial function v(0, x) = v˜ and vˆ, respectively. Then according to Lemma 2.4 there exists a pair of T0 -periodic solutions v¯ and v satisfying ¯ vˆ ≤ v(t, x) ≤ v¯(t, x) ≤ v˜, (t, x) ∈ Q, ¯ D μ ∂¯ v − Δ¯ v= f (M v¯, M ) − ϕ¯ v , (t, x) ∈ Q, (3.7) ∂t M M ∂v D μ ¯− Δv = f (M v, M ) − ϕv, (t, x) ∈ Q, (3.8) ¯ ¯ ∂t M ¯ M ∂¯ v ∂v = ¯ = 0, (t, x) ∈ ∂Q. ∂n ∂n √ Next, we will show that v¯ = v if (3.6) holds. Denote by φ = (¯ v − v) M , by subtracting (3.8) ¯ ¯

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√ from (3.7) and then multiplying the result by φ M , we have φ μ 1 ∂φ φ φ = D √ Δ( √ ) + √ φ[f (M v¯, M ) − f (M v, M )] − ϕφ2 . (3.9) ¯ ∂t 2 M M M Due to the periodicity of M (t, x), v¯ and v, the function φ is also T0 -periodic. Integrating ¯ equation (3.9) over QT0 , and applying Green formula, (3.6) and C1 , we obtain

T0

T0

∂φ φ 0 = φ dxdt = − D|∇( √ )|2 dxdt M 0 Ω ∂t 0 Ω

T0

1 μφ √ [f (M v¯, M ) − f (M v, M )] − ϕφ2 )dxdt + ¯ 2 M 0 Ω

T0

T0

φ 1 = − D|∇( √ )|2 dxdt + (μf1 (M ε, M ) − ϕ)φ2 dxdt 2 M 0 Ω 0 Ω

T0

T0

φ 1 D|∇( √ )|2 dxdt − [ ϕmin − βμmax ] φ2 dxdt ≤ 0. ≤ − 2 M 0 Ω 0 Ω It is easy to see that v¯ − v = 0, and we obtain a unique periodic solution v ∗ ≡ v¯ = v of (2.10)¯ ¯ v , v˜ (2.11) in the ˆ v , v˜ , Moreover, according to Lemma 2.4, for any initial function v0 (x) ∈ ˆ ¯ The proof is the solution of (2.10)-(2.11) satisﬁes v(t, x) → v ∗ (t, x) as t → +∞ for any x ∈ Ω. complete.

3.3

Global attractivity of the stationary solutions

In the case where M and μ are time-independent functions, equation (1.3) can be rewritten as

u(t, x) ∂u(t, x) = DΔ + μ(x)f (u(t, x), M (x)), (t, x) ∈ Q ∂t M (x) We ﬁrst assume that

(3.10)

(F4 ) f (M, M ) = 0 for x ∈ Ω and f (u, M )(u − M ) < 0 for u = M and (t, x) ∈ Q. Thus M (x) is a stationary solution of (3.10) with (1.4), that is, u(x) = M (x) satiﬁes the following elliptic boundary value problem ⎧ u(x) ⎪ −DΔ( M(x) ) = μ(x)f (u(x), M (x)), x ∈ Ω, ⎪ ⎨ ⎪ ⎪ ⎩

(3.11) u(x) ∂( M ) (x) ∂n

= 0, x ∈ ∂Ω.

By (F2 ), we know that f (0, M ) = 0, noting (F4 ), we can factorize f (u, M ) as f (u, M ) = u(u − M )g(u, M ). Based on the previous assumptions, it is clear that f (u, M ) is continuous in t and x for (t, x) ∈ Q. Hence we can suppose (G) There exists a strictly positive continuous function γ(t, x) such that |g(u, M )| ≤ γ(t, x)/M (t, x) for (t, x) ∈ Q. In what follows, we will show that the solution of (3.10) with (1.5) converges to M (x) and estimate the convergence rate.

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Set η = max{sup u0 (x), sup M (x)}, ¯ x∈Ω

¯ x∈Ω

wτ = min{ inf u(τ, x), inf M (x)}, for some ﬁxed moment τ > 0, ¯ x∈Ω

¯ x∈Ω

where u(τ, x) satisﬁes (3.10) with (1.4) and (1.5), by assumptions and Theorem 3.1, we have ¯ and so it is obvious that wτ > 0. Let uη (t, x) and u(τ, x) > 0 for any τ > 0, x ∈ Ω; uwτ (t, x) be the solutions of initial-boundary value problem (3.10)-(1.4) with initial conditions uη (0, x) = η ≥ u0 (x) and uwτ (τ, x) = wτ ≤ u(τ, x), respectively. Denote by δ=

inf {μ(x)γ(t, x)}

¯ (t,x)∈Q

M C = max (η − M )dx, eδτ (M − wτ ) dx . wτ Ω Ω Then we have the following conclusion:

and

Theorem 3.3. Assume that (F1 ), (F2 ), (F4 ) and (G) hold. Then M (x) is a solution of stationary equation (3.11) of initial-boundary problem (3.10) with (1.4) and (1.5). Furthermore, and for any u0 (x) ≥ 0 and u0 (x) is not identically equal to 0, the solution u(t, x) of (3.10) with (1.4) and (1.5) converges to M (x), with the convergence speed estimated as

|u(t, x) − M (x)|dx ≤ Ce−δt . Ω

Proof. The ﬁrst statement in the theorem is obviously true. We now verify the second part. According to Deﬁnition 2.1, uη (t, x) is an upper solution of (3.10) with (1.4) and (1.5), uwτ is a lower solution of problem (3.10)-(1.4) with the initial condition at the moment τ which is u(τ, x). By Theorem 3.1, we have uwτ (t, x) ≤ u(t, x) ≤ uη (t, x), for any (t, x) ∈ [τ, ∞) × Ω, uwτ (t, x) − M (x) ≤ u(t, x) − M (x) ≤ uη (t, x) − M (x), ∀ (t, x) ∈ [τ, ∞) × Ω. Using the same argument as above we obtain uwτ (t, x) − M (x) ≤ 0, uη (t, x) − M (x) ≥ 0.

(3.12) (3.13)

From (3.12) and (3.13) it follows that |u(t, x) − M (x)| ≤ max{uη (t, x) − M (x), M (x) − uwτ (t, x)}, (t, x) ∈ [τ, ∞) × Ω. Integrating both sides of (3.14), we have

|u(t, x) − M (x)|dx ≤ max (uη (t, x) − M (x)), (M (x) − uwτ (t, x)) . Ω

Ω

Ω

(3.14)

(3.15)

Next, denote ξ := uη (t, x) − M (x), and ξ ≥ 0, and applying (3.10) we obtain that ξ satisﬁes ∂ξ ξ = DΔ( ) + μ(x)f (uη , M (x)). (3.16) ∂t M (x) Integrating both sides of (3.16) over Ω and using the Gauss’theorem and Neumann boundary

ξ condition which implies Ω DΔ( M(x) )dx = 0, by (F4 ), we have f (u, M )(u − M ) = u(u − M )2 g(u, M ) < 0,

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then it is obvious that g(u, M ) < 0 for positive solution u. From this it follows that

d ξdx = μf (uη , M )dx = μuη (uη − M )g(uη , M )dx dt Ω Ω Ω

uη ≤ − ξμγ dx ≤ −δ ξdx, M Ω Ω which leads to

ξdx ≤ e−δt ξ(0, x)dx = e−δt (η − M )dx. Therefore, we obtain

Ω

Ω

Ω

(uη (t, x) − M (x)) ≤ e−δt

Ω

Ω

(η − M )dx.

(3.17)

Now we are to estimate the second integral in (3.15). Set ζ = M (x) − uwτ (t, x) and consider ∂ ζM ∂ (M − uwτ )M M 2 ∂uwτ . (3.18) = =− 2 ∂t uwτ ∂t uwτ uwτ ∂t Since uwτ satisﬁes (3.10), thus multiply (3.10) by − uM2

2

wτ

and then integrate its both sides over

Ω, so we obtain

d M2 uwτ ζM M2 dx = −D Δ μf (u , M ) dx. (3.19) dx − w τ 2 dt Ω uwτ M u2wτ Ω uwτ Ω Once again using the Gauss’theorem and Neumann boundary condition which uwτ /M satisﬁes, we have

uwτ M2 Δ dx −D 2 M Ω uwτ 2 2

uwτ M uwτ M 3 ∇ = D ∇ 2 dx ≤ 0. ·∇ dx = −2D 3 uwτ M M Ω Ω uwτ From this and (3.19) it follows that

ζM M2 d dx ≤ − μf (uwτ , M ) 2 dx dt Ω uwτ uwτ Ω

ζM M2 ζM = − μuwτ (uwτ − M )g(uwτ , M ) 2 dx ≤ − μγ dx ≤ −δ dx, u u u wτ Ω Ω Ω wτ wτ which implies

ζM M −δ(t−τ ) (M − uwτ )dx = ζdx ≤ dx ≤ e (M − wτ ) dx. (3.20) u w τ Ω Ω Ω wτ Ω Finally, combining (3.15),(3.17) and (3.20), we obtain

|u(t, x) − M (x)|dx ≤ Ce−δt , ∀ t ∈ [τ, +∞). Let t → +∞, then

Ω

lim

t→+∞

Ω

|u(t, x) − M (x)|dx = 0.

(3.21)

for any nonegative initial function u0 (x) (there exists at least one point in Ω such that u0 (x) > 0). (3.21) implies the uniqueness of the positive steady state of (3.10) with (1.4) and (1.5). The proof is complete.

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Vol. 28, No. 3

Applications

Applying the obtained results in the previous sections, we can directly conclude the related properties of some biological models such as carrying capacity-driven diﬀusion models with the Gompertz, Logistic, Gilpin-Ayala growth laws and so on. As for the model with Logistic growth law, we refer to Ref. [4] for the relevant conclusions. We want to leave the derivation of the relevant properties for the Gilpin-Ayala Model with a carrying capacity driven diﬀusion to interested readers. Here we only present the main results on the Gompertz equation M (t, x) ∂u(t, x) u(t, x) = DΔ( ) + μ(t, x)u(t, x) ln , (t, x) ∈ Q (4.1) ∂t M (t, x) u(t, x) with the boundary value condition u ) ∂( M = 0, (t, x) ∈ ∂Q (4.2) ∂n and the initial condition u(0, x) = u0 (x), x ∈ Ω. M(t,x) u(t,x)

Evidently, f (u(t, x), M (t, x)) = u(t, x) ln

(4.3)

satisﬁes (F1 ) − (F4 ) and G. As in section 2,

set

u(t, x) 1 ∂M (t, x) , ϕ(t, x) = . M (t, x) M (t, x) ∂t Through a simple computation, we know that the function v(t, x) =

ϕ(t,x)

u ˇ(t, x) = M (t, x)e− μ(t,x) is the function deﬁned by (3.1), and

(4.4)

Mmax exp(ϕmax /μmin ) − 1. Mmin By using Theorems 3.1-3.3 and their process of being proved, we directly reach the following corollaries: ϕ(t,x)

vˇ(t, x) = e− μ(t,x) , β = ln

Corollary 4.1. If let u0 (x) ∈ C(Ω), u0 (x) ≥ 0 in Ω and there exists Ω1 ∈ Ω such that u0 (x) > 0, then there exists a unique positive solution u(t, x) to problem (4.1)-(4.3) and u(t, x) < M (t, x). Corollary 4.2. If M (t, x) and μ(t, x) have the same period T0 . there exists a unique periodic solution u∗ (t, x) of (4.1)-(4.2) in the interval Mmin e

max −ϕ μ min

ϕmin

, Mmax e− μmax

provided that 1 ϕmin − μmax β > 0, f or any (t, x) ∈ Q. 2 Moreover, for any initial function u0 (x) satisfying u0 (x) ∈ Mmin e

max −ϕ μ min

ϕmin

, Mmax e− μmax in Ω,

the corresponding solution of (4.1) − (4.3) satisfies u(t, x) → u∗ (t, x)

¯ as t → +∞ f or x ∈ Ω.

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Next, we assume that M (t, x) and μ(t, x) are time-independent functions. Equation (4.1) reads in this case: M (x) ∂u(t, x) u(t, x) = DΔ( ) + μ(x)u(t, x) ln , (t, x) ∈ Q. (4.6) ∂t M (x) u(t, x) M(x) We have the expansion of ln u(t,x) M (x) M 1 M =( − 1) − 2 ( − 1)2 , u(t, x) u u −1 where lies between 1 and M u . It is obvious that f (u, M ) ≤ M − u = u(u − M )( u ). Then, by ln

(G) and Corollary 4.1, we have that |g(u, M )| = u1 = M u /M ≤ 1/M which leads to γ(t, x) = 1. Then δ = inf x∈Q¯ μ(x) and

M δτ C = max (η − M )dx, e (M − wτ ) dx , wτ Ω Ω where η, wτ , uη and uwτ deﬁned as in Section 3.3. Now we have Corollary 4.3. The function M (x) is a stationary solution of (4.6) with (4.2). Furthermore, for any u0 (x) ≥ 0 and u0 (x) is not identically equal to 0, the solution u(t, x) of (4.6) with (4.2)-(4.3) converges to M (x), with the convergence speed estimated as

|u(t, x) − M (x)|dx ≤ Ce−δt . Ω

Acknowledgments. The authors would like to thank the anonymous referees for their valuable comments, which greatly improved the exposition of the paper.

References [1] E Braverman, L Braverman. Optimal harvesting of diﬀusive models in a nonhomogeneous environment, Nonlinear Anal-Theor, 2009, 71: e2173-e2181. [2] A Chakraborty, M Singh, D Lucy, P Ridland. Predator-prey model with prey-taxis and diﬀusion, Math Comput Modelling, 2007, 46: 482-498. [3] R S Cantrell, C Cosner. Diﬀusive logistic equations with indeﬁnite weights: Population models in disrupted environments II, SIAM J Math Anal, 1991, 22: 1043-1069. [4] L Korobenko, E Braverman. A logistic model with a carrying capacity driven diﬀusion, Can Appl Math Q, 2009, 17: 85-104. [5] K Kawasaki, A Mochizuki, M Matsushita, T Umeda, N Shigesada. Modeling spatio-temporal patterns generated by Baccilus subtilis, J Theoret Biol, 1997, 188: 177-185. [6] L Korobenko, E Braverman. On permanence and stability of a logistic model with harvesting and a carrying capacity dependent diﬀusion, J Nonlinear Syst Appl, 2011, 1-2: 9-15. [7] L Korobenko, Md Kamrujjaman, E Braverman. Persistence and extinction in spatial models with a carrying capacity driven diﬀusion and harvesting, J Math Anal Appl, 2013, 399: 352-368.

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[8] J D Murray. Mathematical Biology II: Spatial Models and Biomedical Applications, 3rd edition, Springer, New York, 2003. [9] M Ma, D M Yang, H S Tang. Traveling fronts of the volume-ﬁlling chemotaxis model with general kinetics, Appl Math Comput, 2010, 216: 3162-3171. [10] C H Ou, W Yuan. Traveling wavefronts in a volume-ﬁlling chemotaxis model, SIAM Appl Dyn Syst, 2009, 8: 390-416. [11] K Painter, T Hillen. Volume-ﬁlling and quorum-sensing in models for chemosensitive movement, Can Appl Math Q, 2002, 10: 501-543. [12] C V Pao. Quasilinear parabolic and elliptic equations with nonlinear boundary conditions, Nonlinear Anal, 2007, 66: 639-662. [13] C V Pao. Nonlinear parabolic and Elliptic Equations, New York, Plenum, 1992. [14] Y Wu, X Q Zhao. The existence and stability of traveling waves with transition layers for some singular cross-diﬀusion systems, Phys D, 2005, 200:325-358. [15] X Q Zhao. Dynamical Systems in Population Biology, Springer-Verlag, New York, 2003. [16] L Zhou, Y Fu. Existence and stability of periodic quasisolutions in nonlinear parabolic systems with discrete delays, J Math Anal Appl, 2000, 250: 139-161.

1

Department of Mathematics, College of Sciences, China Jiliang University, Hangzhou 310018, China. Email: [email protected]

2

School of Mathematics and Physics, University of South China, Hengyang 421001, China.

A reaction-diﬀusion model with nonlinearity driven diﬀusion MA Man-jun1

HU Jia-jia1

ZHANG Jun-jie2

TAO Ji-cheng1

Abstract. In this paper, we deal with the model with a very general growth law and an M driven diﬀusion

∂u(t, x) u(t, x) = DΔ( ) + μ(t, x)f (u(t, x), M (t, x)). ∂t M (t, x)

For the general case of time dependent functions M and μ, the existence and uniqueness for positive solution is obtained. If M and μ are T0 -periodic functions in t, then there is an attractive positive periodic solution. Furthermore, if M and μ are time-independent, then the non-constant stationary solution M (x) is globally stable. Thus, we can easily formulate the conditions deriving the above behaviors for speciﬁc population models with the logistic growth law, Gilpin-Ayala growth law and Gompertz growth law, respectively. We answer an open problem proposed by L. Korobenko and E. Braverman in [Can. Appl. Math. Quart. 17(2009) 85-104].

§1

Introduction

The reaction-diﬀusion equation has been widely used to model spatial propagation or spreading of biological species. The original and simplest reaction-diﬀusion equation ut = Δu + f (u), x ∈ RN

(1.1)

has been continually improved in order to model real situation better. From a mass of the biological experiments it is known that the diﬀusion term or the reaction term greatly aﬀects population density and their dynamical behaviors. Therefore, the alternative types of diﬀusion and reaction were considered in many publications, for example, see [1-11, 14-15]. Various diﬀusion types, in particular nonlinear, were considered in population dynamics, see [1,4-8,12,14] and references therein. Some general approach to the equation with a nonlinear diﬀusion term ∇ · (D(u)∇(u)) was developed in [12]. In [4], the authors investigated the existence of positive solutions, the existence and the global attractivity of positive periodic Received: 2011-11-15. MR Subject Classiﬁcation: 35K55, 35K57, 35K45, 35K50. Keywords: general form of growth law, nonlinearity-driven diﬀusion, periodic solution, global attractivity, rate of convergence. Digital Object Identiﬁer(DOI): 10.1007/s11766-013-2966-4. Supported by the National Natural Science Foundation of China (11271342).

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solutions for a Logistic model with a carrying capacity driven diﬀusion as follows: u(t, x) ∂u(t, x) u(t, x) = DΔ + μ(t, x)u(t, x) 1 − . (1.2) ∂t M (t, x) M (t, x) The type of diﬀusion in (1.2) was ﬁrst considered in [1] motivated by the choice of optimal harvesting strategies. The detailed biological signiﬁcance of (1.2) can be found in [4] and its references. Compared with (1.2), Korobenko and Braverman [4] introduced a more general form of population model with nonlinearity-driven diﬀusion as follows u(t, x) ∂u(t, x) = DΔ + μ(t, x)f (u(t, x), M (t, x)), t > 0, x ∈ Ω. (1.3) ∂t M (t, x) Researching the dynamics of (1.3) was presented as an open problem in [4]. In this paper, we aim to investigate (1.3) with the Neumann boundary condition u(t,x) ∂ M(t,x) = 0, t > 0, x ∈ ∂Ω (1.4) ∂n and the initial condition u(0, x) = u0 (x) ≥ 0, x ∈ Ω, (1.5) where the constant D > 0 is a diﬀusive coeﬃcient, Ω is an open nonempty bounded domain of RN (usually, N = 1, 2 or 3) with ∂Ω ∈ C 1+α , 0 < α < 1, n is the exterior normal to the boundary ∂Ω, boundary condition (1.4) means that the quantity u(t, x)/M (t, x) is closed and there is no ﬂux through the isolated boundaries or that immigration to the domain is compensated by emigration. Throughout this paper, we set QT = (0, T ] × Ω, ∂QT = (0, T ] × ∂Ω for any T > 0, ¯ = [0, ∞) × Ω. ¯ Q = (0, ∞) × Ω, ∂Q = (0, ∞) × ∂Ω, Q ¯ ¯ C(QT ) denotes the set of functions which are continuous in (t, x) ∈ QT and C 1,2 (QT ) denotes the set of functions which are once continuously diﬀerentiable in t ∈ (0, T ] and twice continuously diﬀerentiable in x ∈ Ω. Furthermore, M (t, x) is H¨ older continuous in x, continuously ¯ μ(t, x) is continuous in diﬀerentiable periodic in t and strictly positive for any (t, x) ∈ Q; ¯ f (u, M ) is locally Lipschitz (t, x) ∈ Q, periodic in t and strictly positive for any (t, x) ∈ Q; continuous with respect to u uniformly for (t, x) ∈ QT . The present paper is organized as follows. In the next section we give some deﬁnitions and lemmas. Main results and their proofs are presented in Section 3. As applications, we directly conclude some related consequences for the model with Gompertz growth law in Section 4.

§2

Preliminaries

This section is devoted to some deﬁnitions and important lemmas which plays vitally important role in the derivation of the main results in this paper. These lemmas can easily be obtained from the basic theories of partial diﬀerential equations and the related references. Therefore, we omit the proofs.

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We ﬁrst consider the following equation ∂u(t, x) u(t, x) = DΔ( ) + μ(t, x)f (u(t, x), M (t, x)), (t, x) ∈ QT , (2.1) ∂t M (t, x) with boundary condition (1.4) and initial condition (1.5). We set u(t, x) . (2.2) v(t, x) = M (t, x) ¯ T , v(t, x) is well deﬁned. Then (2.1), Because M (t, x) is positive and bounded from above in Q (1.4) and (1.5) can be rewritten as ⎧ ∂v μ(t,x) D ⎪ ∂t − M Δv = M(t,x) f (M (t, x)v(t, x), M (t, x)) − ϕ(t, x)v(t, x), (t, x) ∈ QT , ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ∂v(t,x) = 0, (t, x) ∈ ∂QT , ∂n ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ u0 (x) v(0, x) = v0 (x) = M(0,x) , x ∈ Ω, where ϕ(t, x) =

∂M(t,x) 1 . M(t,x) ∂t

(2.3)

It is easy to see that the operator

D Δ M (t, x) is uniformly elliptic, with H¨ older continuous coeﬃcients in QT . L :=

(2.4)

¯ T ) ∩ C 1,2 (QT ) is called an upper solution of (2.3) Definition 2.1. A function v¯(t, x) ∈ C(Q if it satisﬁes the following inequalities: ⎧ ∂¯v μ(t,x) D v ≥ M(t,x) f (M (t, x)¯ v (t, x), M (t, x)) − ϕ(t, x)¯ v (t, x), (t, x) ∈ QT , ⎪ ⎪ ∂t − M Δ¯ ⎪ ⎪ ⎪ ⎪ ⎨ ∂¯ v (t,x) (2.5) ≥ 0, (t, x) ∈ ∂QT , ∂n ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ v¯(0, x) ≥ v0 (x), x ∈ Ω. ¯ T ) ∩ C 1,2 (QT ) is called a lower solution of (2.3) if it satisﬁes the A function v(t, x) ∈ C(Q ¯ following inequalities: ⎧ ∂v μ(t,x) D ≤ M(t,x) f (M (t, x)v(t, x), M (t, x)) − ϕv(t, x), (t, x) ∈ QT , ⎪ ¯ − M Δv ⎪ ∂t ⎪ ¯ ¯ ¯ ⎪ ⎪ ⎪ ⎨ ∂ v(t,x) (2.6) ¯∂n ≤ 0, (t, x) ∈ ∂QT , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ v(0, x) ≤ v0 (x), x ∈ Ω. ¯ The pair of functions v¯, v is said to be ordered if v¯ ≥ v in QT . We set v, v¯ ≡ {u ∈ ¯ ¯ ¯ ¯ C(QT ) : v ≤ u ≤ v¯). Then, in the sector v, v¯ there exist some bounded functions c ≡ c(t, x) ¯ ¯ ¯ ¯ and c¯ ≡ c¯(t, x) such that the function μ(t, x) f (M (t, x)v(t, x), M (t, x)) − ϕ(t, x)v(t, x) F (t, x, v) = M (t, x) satisﬁes −c(v1 − v2 ) ≤ F (t, x, v1 ) − F (t, x, v2 ) ≤ c¯(v1 − v2 ), v ≤ v2 ≤ v1 ≤ v¯, (t, x) ∈ QT . ¯ ¯

(2.7)

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Then function G(t, x, v) = c · v + F (t, x, v) is monotonically nondecreasing in v for v ≤ v ≤ v¯ ¯ ¯ and (t, x) ∈ QT . Now we are going to construct an upper and a lower sequences for (2.3) which will converge to an unique solution of problem (2.3). Definition 2.2. Consider a sequence {v (k) }∞ k=0 deﬁned by the following iteration process ⎧ (k) ∂v (t,x) D ⎪ − M(t,x) Δv (k) (t, x) + cv (k) (t, x) = G(t, x, v (k−1) ), (t, x) ∈ QT , ⎪ ⎪ ∂t ⎪ ¯ ⎪ ⎪ ⎪ ⎨ ∂v (k) (t,x) (2.8) = 0, (t, x) ∈ ∂QT , ∂n ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ v (k) (0, x) = v (x), x ∈ Ω. 0

Denote the sequence with the initial iteration v (0) = v by {v(k) }∞ k=0 (the lower sequence), ¯ ¯ (0) (k) ∞ v }k=0 (the upper sequence). and the sequence with v = v¯ by {¯ In view of the above assumptions and deﬁnitions, the three lemmas below can be immediately derived from [13]. v (k) } of (2.3) introduced in Definition 2.2 are well Lemma 2.1. The two sequences {v(k) } and {¯ ¯ defined and v(k) , v¯(k) are in C α (QT ), 0 < α ≤ 1, for each k. ¯ Lemma 2.2. If v, v¯ are ordered lower and upper solutions of (2.3) introduced in Definition ¯ 2.1, then the sequences {v(k) } and {¯ v (k) } converge monotonically to a unique solution v of (2.3) ¯ and v ≤ v(k) ≤ v ≤ v¯(k) ≤ v¯. ¯ ¯

(2.9)

To prove the positivity of the solution we will need the strong maximum principle for parabolic equation (2.3). ¯ T ) ∩ C 1,2 (QT ) be such that Lemma 2.3. Let v(t, x) ∈ C(Q ∂v(t, x) − Lv(t, x) ≥ 0, (t, x) ∈ QT ∂t where L is defined by (2.4). If v(t, x) attains a minimum value m0 at some point in QT , then v(t, x) ≡ m0 throughout QT . If v(t, x) attains a minimum at some point (t0 , x0 ) on ∂QT , then ∂v/∂n < 0 at (t0 , x0 ) whenever v(t, x) is not a constant. In order to apply the global attractivity result in [16] to the following problem: ⎧ ∂v μ(t,x) D ⎪ ⎨ ∂t − M Δv = M(t,x) f (M (t, x)v(t, x), M (t, x)) − ϕ(t, x)v(t, x), (t, x) ∈ Q, ⎪ ⎩

(2.10) ∂v(t,x) ∂n

= 0, (t, x) ∈ ∂Q, with the initial condition v(0, x) = v0 (x), x ∈ Ω, we need further supposed that

(2.11)

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(T1 ) both M (t, x) and μ(t, x) are periodic bounded functions with the same period T0 for (t, x) ∈ Q; (C1 ) f (u, M ) is continuous in M for (t, x) ∈ Q; For a given pair of constant upper and lower solutions u ˜ and u ˆ, the function f (u, M ) is continuously diﬀerentiable in u for u ∈ ˆ u, u ˜ . We have the following result: Lemma 2.4. Let v˜ ≥ vˆ be a pair of constant upper and lower solution of (2.10). Then there exists a pair of T0 -periodic solutions v¯ and v of (2.10) with vˆ ≤ v ≤ v¯ ≤ v˜. Moreover, for ¯ ¯ v , v˜ in Ω, the corresponding solution v(x, t) of any initial function v0 (x) satisfying v0 (x) ∈ ˆ (2.10)-(2.11) satisfies ¯ v ≤ lim inf v ≤ lim sup v ≤ v¯, f or any x ∈ Ω. t→∞ ¯ t→∞ Furthermore, if v = v¯ ≡ v ∗ , then v ∗ is the unique T0 -periodic solution in ˆ v , v˜ which satisfies ¯ ∗ ¯ lim |v(t, x) − v (t, x)| = 0, f or any x ∈ Ω. t→∞

Compared with Lemma 2.4, the proof of a more general result can be found in [16, Theorem 3.2].

§3

Main results

In this section we will formulate the conditions for the existence and uniqueness of positive solution and an attracting positive periodic solution as well as the global attractivity of the stationary solution.

3.1

Existence and uniqueness of positive solution

We are going to study the existence and uniqueness of positive solution for problem (2.1) with (1.4) and (1.5). To proceed, we assume that (F1 ) there exists an unique function vˇ(t, x) satisfying F (t, x, vˇ) = 0; moreover, vˇ(t, x) can be described as vˇ(t, x) = H(μ(t, x), ϕ(t, x), M (t, x)), for (t, x) ∈ QT ,

(3.1)

where F (t, x, v) is deﬁned in Section 2. By (2.2), we have u ˇ(t, x) = M (t, x)ˇ v (t, x). (F2 ) For ∀ (t, x) ∈ QT , s −→

f (s,M) s

is bounded and decreasing in s > 0.

Examples of functions f (u, M ) satisfying (F1 ) and (F2 ) are functions f (u, M ) = u(1−u/M ), f (u, M ) = u(t, x) ln(M/u) and f (u, M ) = u(μ(x) − (u/M )θ ), just to name a few. Now we state the main result of this subsection.

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Theorem 3.1. Under the assumptions (F1 ) and (F2 ), if let u0 (x) ∈ C(Ω), u0 (x) ≥ 0 in Ω and there exists Ω1 ⊂ Ω such that u0 (x) > 0 for x ∈ Ω1 , then there exists a unique positive solution u(t, x) to problem (2.1) with (1.4) and (1.5). Proof. By substitution (2.2), seeking a unique positive solution u(t, x) to problem (2.1) with (1.4) and (1.5) is equivalent to seeking a unique positive solution v(t, x) to problem (2.3). Therefore, according to Lemma 2.2, in order to show the existence of the unique solution of (2.3) we only need to construct an ordered pair of upper and lower solutions of (2.3). We take v¯ = max{sup(v(0, x)), x∈Ω

sup

H(μ(t, x), ϕ(t, x), M (t, x))} > 0,

(t,x)∈QT

where H(μ(t, x), ϕ(t, x), M (t, x)) deﬁned in (3.1). Therefore, from (F2 ) it follows that D μ(t, x) ∂¯ v − Δ¯ v = 0, f (M (t, x)¯ v , M (t, x)) − ϕ(t, x)¯ v ≤ 0, (3.2) ∂t M (t, x) M (t, x) ∂¯ v = 0 and v¯ ≥ v0 (x), So the ﬁrst inequality of (2.5) in Deﬁnition 2.1 is satisﬁed. Furthermore, ∂n

therefore v¯ is an upper solution of (2.3) by Deﬁnition 2.1. The function v ≡ 0 is obviously a lower ¯ solution. By Lemma 2.2, we immediately draw a conclusion that there exists a unique solution of problem (2.3) satisfying v(t, x) ∈ v, v¯ . Making the inverse substitution we have a unique ¯ solution to (2.1) with (1.4) and (1.5) which is u(t, x) = M (t, x)v(t, x), 0 ≤ u(t, x) ≤ M (t, x)¯ v.

Let us now turn to the proof of the positivity of the solution for any nonnegative initial function v0 (x). We consider u(t, x) ωt e , ψ(t, x) = v(t, x)eωt = M (t, x) v,M(t,x))| where ω = sup(t,x)∈QT |ϕ(t, x)|+sup(t,x)∈QT μ(t,x)|f (M(t,x)¯ , then by substituting v(t, x) = M(t,x)¯ v −ωt into (2.3), we get ψ(t, x)e ⎧ ∂ψ μ D ωt ⎪ ≥ 0, (t, x) ∈ QT , ⎪ ∂t − M Δψ = [ M f (M v, M ) − ϕv + ωv]e ⎪ ⎪ ⎪ ⎪ ⎨ ∂ψ (3.3) ∂n = 0, (t, x) ∈ ∂QT , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ψ(0, x) = v(0, x) ≥ 0, x ∈ Ω.

Since v(t, x) ≥ v = 0, ψ(t, x) is nonnegative. Next, we will prove that it is impossible that there ¯ ¯ T such that ψ(t, x) = 0. Suppose to the contrary. Assume ψ(t, x) attains a exists a point in Q zero value at some point (t0 , x0 ). There are two cases to discuss. Case i: If (t0 , x0 ) ∈ QT , then Lemma 2.3 leads to ψ(t, x) ≡ 0 in QT . However, according ¯ T ) since v(t, x) is a to the assumption of Theorem 3.1, ψ(0, x) > 0 in Ω1 and ψ(t, x) ∈ C(Q solution of (2.3). So we have ψ(t, x) > 0 in Ω1 for some t > 0, thus ψ(t, x) is not identically equal to zero in QT . This contradiction indicates ψ(t, x) > 0 in QT . Case ii: If (t0 , x0 ) ∈ ∂QT , then by Lemma 2.3 we have ∂ψ(t, x)/∂n|(t0 ,x0 ) < 0 which contradicts (3.3). Thus, we also obtain ψ(t, x) > 0 in ∂QT . Therefore, ψ(t, x) is positive. the proof of Theorem 3.1 is complete.

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Existence of an attracting positive periodic solution

In this section, we consider the existence of an attracting positive periodic solution for (1.3)-(1.5). Through using the same substitution v(t, x) = u(t, x)/M (t, x), problem (1.3)-(1.5) is equivalent to (2.10)-(2.11). We ﬁrst set ¯ ¯ gmin = inf{g(t, x), (t, x) ∈ Q}, gmax = sup{g(t, x), (t, x) ∈ Q} for any bounded continuous periodic function g(t, x). In addition, we suppose that ¯ (F3 ) The function H(μ, ϕ, M ) > 0 for (t, x) ∈ Q. Denote vˆ = Hmin , v˜ = Hmax

(3.4)

and β = sup

sup f1 (M ε, M ),

¯ ε∈ˆ v ,˜ v (t,x)∈Q

(3.5)

where f1 (·, ·) stands for the partial derivation of f with respect to the ﬁrst variable. The main result of this section is stated as follows: Theorem 3.2. Under assumptions (T1 ), (C1 ) and (F1 ) − (F3 ), there exists a unique periodic v , v˜ provided that solution v ∗ (t, x) of (2.10) in the interval ˆ 1 ϕmin − βμmax > 0. (3.6) 2 Moreover, for any initial function v0 (x) satisfying v0 (x) ∈ ˆ v , v˜ in Ω, the corresponding solution of (2.10) − (2.11) satisfies ¯ v(t, x) → v ∗ (t, x) as t → +∞ f or x ∈ Ω. Proof. By Lemma 2.4, ﬁrstly it is necessary to ﬁnd a pair of constant upper and lower solutions of (2.10). From (3.1), (3.4) and (F2 ) , it follows that D f (M vˇ, M ) μ ∂˜ v − Δ˜ v = 0 = v˜ μ −ϕ ≥ f (M v˜, M ) − ϕ˜ v, ∂t M M vˇ M ∂ˆ v D f (M vˇ, M ) μ − Δˆ v = 0 = vˆ μ −ϕ ≤ f (M vˆ, M )) − ϕˆ v. ∂t M M vˇ M The above two inequalities indicate that v˜, vˆ are the constant upper and lower solutions of (2.10)-(2.11) with the initial function v(0, x) = v˜ and vˆ, respectively. Then according to Lemma 2.4 there exists a pair of T0 -periodic solutions v¯ and v satisfying ¯ vˆ ≤ v(t, x) ≤ v¯(t, x) ≤ v˜, (t, x) ∈ Q, ¯ D μ ∂¯ v − Δ¯ v= f (M v¯, M ) − ϕ¯ v , (t, x) ∈ Q, (3.7) ∂t M M ∂v D μ ¯− Δv = f (M v, M ) − ϕv, (t, x) ∈ Q, (3.8) ¯ ¯ ∂t M ¯ M ∂¯ v ∂v = ¯ = 0, (t, x) ∈ ∂Q. ∂n ∂n √ Next, we will show that v¯ = v if (3.6) holds. Denote by φ = (¯ v − v) M , by subtracting (3.8) ¯ ¯

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√ from (3.7) and then multiplying the result by φ M , we have φ μ 1 ∂φ φ φ = D √ Δ( √ ) + √ φ[f (M v¯, M ) − f (M v, M )] − ϕφ2 . (3.9) ¯ ∂t 2 M M M Due to the periodicity of M (t, x), v¯ and v, the function φ is also T0 -periodic. Integrating ¯ equation (3.9) over QT0 , and applying Green formula, (3.6) and C1 , we obtain

T0

T0

∂φ φ 0 = φ dxdt = − D|∇( √ )|2 dxdt M 0 Ω ∂t 0 Ω

T0

1 μφ √ [f (M v¯, M ) − f (M v, M )] − ϕφ2 )dxdt + ¯ 2 M 0 Ω

T0

T0

φ 1 = − D|∇( √ )|2 dxdt + (μf1 (M ε, M ) − ϕ)φ2 dxdt 2 M 0 Ω 0 Ω

T0

T0

φ 1 D|∇( √ )|2 dxdt − [ ϕmin − βμmax ] φ2 dxdt ≤ 0. ≤ − 2 M 0 Ω 0 Ω It is easy to see that v¯ − v = 0, and we obtain a unique periodic solution v ∗ ≡ v¯ = v of (2.10)¯ ¯ v , v˜ (2.11) in the ˆ v , v˜ , Moreover, according to Lemma 2.4, for any initial function v0 (x) ∈ ˆ ¯ The proof is the solution of (2.10)-(2.11) satisﬁes v(t, x) → v ∗ (t, x) as t → +∞ for any x ∈ Ω. complete.

3.3

Global attractivity of the stationary solutions

In the case where M and μ are time-independent functions, equation (1.3) can be rewritten as

u(t, x) ∂u(t, x) = DΔ + μ(x)f (u(t, x), M (x)), (t, x) ∈ Q ∂t M (x) We ﬁrst assume that

(3.10)

(F4 ) f (M, M ) = 0 for x ∈ Ω and f (u, M )(u − M ) < 0 for u = M and (t, x) ∈ Q. Thus M (x) is a stationary solution of (3.10) with (1.4), that is, u(x) = M (x) satiﬁes the following elliptic boundary value problem ⎧ u(x) ⎪ −DΔ( M(x) ) = μ(x)f (u(x), M (x)), x ∈ Ω, ⎪ ⎨ ⎪ ⎪ ⎩

(3.11) u(x) ∂( M ) (x) ∂n

= 0, x ∈ ∂Ω.

By (F2 ), we know that f (0, M ) = 0, noting (F4 ), we can factorize f (u, M ) as f (u, M ) = u(u − M )g(u, M ). Based on the previous assumptions, it is clear that f (u, M ) is continuous in t and x for (t, x) ∈ Q. Hence we can suppose (G) There exists a strictly positive continuous function γ(t, x) such that |g(u, M )| ≤ γ(t, x)/M (t, x) for (t, x) ∈ Q. In what follows, we will show that the solution of (3.10) with (1.5) converges to M (x) and estimate the convergence rate.

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Set η = max{sup u0 (x), sup M (x)}, ¯ x∈Ω

¯ x∈Ω

wτ = min{ inf u(τ, x), inf M (x)}, for some ﬁxed moment τ > 0, ¯ x∈Ω

¯ x∈Ω

where u(τ, x) satisﬁes (3.10) with (1.4) and (1.5), by assumptions and Theorem 3.1, we have ¯ and so it is obvious that wτ > 0. Let uη (t, x) and u(τ, x) > 0 for any τ > 0, x ∈ Ω; uwτ (t, x) be the solutions of initial-boundary value problem (3.10)-(1.4) with initial conditions uη (0, x) = η ≥ u0 (x) and uwτ (τ, x) = wτ ≤ u(τ, x), respectively. Denote by δ=

inf {μ(x)γ(t, x)}

¯ (t,x)∈Q

M C = max (η − M )dx, eδτ (M − wτ ) dx . wτ Ω Ω Then we have the following conclusion:

and

Theorem 3.3. Assume that (F1 ), (F2 ), (F4 ) and (G) hold. Then M (x) is a solution of stationary equation (3.11) of initial-boundary problem (3.10) with (1.4) and (1.5). Furthermore, and for any u0 (x) ≥ 0 and u0 (x) is not identically equal to 0, the solution u(t, x) of (3.10) with (1.4) and (1.5) converges to M (x), with the convergence speed estimated as

|u(t, x) − M (x)|dx ≤ Ce−δt . Ω

Proof. The ﬁrst statement in the theorem is obviously true. We now verify the second part. According to Deﬁnition 2.1, uη (t, x) is an upper solution of (3.10) with (1.4) and (1.5), uwτ is a lower solution of problem (3.10)-(1.4) with the initial condition at the moment τ which is u(τ, x). By Theorem 3.1, we have uwτ (t, x) ≤ u(t, x) ≤ uη (t, x), for any (t, x) ∈ [τ, ∞) × Ω, uwτ (t, x) − M (x) ≤ u(t, x) − M (x) ≤ uη (t, x) − M (x), ∀ (t, x) ∈ [τ, ∞) × Ω. Using the same argument as above we obtain uwτ (t, x) − M (x) ≤ 0, uη (t, x) − M (x) ≥ 0.

(3.12) (3.13)

From (3.12) and (3.13) it follows that |u(t, x) − M (x)| ≤ max{uη (t, x) − M (x), M (x) − uwτ (t, x)}, (t, x) ∈ [τ, ∞) × Ω. Integrating both sides of (3.14), we have

|u(t, x) − M (x)|dx ≤ max (uη (t, x) − M (x)), (M (x) − uwτ (t, x)) . Ω

Ω

Ω

(3.14)

(3.15)

Next, denote ξ := uη (t, x) − M (x), and ξ ≥ 0, and applying (3.10) we obtain that ξ satisﬁes ∂ξ ξ = DΔ( ) + μ(x)f (uη , M (x)). (3.16) ∂t M (x) Integrating both sides of (3.16) over Ω and using the Gauss’theorem and Neumann boundary

ξ condition which implies Ω DΔ( M(x) )dx = 0, by (F4 ), we have f (u, M )(u − M ) = u(u − M )2 g(u, M ) < 0,

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then it is obvious that g(u, M ) < 0 for positive solution u. From this it follows that

d ξdx = μf (uη , M )dx = μuη (uη − M )g(uη , M )dx dt Ω Ω Ω

uη ≤ − ξμγ dx ≤ −δ ξdx, M Ω Ω which leads to

ξdx ≤ e−δt ξ(0, x)dx = e−δt (η − M )dx. Therefore, we obtain

Ω

Ω

Ω

(uη (t, x) − M (x)) ≤ e−δt

Ω

Ω

(η − M )dx.

(3.17)

Now we are to estimate the second integral in (3.15). Set ζ = M (x) − uwτ (t, x) and consider ∂ ζM ∂ (M − uwτ )M M 2 ∂uwτ . (3.18) = =− 2 ∂t uwτ ∂t uwτ uwτ ∂t Since uwτ satisﬁes (3.10), thus multiply (3.10) by − uM2

2

wτ

and then integrate its both sides over

Ω, so we obtain

d M2 uwτ ζM M2 dx = −D Δ μf (u , M ) dx. (3.19) dx − w τ 2 dt Ω uwτ M u2wτ Ω uwτ Ω Once again using the Gauss’theorem and Neumann boundary condition which uwτ /M satisﬁes, we have

uwτ M2 Δ dx −D 2 M Ω uwτ 2 2

uwτ M uwτ M 3 ∇ = D ∇ 2 dx ≤ 0. ·∇ dx = −2D 3 uwτ M M Ω Ω uwτ From this and (3.19) it follows that

ζM M2 d dx ≤ − μf (uwτ , M ) 2 dx dt Ω uwτ uwτ Ω

ζM M2 ζM = − μuwτ (uwτ − M )g(uwτ , M ) 2 dx ≤ − μγ dx ≤ −δ dx, u u u wτ Ω Ω Ω wτ wτ which implies

ζM M −δ(t−τ ) (M − uwτ )dx = ζdx ≤ dx ≤ e (M − wτ ) dx. (3.20) u w τ Ω Ω Ω wτ Ω Finally, combining (3.15),(3.17) and (3.20), we obtain

|u(t, x) − M (x)|dx ≤ Ce−δt , ∀ t ∈ [τ, +∞). Let t → +∞, then

Ω

lim

t→+∞

Ω

|u(t, x) − M (x)|dx = 0.

(3.21)

for any nonegative initial function u0 (x) (there exists at least one point in Ω such that u0 (x) > 0). (3.21) implies the uniqueness of the positive steady state of (3.10) with (1.4) and (1.5). The proof is complete.

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Applications

Applying the obtained results in the previous sections, we can directly conclude the related properties of some biological models such as carrying capacity-driven diﬀusion models with the Gompertz, Logistic, Gilpin-Ayala growth laws and so on. As for the model with Logistic growth law, we refer to Ref. [4] for the relevant conclusions. We want to leave the derivation of the relevant properties for the Gilpin-Ayala Model with a carrying capacity driven diﬀusion to interested readers. Here we only present the main results on the Gompertz equation M (t, x) ∂u(t, x) u(t, x) = DΔ( ) + μ(t, x)u(t, x) ln , (t, x) ∈ Q (4.1) ∂t M (t, x) u(t, x) with the boundary value condition u ) ∂( M = 0, (t, x) ∈ ∂Q (4.2) ∂n and the initial condition u(0, x) = u0 (x), x ∈ Ω. M(t,x) u(t,x)

Evidently, f (u(t, x), M (t, x)) = u(t, x) ln

(4.3)

satisﬁes (F1 ) − (F4 ) and G. As in section 2,

set

u(t, x) 1 ∂M (t, x) , ϕ(t, x) = . M (t, x) M (t, x) ∂t Through a simple computation, we know that the function v(t, x) =

ϕ(t,x)

u ˇ(t, x) = M (t, x)e− μ(t,x) is the function deﬁned by (3.1), and

(4.4)

Mmax exp(ϕmax /μmin ) − 1. Mmin By using Theorems 3.1-3.3 and their process of being proved, we directly reach the following corollaries: ϕ(t,x)

vˇ(t, x) = e− μ(t,x) , β = ln

Corollary 4.1. If let u0 (x) ∈ C(Ω), u0 (x) ≥ 0 in Ω and there exists Ω1 ∈ Ω such that u0 (x) > 0, then there exists a unique positive solution u(t, x) to problem (4.1)-(4.3) and u(t, x) < M (t, x). Corollary 4.2. If M (t, x) and μ(t, x) have the same period T0 . there exists a unique periodic solution u∗ (t, x) of (4.1)-(4.2) in the interval Mmin e

max −ϕ μ min

ϕmin

, Mmax e− μmax

provided that 1 ϕmin − μmax β > 0, f or any (t, x) ∈ Q. 2 Moreover, for any initial function u0 (x) satisfying u0 (x) ∈ Mmin e

max −ϕ μ min

ϕmin

, Mmax e− μmax in Ω,

the corresponding solution of (4.1) − (4.3) satisfies u(t, x) → u∗ (t, x)

¯ as t → +∞ f or x ∈ Ω.

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Next, we assume that M (t, x) and μ(t, x) are time-independent functions. Equation (4.1) reads in this case: M (x) ∂u(t, x) u(t, x) = DΔ( ) + μ(x)u(t, x) ln , (t, x) ∈ Q. (4.6) ∂t M (x) u(t, x) M(x) We have the expansion of ln u(t,x) M (x) M 1 M =( − 1) − 2 ( − 1)2 , u(t, x) u u −1 where lies between 1 and M u . It is obvious that f (u, M ) ≤ M − u = u(u − M )( u ). Then, by ln

(G) and Corollary 4.1, we have that |g(u, M )| = u1 = M u /M ≤ 1/M which leads to γ(t, x) = 1. Then δ = inf x∈Q¯ μ(x) and

M δτ C = max (η − M )dx, e (M − wτ ) dx , wτ Ω Ω where η, wτ , uη and uwτ deﬁned as in Section 3.3. Now we have Corollary 4.3. The function M (x) is a stationary solution of (4.6) with (4.2). Furthermore, for any u0 (x) ≥ 0 and u0 (x) is not identically equal to 0, the solution u(t, x) of (4.6) with (4.2)-(4.3) converges to M (x), with the convergence speed estimated as

|u(t, x) − M (x)|dx ≤ Ce−δt . Ω

Acknowledgments. The authors would like to thank the anonymous referees for their valuable comments, which greatly improved the exposition of the paper.

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[8] J D Murray. Mathematical Biology II: Spatial Models and Biomedical Applications, 3rd edition, Springer, New York, 2003. [9] M Ma, D M Yang, H S Tang. Traveling fronts of the volume-ﬁlling chemotaxis model with general kinetics, Appl Math Comput, 2010, 216: 3162-3171. [10] C H Ou, W Yuan. Traveling wavefronts in a volume-ﬁlling chemotaxis model, SIAM Appl Dyn Syst, 2009, 8: 390-416. [11] K Painter, T Hillen. Volume-ﬁlling and quorum-sensing in models for chemosensitive movement, Can Appl Math Q, 2002, 10: 501-543. [12] C V Pao. Quasilinear parabolic and elliptic equations with nonlinear boundary conditions, Nonlinear Anal, 2007, 66: 639-662. [13] C V Pao. Nonlinear parabolic and Elliptic Equations, New York, Plenum, 1992. [14] Y Wu, X Q Zhao. The existence and stability of traveling waves with transition layers for some singular cross-diﬀusion systems, Phys D, 2005, 200:325-358. [15] X Q Zhao. Dynamical Systems in Population Biology, Springer-Verlag, New York, 2003. [16] L Zhou, Y Fu. Existence and stability of periodic quasisolutions in nonlinear parabolic systems with discrete delays, J Math Anal Appl, 2000, 250: 139-161.

1

Department of Mathematics, College of Sciences, China Jiliang University, Hangzhou 310018, China. Email: [email protected]

2

School of Mathematics and Physics, University of South China, Hengyang 421001, China.