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Traveling wave fronts and minimal wave speed for a delayed non-autonomous Fisher equation without quasimonotonicity
Linear Algebra Letters and its 49 Applications 466 (2015) 102–116 Applied Mathematics (2015) 91–99
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LinearMathematics Algebra andLetters its Applications Applied www.elsevier.com/locate/laa www.elsevier.com/locate/aml
problem of Jacobi Traveling waveInverse fronts eigenvalue and minimal wave speed for amatrix delayed withFisher mixedequation data without quasimonotonicity✩ non-autonomous 1 Ying WeiLiu Yanling Tian ∗ , Zhengrong School of Mathematics,Department South China University, Guangzhou, 510631, PR Chinaand Astronautics, of Normal Mathematics, Nanjing University of Aeronautics Department of Mathematics, South China of Technology, PR China Nanjing 210016, PRUniversity China
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Article history: Article history: Inathis paper, the inverse eigenvalue problem of reconstructing Traveling wave front for delayed non-autonomous diffusion Fisher equation without 2014 Received 26 March 2015Received 16 January quasi-monotonicity Jacobi matrix its Iteigenvalues, leading principal is aconsidered in thefrom paper. is indicateditsthat, although the Accepted 20 September 2014 Accepted 5 May 2015 submatrix and part of theand eigenvalues its arbitrarily, submatrix equation is non-autonomous, not quasi-monotonous the delay isoflarge online 22 October 2014 Available online 15 MayAvailable 2015 considered. necessaryofand sufficient wave conditions for both the minimal waveis speed and theThe monotonicity the traveling fronts are Submitted by Y. Wei existence uniqueness the solution obtained, which is thethe same as those and of the traditionalofFisher equation.are derived. Keywords: Furthermore, a numerical algorithm and © 2015 Elsevier Ltd. Allsome rightsnumerical reserved. Traveling wave fronts MSC: examples are given. 15A18 Minimal wave speed © 2014 Published by Elsevier Inc. 15A57 Coupled upper–lower solutions Monotonicity Keywords: Jacobi matrix Eigenvalue Inverse problem 1. Introduction Submatrix
Reaction–diffusion equations with delays often arise in biology and other disciplines. (See [1–8].) In the traditional Fisher equation if the effects of both delay and disposition are taken into consideration, together with the consideration that the reaction term changes over time, then the equation for the population density u ≡ u(t, x) is governed by a non-autonomous parabolic equation without quasi-monotonicity as follows, ∂u = D∆u + r(t)u (1 − u − au(t − τ, x)) , ∂t where D > 0, 0 ≤ a < 1, the function r(t) satisfies the following condition (R),
(1.1)
(R) r(t) > 0 is continuous and 0 < inf t∈R r(t) = limt→−∞ r(t) = r− ≤ supt∈R r(t) = r+ < +∞. E-mail address: [email protected].
1 1 Tel.: +86 13914485239. u ≡ 0 and u ≡ 1+a are two equilibria of (1.1). We wonder whether there exists a traveling wave front connecting the two http://dx.doi.org/10.1016/j.laa.2014.09.031 equilibria or not.
0024-3795/© 2014 Published by Elsevier Inc.
Research supported by the National Natural Science Foundation of China (10571064), Natural Science Foundation of Guangdong Province (10151063101000003), and the National Natural Science Foundation of China (11171115). ∗ Corresponding author at: School of Mathematics, South China Normal University, Guangzhou, 510631, PR China. ✩
E-mail address: [email protected] (Y. Tian).
http://dx.doi.org/10.1016/j.aml.2015.05.002 0893-9659/© 2015 Elsevier Ltd. All rights reserved.
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To investigate the non-autonomous system (1.1), we recall some results for autonomous system, i.e., r is a constant. For the traditional diffusion Fisher equation ∂u = D∆u + ru (1 − u) , ∂t
(1.2)
√ it is well known that there is a traveling wave front connecting 0 and 1 if c ≥ 2 Dr and there is no such √ √ wave front if c < 2 Dr, i.e., 2 Dr is the minimal wave speed (see [9–12] and so on). In view of that (1.1) is a equation without monotonicity, we found that Zou et al. have obtained the existence of the traveling wave fronts by using some exponential quasi-monotonicity condition (EQM) in [5,6] and Gomez et al. have obtained the existence of such fronts by using a new iteration scheme different from [5] when they considered the equation without monotonicity. But both of them demand the delay τ is sufficiently small and have not √ proved that 2 Dr is the minimal wave speed. By using the similar ways in [5] and [13], the following equation ∂u = D∆u + ru (1 − u − au(t − τ )) ∂t
(1.3)
1 where 0 < a < 1, has the traveling wave front connecting 0 to a+1 under the same restricts. The question of minimal wave speed cannot be solved yet. Comparing with the above results, there is fewer result on the topic for (1.1) since it is a non-autonomous equation. Thus we consider the following problem: whether (1.1) has the traveling wave fronts connecting 1 0 and 1+a or not, whether there is the minimal wave speed for (1.1) or not. To solve the two problems, we adopt the following strategies. First, we use the similar way in [13] to obtain the wave system of (1.1). Second, we use the Schauder fixed point theory to establish the existence of the traveling wave solution of (1.1). But the monotonicity of such solutions cannot be detected, so we prove that √ the solutions possess the monotonicity by using some analysis technics thirdly. Finally, we prove that 2 Dr− is the minimal wave speed by using the comparison method. Moreover, a condition on function r(t) for the non-existence of the traveling wave solution of (1.1) is obtained at the last theorem. So we believe that our results are interesting and meaningful. Our paper is organized as follows. Some useful lemmas are given in Section 2. The existence and the monotonicity are obtained in Section 3. Especially, the minimal speed is also proved.
2. Preliminaries Some preliminaries are listed as follows for convenience. Lemma 2.1 is from [10]. Lemma 2.1. Consider the equation as follows: ∂w = d∆w + r¯w(1 − M w(t, x)), ∂t w(0, x) = w0 (x), x ∈ R.
t > 0, x ∈ R,
(2.1)
If w0 (x) is continuous with w0 (x) ≥ 0. Then the following statements are valid. √ r. (i) If w0 (x) ≡ 0 for x outside a bounded interval, then limt→+∞,|x|>ct w(t, x; w0 ) = 0 for each c > 2 d¯ √ 1 (ii) If w0 (x) ̸≡ 0 for x ∈ R, then limt→+∞,|x|
(t ̸= tj ),
′ − y ′ (t+ j ) − y (tj ) = βj ,
j = 1, 2, . . . , m
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where {tj } is a finite increasing sequence, f : R → R is bounded and continuous at every t ̸= tj . Assume that dz 2 + az + b = 0 has one negative and one positive roots λ < 0 < µ. Then t +∞ 1 ϱ(t, s)f (s)ds ϱ(t, s)f (s)ds + y(t) = d(µ − λ) −∞ t m 1 − min{eλ(t−tj ) , eµ(t−tj ) }βj , (µ − λ) k=1
where eλ(t−s) , ϱ(t, s) = eµ(t−s) ,
s ≤ t, s ≥ t.
The lemma is easy to be proved by direct calculation, we omit the proof here. 3. Traveling wave solutions of (1.1) We seek the wave solutions of (1.1) with the form ψ(x + ct) with ψ(−∞) = 0, ψ(+∞) = s = x + ct, ψ(s) = ψ(x + ct) = φx (t), then by using the similar way in [13] there is ∂ψ dψ dφ =c = , ∂t ds dt substitute ψ(x + ct) into the system (1.1), then
1 1+a .
Set
∂2ψ 1 d2 φ = , ∂x2 c2 dt2
D ′′ φ (t) − φ′ (t) + r(t)φ(t) (1 − φ(t) − aφ(t − τ )) = 0. c2 Denote φ(t − τ ) by φτ (t), together with ψ(−∞) = 0, ψ(+∞) = of (1.1)
1 1+a ,
we obtain the following wave system
D ′′ φ (t) − φ′ (t) + r(t)φ(t) (1 − φ − aφτ ) = 0, c2
(3.1)
and φ(−∞) = 0,
φ(+∞) =
1 . 1+a
(3.2)
We plan to investigate the solution of (3.1)–(3.2) by using weak coupled upper–lower solutions and Schauder fixed-point theory. Hence, the definition of a pair of weak coupled upper–lower solutions are given as follows. Definition 3.1. Define Λ := {ψ : R → R, ψ ′ and ψ ′′ exist almost everywhere and they are essentially bounded on R}. ¯ φ ∈ Λ. A pair of continuous functions φ, ¯ φ are called weak coupled upper and lower solutions Suppose that φ, of (3.1)–(3.2) if they satisfy D ¯′′ ¯′ ¯ − φ¯ − aφ ) ≤ 0, φ − φ + r(t)φ(1 τ c2 D ′′ ′ φ − φ + r(t)φ(1 − φ − aφ¯τ ) ≥ 0, c2
t ∈ R \ {Ti : i = 1, . . . , m}, φ¯′ (t+ ) ≤ φ¯′ (t− ), ′
+
(3.3) ′
−
t ∈ R \ {Ti : i = 1, . . . , m}, φ (t ) ≥ φ (t ).
Define χ ¯ :=
1 ϕ ∈ Λ : 0 < ϕ < 1, lim ϕ(t) = 0, lim ϕ(t) = t→−∞ t→+∞ 1+a
.
(3.4)
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In what follows, we assume that the weak coupled upper and lower solutions of (3.1) are given so that ¯ and φ(t), φ(t) ¯ ∈ χ. ¯ Next we define φ(t) ≤ φ(t) ¯ χ = {ϕ ∈ C(−∞, +∞) : φ(t) ≤ ϕ(t) ≤ φ(t)},
(3.5)
and H(t, φ1 , φ2 ) = Lφ1 + r(t)φ1 (1 − φ1 − aφ2 ),
L = 2r+ + a.
Obviously, χ is a Banach space when it is equipped with the super-norm. H(t, φ1 , φ2 ) is nondecreasing in φ1 and nonincreasing in φ2 if φi ∈ χ, (i = 1, 2). Eq. (3.1) can be written as D ′′ φ − φ′ − Lφ + H(t, φ, φτ ) = 0, c2
t ∈ R.
D ′′ z − z1′ − Lz1 + H(t, φ, φτ ) = 0, c2 1
t ∈ R.
So we consider the system
Set λ1 < 0 < λ2 are the roots of the equation 1 F (φ) = D (λ 2 − λ1 ) c2
D 2 c2 z
(3.6)
− z − L = 0, define
+∞
k(t, h)H(h, φ, φτ )dh,
−∞
k(t, s) =
eλ1 (t−s) , eλ2 (t−s) ,
s ≤ t, s ≥ t.
(3.7)
It is easy to verify that F (φ) is the solution of (3.6). To seek the solution of (3.1)–(3.2), we plan to prove that the operator F has a fixed point in the set χ. Lemma 3.1. There is a function U ∈ χ such that F (U ) = U , satisfying (3.1)–(3.2). Proof. Firstly, we claim F (χ) ⊂ χ. Consider the system D ′′ ¯ φ ) = 0, t ∈ R, z¯ − z¯′ − L¯ z + H(t, φ, τ c2 D ′′ z − z ′ − Lz + H(t, φ, φ¯τ ) = 0, t ∈ R. c2 From Lemma 2.2 and the definition of weak upper–lower solutions, there are m
z¯ − φ¯ ≤
1 min{eλ1 (t−tj ) , eλ2 (t−tj ) }(φ¯′ (t+ ) − φ¯′ (t− )) ≤ 0, (λ2 − λ1 ) k=1 m
1 z−φ≥ min{eλ1 (t−tj ) , eλ2 (t−tj ) }(φ′ (t+ ) − φ′ (t− )) ≥ 0. (λ2 − λ1 ) k=1
For any given φ ∈ χ, we have 1 F (φ) ≤ D c2 (λ2 − λ1 )
1 D (λ 2 2 − λ1 ) c
F (φ) ≥ thus F (χ) ⊂ χ.
+∞
−∞ +∞
−∞
¯ φ )dh = z¯ ≤ φ, ¯ k(t, h)H(h, φ, τ k(t, h)H(h, φ, φ¯τ )(h)dh = z ≥ φ,
(3.8)
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Secondly, we claim that F : χ → χ is continuous with respect to the super-norm. By direct calculation, for any φ1 , φ2 ∈ χ, there is +∞ 1 k(t, h)|H(h, φ1 , φ1τ ) − H(h, φ2 , φ2τ )|(h)dh |F (φ1 )(t) − F (φ2 )(t)| = D c2 (λ2 − λ1 ) −∞ +∞ (5 + a)r+ (5 + a)r+ ≤ D k(t, h)∥φ1 − φ2 ∥dh ≤ ∥φ1 − φ2 ∥. L c2 (λ2 − λ1 ) −∞ Thus, if ∥φ1 − φ2 ∥ → 0, then ∥F (φ1 ) − F (φ2 )∥ → 0, i.e., F is continuous with respect to the super-norm. Thirdly, we claim F (χ) is uniform-bounded and equiv-continuous, then F : χ → χ is compact. F (χ) is uniform bounded obviously. We claim that F (χ) is equiv-continuous, i.e., for any ε > 0, there is δ > 0 such that for |t′ − t| < δ, |F (φ)(t′ ) − F (φ)(t)| < ε is valid. Suppose t > t′ , then there is t1 ∈ (t′ , t) such that eλ1 (t−h) < eλ2 (t −h) (h < t1 ), ′
eλ1 (t−h) > eλ2 (t −h) (h > t1 ), ′
eλ1 (t−h) = eλ2 (t −h) (h = t1 ). ′
¯ > 0 such that for any t ∈ R, |H(t, φ, φτ )| < H ¯ exists if φ ∈ χ, then In view of that there is H +∞ [k(t, h) − k(t′ , h)] H(h, φ(h), φτ (h))dh |F (φ)(t′ ) − F (φ)(t)| = ¯ ≤H
−∞ +∞ −∞
|k(t, h) − k(t′ , h)| dh ′
¯ H ≤ D c2 (λ2 − λ1 )
t
(e
λ1 (t′ −h)
−e
λ1 (t−h)
t1
)dh +
(e
λ2 (t′ −h)
−e
λ1 (t−h)
)dh
t′
−∞
t +∞ ¯ H λ1 (t−h) λ2 (t′ −h) λ2 (t−h) λ2 (t′ −h) )ds + D (e −e )dh + (e −e t1 t c2 (λ2 − λ1 ) ¯ ′ 1 λ1 (t−t1 ) 1 2H (e − 1) + (1 − eλ2 (t −t1 ) ) . = D λ1 λ2 c2 (λ2 − λ1 ) So for any ε > 0, there is δ(ε) small enough that |F (φ)(t′ )−F (φ)(t)| < ε provided |t′ −t| < δ, then F : χ → χ is compact. Finally, we can find U ∈ χ such that F (U ) = U . By direct calculation, U satisfies (3.1)–(3.2). Thus the lemma is proved. Moreover, we can prove U is monotonous. Lemma 3.2. Suppose U (t) is the solution of (3.1)–(3.2), then it is monotonous. Proof. It is sufficient to prove that U (t) has no maximal value point. If not, there is t¯ such that U ′ (t¯) = 0, U ′′ (t¯) < 0, which implies g(t¯) > 0, where g(t) = rU (t)(1 − U (t) − aU (t − τ )). Hence there is l1 < t¯ < l2 c2 such that such that g(t) > 0 and U (l1 ) = U (l2 ) < U (t¯) for t ∈ [l1 , l2 ]. This means that there is 0 < A < 4D g(t) > AU (t) for t ∈ [l1 , l2 ]. Consequently, we have D ′′ U − U ′ + AU = l(t), c2 where l(t) ≤ 0 for t ∈ [l1 , l2 ]. Set 0 < ν1 < ν2 as the roots of cD2 λ2 − λ + A = 0, note from calculations that l1 1 U (t) = U (l1 ) + D [eν1 (t−h) − eν2 (t−h) ]l(h)dh for t ∈ [l1 , l2 ], (ν − ν ) 2 t 2 1 c then there is 1 U (l2 ) − U (l1 ) = D c2 (ν2 − ν1 )
l1
[eν1 (l2 −h) − eν2 (l2 −h) ]l(h)dh > 0,
l2
which is a contradiction, then the theorem is proved.
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Hence Lemmas 3.1–3.2 can lead to the following theorem. ¯ φ), where φ, ¯ φ ∈ χ. Theorem 3.1. Suppose that (3.1) has a pair of coupled upper and lower solutions (φ, ¯ 1 Then (3.1)–(3.2) have a monotonous solution, which is the traveling wave front of (1.1) connecting 0 and 1+a . ¯ φ) of (3.1) in χ Thus our main task is to find a pair of coupled upper and lower solutions (φ, ¯ in the remainder of this section. √ Suppose that τ > 0 and a ∈ (0, 1), choose γˆ < r− 1−a a), choose s > 0 such that 1+a (1 − −1 + s=
1 + 4 cD2 γˆ 2 cD2
<
− ln a . 4τ
(3.9)
¯ Let Since limt→−∞ r(t) = r− , then for any ϵ > 0, there is ξ¯ < 0 such that r(t) < r− + ϵ for t < ξ. c2 − c c2 − 4D(r− + ϵ) − ¯ c ≥ 2 D(r + ϵ), . (3.10) λ1 = 2D 1 Select ξ < ξ¯ and 0 < α < a such that 1 − αesτ > 0 and 0 < 1+a (1 − αe−sξ ) < 1 − a, set B = 1 − a − D 1 −sξ ), 0 < κ1 < κ2 are the roots of the equation c2 κ2 − κ + r− B = 0, κ1 + γ < κ2 , choose δ > 0 1+a (1 − αe 1 δ such that 1+a (1 − αe−sξ ) = 1+a (1 − αeγξ )eκ1 ξ . We define 1 (1 − αe−st ), 1+a φ= δ (1 − αeγt )eκ1 t , 1+a
1 (1 + αe−st ), t > ξ 1 + a ¯ φ= , 1 + α eλ¯ 1 t , t ≤ ξ. 1+a Remark 3.1. If (3.9) is valid, then 1 − aeτ s > 1 −
√
t>ξ (3.11) t ≤ ξ.
a.
¯ ¯ ¯ It is easy to verify that φ(t), φ(t) ∈ χ, ¯ φ(t) ≤ φ(t). Next we verify that (φ(t), φ(t)) are a pair of weak coupled upper and lower solutions of (3.1). ¯ − φ¯ − aφ ) ≤ 0. Step 1: Claim that cD2 φ¯′′ − φ¯′ + r(t)φ(1 τ For t > ξ, (3.11) and direct calculation show φ¯ =
1 (1 + αe−st ), 1+a
φ¯′ =
α (−s)e−st , 1+a
φ¯′′ =
α 2 −st s e , 1+a
φ=
1 (1 − αe−st ), 1+a
then Remark 3.1 implies D ¯′′ ¯′ α aαesτ ¯ − φ¯ − aφ ) = α e−st D s2 + s − r(t)φ¯ φ − φ + r(t) φ(1 − e−st τ c2 1+a c2 1+a 1+a α −st D 2 1 − aesτ α −st e e ≤ s + s − r− 2 1+a c 1+a 1+a √ α −st D 2 α −st D 2 a −1 − = e s + s − r ≤ e s + s − γ ˆ = 0. 1+a c2 1+a 1+a c2 For t ≤ ξ, 1 + α λ¯ 1 t φ¯ = e , 1+a
1 + α ¯ λ¯ 1 t φ¯′1 = λ1 e , 1+a
1 + α ¯ 2 λ¯ 1 t φ¯′′ = λ e , 1+a 1
then from (3.10) D ¯′′ ¯′ ¯ − φ¯ − aφ ) ≤ D φ¯′′ − φ¯′ + (r− + ϵ)φ¯ = 1 + α eλ¯ 1 t D λ ¯2 − λ ¯ 1 + (r− + ϵ) = 0. φ − φ + r(t) φ(1 τ c2 c2 1+a c2 1
Y. Tian, Z. Liu / Applied Mathematics Letters 49 (2015) 91–99
Step 2: Claim For t > ξ + τ ,
D ′′ c2 φ
97
− φ′ + r(t)φ(1 − φ − aφ¯τ ) ≥ 0. 1 α (1 − αe−st ), φ′ = se−st , 1+a 1+a −α 2 −st 1 φ′′ = s e , φ¯τ = (1 + αe−s(t−τ ) ), 1+a 1+a φ=
then from Remark 3.1, D ′′ ′ ¯τ ) = α e−st − D s2 − s + r(t)φ α (1 − aesτ ) e−st φ − φ − a φ − φ + r(t)φ(1 c2 1+a c2 1+a √ α −st D 1−α ≥ e − 2 s2 − s + r− (1 − a) 1+a c 1+a α −st D ≥ e − 2 s2 − s + γˆ = 0. 1+a c For ξ < t ≤ ξ + τ , φ=
1 (1 − αe−st ), 1+a
1 + α λ¯ 1 (t−τ ) , φ¯τ = e 1+a
then D ′′ α −st aα λ¯ 1 (t−τ ) α −st D 2 ′ ¯ φ − φ + r(t)φ(1 − φ − aφτ ) = e − 2 s − s + r(t)φ e − e c2 1+a c 1+a 1+a α ¯ −st D α aαeτ s ≥ e − 2 s2 − s + r− φ − e−st 1+a c 1+a 1+a √ α −st D α 1−α ≥ e − 2 s2 − s + r− (1 − a)e−st 1+a c 1+a 1+a D α −st e − 2 s2 − s + γˆ = 0. ≥ 1+a c For t ≤ ξ, φ =
δ 1+a (1
− αeγt )eκ1 t , then
D ′′ φ − φ′ + r(t)φ(1 − φ − aφ¯τ ) c2 δ D 2 D δ κ1 t (κ1 +γ)t 2 ≥ e κ − κ1 − e (κ1 + γ) − (κ1 + γ) + r− Bφ 1+a c2 1 1+a c2 δ D 2 δ D κ1 t (κ1 +γ)t − 2 − ≥ e κ − κ1 + r B − e (κ1 + γ) − (κ1 + γ) + r B ≥ 0. 1+a c2 1 1+a c2 ¯ φ) which is given by (3.11) are a pair of weak After the above two steps, we have the conclusion that (φ, − coupled √ wave front of (1.1) exists if upper–lower solutions of (3.1) if c > 2 D(r + ϵ). Then the traveling c > 2 D(r− + ϵ). In view of ϵ√is arbitrary, then such solution exists for c > 2 Dr− . Moreover, the following theorem indicates that c∗ = 2 Dr− is the minimal wave speed of (1.1). √ − Theorem 3.2. Assume that a ∈ [0, 1], then for c ≥ 2 Dr √ , condition (R) holds, then (1.1) has a traveling 1 wave front with speed c connected 0 with 1+a . For c < 2 Dr− , there is no such traveling wave fronts. That √ means c∗ = 2 Dr− is the minimal wave speed. √ Proof. For c > 2 Dr−√ , a ∈ [0, 1), the traveling wave front exists from the above detailed discussion and Theorem 3.1. For c = 2 Dr− or a = 1, we can use the similar way in [7] to obtain the existence because the wave solution is monotonous. Thus the first conclusion is valid. We use reduction to absurdity to prove the
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second conclusion of this theorem. Suppose there is c1 < c∗ such that U (x + c1 t) is the traveling wavefronts 1 connecting 0 with 1+a . Let u0 (0, x) = U (x), then u0 (0, x) > 0 and u(t, x; u0 ) = U (x + c1 t). Since U (s) is increasing for s = x + ct, then u(t − τ, x) = U (x + c1 (t − τ )) < U (x + c1 t) = u(t, x), then ∂u = D∆u + r(t)u(1 − u − au(t − τ )) ≥ D∆u + r− u(1 − u − au). ∂t Choose v(t, x) as the solution of ∂u = D∆u + r− u(1 − (a + 1)u) ∂t with v(t, x) ̸≡ 0, 0 ≤ v(t, x) ≤ U (x + c1 t) = u(t,√x) for t ∈ [0, τ ], then U (x + c1 t) ≥ v(t, x) by using comparison principle. Thus for 0 < c1 < c2 < c∗ = 2 Dr− , from Lemma 2.1, there is 0 = U (−∞) =
lim inf
t→+∞, x=−c2 t
u(t, x) ≥
lim inf
t→+∞, x=−c2 t
v(t, x) =
1 , 1+a
which is a contradiction. The proof is complete. Finally, we give a condition on r(t) for the non-existence of the traveling wave front for (1.1). Theorem 3.3. Assume that a ∈ [0, 1) and limt→+∞ r(t) = +∞ hold, then (1.1) has no traveling wave front 1 connected 0 with 1+a . Proof. We use reduction to absurdity to prove the statement. Suppose that there is a traveling wave front 1 U (x + c˜t) connecting 0 to 1+a with speed c˜. Since limt→+∞ r(t) = +∞, then there is T1 > 0 such that r(t) > M with 2 DM (1 − a) > c˜ for t > T1 . In view of ∂u = D∆u + r(t)u(1 − u − au(t − τ )) ≥ D∆u + M u(1 − a − u), ∂t then choose v(t, x) as the solution of
t > T1 , x ∈ R,
∂u = D∆u + M u(1 − a − u), t > T1 , x ∈ R ∂t with v(T1 , x) ̸≡ 0, 0 ≤ v(T1 , x) ≤ U (x + c˜T1 ) for x ∈ R, then U (x + c˜t) ≥ v(t, x) by using comparison principle. Thus for c˜ < c1 < 2 DM (1 − a), from Lemma 2.1, there is 0 = U (−∞) =
lim inf
t→+∞, x=−c1 t
u(t, x) ≥
lim inf
t→+∞, x=−c1 t
v(t, x) = 1 − a,
which is a contradiction. Hence the conclusion is valid. References [1] J.D. Murray, Mathematical Biology, I & II, Springer-Verlag, New York, 2002. [2] K.W. Schaaf, Asymptotic behavior and traveling wave solutions for parabolic functional differential equations, Trans. Amer. Math. Soc. 302 (1987) 587–615. [3] H.L. Smith, X.-Q. Zhao, Global asymptotic stability of traveling waves in delayed reaction–diffusion equations, SIAM J. Math. Anal. 31 (2000) 514–534. [4] Y.L. Tian, P.X. Weng, Asymptotic Patterens of a reaction–diffusion equation with nonlinear-nonlocal functional response, IMA J. Appl. Math. 78 (2013) 70–101. [5] J.H. Wu, X.F. Zou, Traveling wave fronts of reaction–diffusion systems with delay, J. Dynam. Differential Equations 13 (2001) 651–687. [6] J.H. Wu, X.F. Zou, Erratum to “Traveling wave fronts of reaction–diffusion system with delays” [J. Dynam. Differential Equations 13 (2001) 651, 687], http://dx.doi.org/10.1023/A:1016690424892. [7] X.-Q. Zhao, D.-M. Xiao, The asymptotic speed of spread and traveling waves for a vector disease model, J. Dynam. Differential Equations 18 (2006) 1001–1019. [8] X.-Q. Zhao, W.D. Wang, Fisher waves in an epidemic model, Discrete Contin. Dyn. Syst. Ser. B 4 (2004) 1117–1128. [9] M.J. Ablowitz, A. Zeppetella, Explicit solutions of Fisher’s equation for a special wave speed, Bull. Math. Biol. 41 (1979) 835–840.
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99
[10] D.G. Aronson, H.F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, in: J.A. Goldstein (Ed.), Partial Differential Equations and Related Topics, in: Lecture Notes in Mathematics, vol. 446, Springer, Berlin, 1975, pp. 5–49. [11] R.A. Fisher, The wave of advance of advantageous genes, Ann. Eugenics 7 (1937) 335–369. [12] A.N. Kolmogorov, I.G. Petrowsky, N.S. Piscounov, Etude de lequation de la diffusion aveccroissance de la quantite de matiere et son application a un problem biologique, Bull. Univ. dEtat a Moscou, Ser. Intern., A 1 (1937) 1–26. [13] A. Gomez, S. Trofimchuk, Monotone traveling wavefronts of the KPP-Fisher delayed equation, J. Differential Equations 250 (2011) 1767–1787.
Contents lists at ScienceDirect Contents lists available at available ScienceDirect
LinearMathematics Algebra andLetters its Applications Applied www.elsevier.com/locate/laa www.elsevier.com/locate/aml
problem of Jacobi Traveling waveInverse fronts eigenvalue and minimal wave speed for amatrix delayed withFisher mixedequation data without quasimonotonicity✩ non-autonomous 1 Ying WeiLiu Yanling Tian ∗ , Zhengrong School of Mathematics,Department South China University, Guangzhou, 510631, PR Chinaand Astronautics, of Normal Mathematics, Nanjing University of Aeronautics Department of Mathematics, South China of Technology, PR China Nanjing 210016, PRUniversity China
article
i n faor t i c l e
ianbfs ot r a c t
a b s t r a c t
Article history: Article history: Inathis paper, the inverse eigenvalue problem of reconstructing Traveling wave front for delayed non-autonomous diffusion Fisher equation without 2014 Received 26 March 2015Received 16 January quasi-monotonicity Jacobi matrix its Iteigenvalues, leading principal is aconsidered in thefrom paper. is indicateditsthat, although the Accepted 20 September 2014 Accepted 5 May 2015 submatrix and part of theand eigenvalues its arbitrarily, submatrix equation is non-autonomous, not quasi-monotonous the delay isoflarge online 22 October 2014 Available online 15 MayAvailable 2015 considered. necessaryofand sufficient wave conditions for both the minimal waveis speed and theThe monotonicity the traveling fronts are Submitted by Y. Wei existence uniqueness the solution obtained, which is thethe same as those and of the traditionalofFisher equation.are derived. Keywords: Furthermore, a numerical algorithm and © 2015 Elsevier Ltd. Allsome rightsnumerical reserved. Traveling wave fronts MSC: examples are given. 15A18 Minimal wave speed © 2014 Published by Elsevier Inc. 15A57 Coupled upper–lower solutions Monotonicity Keywords: Jacobi matrix Eigenvalue Inverse problem 1. Introduction Submatrix
Reaction–diffusion equations with delays often arise in biology and other disciplines. (See [1–8].) In the traditional Fisher equation if the effects of both delay and disposition are taken into consideration, together with the consideration that the reaction term changes over time, then the equation for the population density u ≡ u(t, x) is governed by a non-autonomous parabolic equation without quasi-monotonicity as follows, ∂u = D∆u + r(t)u (1 − u − au(t − τ, x)) , ∂t where D > 0, 0 ≤ a < 1, the function r(t) satisfies the following condition (R),
(1.1)
(R) r(t) > 0 is continuous and 0 < inf t∈R r(t) = limt→−∞ r(t) = r− ≤ supt∈R r(t) = r+ < +∞. E-mail address: [email protected].
1 1 Tel.: +86 13914485239. u ≡ 0 and u ≡ 1+a are two equilibria of (1.1). We wonder whether there exists a traveling wave front connecting the two http://dx.doi.org/10.1016/j.laa.2014.09.031 equilibria or not.
0024-3795/© 2014 Published by Elsevier Inc.
Research supported by the National Natural Science Foundation of China (10571064), Natural Science Foundation of Guangdong Province (10151063101000003), and the National Natural Science Foundation of China (11171115). ∗ Corresponding author at: School of Mathematics, South China Normal University, Guangzhou, 510631, PR China. ✩
E-mail address: [email protected] (Y. Tian).
http://dx.doi.org/10.1016/j.aml.2015.05.002 0893-9659/© 2015 Elsevier Ltd. All rights reserved.
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Y. Tian, Z. Liu / Applied Mathematics Letters 49 (2015) 91–99
To investigate the non-autonomous system (1.1), we recall some results for autonomous system, i.e., r is a constant. For the traditional diffusion Fisher equation ∂u = D∆u + ru (1 − u) , ∂t
(1.2)
√ it is well known that there is a traveling wave front connecting 0 and 1 if c ≥ 2 Dr and there is no such √ √ wave front if c < 2 Dr, i.e., 2 Dr is the minimal wave speed (see [9–12] and so on). In view of that (1.1) is a equation without monotonicity, we found that Zou et al. have obtained the existence of the traveling wave fronts by using some exponential quasi-monotonicity condition (EQM) in [5,6] and Gomez et al. have obtained the existence of such fronts by using a new iteration scheme different from [5] when they considered the equation without monotonicity. But both of them demand the delay τ is sufficiently small and have not √ proved that 2 Dr is the minimal wave speed. By using the similar ways in [5] and [13], the following equation ∂u = D∆u + ru (1 − u − au(t − τ )) ∂t
(1.3)
1 where 0 < a < 1, has the traveling wave front connecting 0 to a+1 under the same restricts. The question of minimal wave speed cannot be solved yet. Comparing with the above results, there is fewer result on the topic for (1.1) since it is a non-autonomous equation. Thus we consider the following problem: whether (1.1) has the traveling wave fronts connecting 1 0 and 1+a or not, whether there is the minimal wave speed for (1.1) or not. To solve the two problems, we adopt the following strategies. First, we use the similar way in [13] to obtain the wave system of (1.1). Second, we use the Schauder fixed point theory to establish the existence of the traveling wave solution of (1.1). But the monotonicity of such solutions cannot be detected, so we prove that √ the solutions possess the monotonicity by using some analysis technics thirdly. Finally, we prove that 2 Dr− is the minimal wave speed by using the comparison method. Moreover, a condition on function r(t) for the non-existence of the traveling wave solution of (1.1) is obtained at the last theorem. So we believe that our results are interesting and meaningful. Our paper is organized as follows. Some useful lemmas are given in Section 2. The existence and the monotonicity are obtained in Section 3. Especially, the minimal speed is also proved.
2. Preliminaries Some preliminaries are listed as follows for convenience. Lemma 2.1 is from [10]. Lemma 2.1. Consider the equation as follows: ∂w = d∆w + r¯w(1 − M w(t, x)), ∂t w(0, x) = w0 (x), x ∈ R.
t > 0, x ∈ R,
(2.1)
If w0 (x) is continuous with w0 (x) ≥ 0. Then the following statements are valid. √ r. (i) If w0 (x) ≡ 0 for x outside a bounded interval, then limt→+∞,|x|>ct w(t, x; w0 ) = 0 for each c > 2 d¯ √ 1 (ii) If w0 (x) ̸≡ 0 for x ∈ R, then limt→+∞,|x|
(t ̸= tj ),
′ − y ′ (t+ j ) − y (tj ) = βj ,
j = 1, 2, . . . , m
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where {tj } is a finite increasing sequence, f : R → R is bounded and continuous at every t ̸= tj . Assume that dz 2 + az + b = 0 has one negative and one positive roots λ < 0 < µ. Then t +∞ 1 ϱ(t, s)f (s)ds ϱ(t, s)f (s)ds + y(t) = d(µ − λ) −∞ t m 1 − min{eλ(t−tj ) , eµ(t−tj ) }βj , (µ − λ) k=1
where eλ(t−s) , ϱ(t, s) = eµ(t−s) ,
s ≤ t, s ≥ t.
The lemma is easy to be proved by direct calculation, we omit the proof here. 3. Traveling wave solutions of (1.1) We seek the wave solutions of (1.1) with the form ψ(x + ct) with ψ(−∞) = 0, ψ(+∞) = s = x + ct, ψ(s) = ψ(x + ct) = φx (t), then by using the similar way in [13] there is ∂ψ dψ dφ =c = , ∂t ds dt substitute ψ(x + ct) into the system (1.1), then
1 1+a .
Set
∂2ψ 1 d2 φ = , ∂x2 c2 dt2
D ′′ φ (t) − φ′ (t) + r(t)φ(t) (1 − φ(t) − aφ(t − τ )) = 0. c2 Denote φ(t − τ ) by φτ (t), together with ψ(−∞) = 0, ψ(+∞) = of (1.1)
1 1+a ,
we obtain the following wave system
D ′′ φ (t) − φ′ (t) + r(t)φ(t) (1 − φ − aφτ ) = 0, c2
(3.1)
and φ(−∞) = 0,
φ(+∞) =
1 . 1+a
(3.2)
We plan to investigate the solution of (3.1)–(3.2) by using weak coupled upper–lower solutions and Schauder fixed-point theory. Hence, the definition of a pair of weak coupled upper–lower solutions are given as follows. Definition 3.1. Define Λ := {ψ : R → R, ψ ′ and ψ ′′ exist almost everywhere and they are essentially bounded on R}. ¯ φ ∈ Λ. A pair of continuous functions φ, ¯ φ are called weak coupled upper and lower solutions Suppose that φ, of (3.1)–(3.2) if they satisfy D ¯′′ ¯′ ¯ − φ¯ − aφ ) ≤ 0, φ − φ + r(t)φ(1 τ c2 D ′′ ′ φ − φ + r(t)φ(1 − φ − aφ¯τ ) ≥ 0, c2
t ∈ R \ {Ti : i = 1, . . . , m}, φ¯′ (t+ ) ≤ φ¯′ (t− ), ′
+
(3.3) ′
−
t ∈ R \ {Ti : i = 1, . . . , m}, φ (t ) ≥ φ (t ).
Define χ ¯ :=
1 ϕ ∈ Λ : 0 < ϕ < 1, lim ϕ(t) = 0, lim ϕ(t) = t→−∞ t→+∞ 1+a
.
(3.4)
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In what follows, we assume that the weak coupled upper and lower solutions of (3.1) are given so that ¯ and φ(t), φ(t) ¯ ∈ χ. ¯ Next we define φ(t) ≤ φ(t) ¯ χ = {ϕ ∈ C(−∞, +∞) : φ(t) ≤ ϕ(t) ≤ φ(t)},
(3.5)
and H(t, φ1 , φ2 ) = Lφ1 + r(t)φ1 (1 − φ1 − aφ2 ),
L = 2r+ + a.
Obviously, χ is a Banach space when it is equipped with the super-norm. H(t, φ1 , φ2 ) is nondecreasing in φ1 and nonincreasing in φ2 if φi ∈ χ, (i = 1, 2). Eq. (3.1) can be written as D ′′ φ − φ′ − Lφ + H(t, φ, φτ ) = 0, c2
t ∈ R.
D ′′ z − z1′ − Lz1 + H(t, φ, φτ ) = 0, c2 1
t ∈ R.
So we consider the system
Set λ1 < 0 < λ2 are the roots of the equation 1 F (φ) = D (λ 2 − λ1 ) c2
D 2 c2 z
(3.6)
− z − L = 0, define
+∞
k(t, h)H(h, φ, φτ )dh,
−∞
k(t, s) =
eλ1 (t−s) , eλ2 (t−s) ,
s ≤ t, s ≥ t.
(3.7)
It is easy to verify that F (φ) is the solution of (3.6). To seek the solution of (3.1)–(3.2), we plan to prove that the operator F has a fixed point in the set χ. Lemma 3.1. There is a function U ∈ χ such that F (U ) = U , satisfying (3.1)–(3.2). Proof. Firstly, we claim F (χ) ⊂ χ. Consider the system D ′′ ¯ φ ) = 0, t ∈ R, z¯ − z¯′ − L¯ z + H(t, φ, τ c2 D ′′ z − z ′ − Lz + H(t, φ, φ¯τ ) = 0, t ∈ R. c2 From Lemma 2.2 and the definition of weak upper–lower solutions, there are m
z¯ − φ¯ ≤
1 min{eλ1 (t−tj ) , eλ2 (t−tj ) }(φ¯′ (t+ ) − φ¯′ (t− )) ≤ 0, (λ2 − λ1 ) k=1 m
1 z−φ≥ min{eλ1 (t−tj ) , eλ2 (t−tj ) }(φ′ (t+ ) − φ′ (t− )) ≥ 0. (λ2 − λ1 ) k=1
For any given φ ∈ χ, we have 1 F (φ) ≤ D c2 (λ2 − λ1 )
1 D (λ 2 2 − λ1 ) c
F (φ) ≥ thus F (χ) ⊂ χ.
+∞
−∞ +∞
−∞
¯ φ )dh = z¯ ≤ φ, ¯ k(t, h)H(h, φ, τ k(t, h)H(h, φ, φ¯τ )(h)dh = z ≥ φ,
(3.8)
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Secondly, we claim that F : χ → χ is continuous with respect to the super-norm. By direct calculation, for any φ1 , φ2 ∈ χ, there is +∞ 1 k(t, h)|H(h, φ1 , φ1τ ) − H(h, φ2 , φ2τ )|(h)dh |F (φ1 )(t) − F (φ2 )(t)| = D c2 (λ2 − λ1 ) −∞ +∞ (5 + a)r+ (5 + a)r+ ≤ D k(t, h)∥φ1 − φ2 ∥dh ≤ ∥φ1 − φ2 ∥. L c2 (λ2 − λ1 ) −∞ Thus, if ∥φ1 − φ2 ∥ → 0, then ∥F (φ1 ) − F (φ2 )∥ → 0, i.e., F is continuous with respect to the super-norm. Thirdly, we claim F (χ) is uniform-bounded and equiv-continuous, then F : χ → χ is compact. F (χ) is uniform bounded obviously. We claim that F (χ) is equiv-continuous, i.e., for any ε > 0, there is δ > 0 such that for |t′ − t| < δ, |F (φ)(t′ ) − F (φ)(t)| < ε is valid. Suppose t > t′ , then there is t1 ∈ (t′ , t) such that eλ1 (t−h) < eλ2 (t −h) (h < t1 ), ′
eλ1 (t−h) > eλ2 (t −h) (h > t1 ), ′
eλ1 (t−h) = eλ2 (t −h) (h = t1 ). ′
¯ > 0 such that for any t ∈ R, |H(t, φ, φτ )| < H ¯ exists if φ ∈ χ, then In view of that there is H +∞ [k(t, h) − k(t′ , h)] H(h, φ(h), φτ (h))dh |F (φ)(t′ ) − F (φ)(t)| = ¯ ≤H
−∞ +∞ −∞
|k(t, h) − k(t′ , h)| dh ′
¯ H ≤ D c2 (λ2 − λ1 )
t
(e
λ1 (t′ −h)
−e
λ1 (t−h)
t1
)dh +
(e
λ2 (t′ −h)
−e
λ1 (t−h)
)dh
t′
−∞
t +∞ ¯ H λ1 (t−h) λ2 (t′ −h) λ2 (t−h) λ2 (t′ −h) )ds + D (e −e )dh + (e −e t1 t c2 (λ2 − λ1 ) ¯ ′ 1 λ1 (t−t1 ) 1 2H (e − 1) + (1 − eλ2 (t −t1 ) ) . = D λ1 λ2 c2 (λ2 − λ1 ) So for any ε > 0, there is δ(ε) small enough that |F (φ)(t′ )−F (φ)(t)| < ε provided |t′ −t| < δ, then F : χ → χ is compact. Finally, we can find U ∈ χ such that F (U ) = U . By direct calculation, U satisfies (3.1)–(3.2). Thus the lemma is proved. Moreover, we can prove U is monotonous. Lemma 3.2. Suppose U (t) is the solution of (3.1)–(3.2), then it is monotonous. Proof. It is sufficient to prove that U (t) has no maximal value point. If not, there is t¯ such that U ′ (t¯) = 0, U ′′ (t¯) < 0, which implies g(t¯) > 0, where g(t) = rU (t)(1 − U (t) − aU (t − τ )). Hence there is l1 < t¯ < l2 c2 such that such that g(t) > 0 and U (l1 ) = U (l2 ) < U (t¯) for t ∈ [l1 , l2 ]. This means that there is 0 < A < 4D g(t) > AU (t) for t ∈ [l1 , l2 ]. Consequently, we have D ′′ U − U ′ + AU = l(t), c2 where l(t) ≤ 0 for t ∈ [l1 , l2 ]. Set 0 < ν1 < ν2 as the roots of cD2 λ2 − λ + A = 0, note from calculations that l1 1 U (t) = U (l1 ) + D [eν1 (t−h) − eν2 (t−h) ]l(h)dh for t ∈ [l1 , l2 ], (ν − ν ) 2 t 2 1 c then there is 1 U (l2 ) − U (l1 ) = D c2 (ν2 − ν1 )
l1
[eν1 (l2 −h) − eν2 (l2 −h) ]l(h)dh > 0,
l2
which is a contradiction, then the theorem is proved.
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Hence Lemmas 3.1–3.2 can lead to the following theorem. ¯ φ), where φ, ¯ φ ∈ χ. Theorem 3.1. Suppose that (3.1) has a pair of coupled upper and lower solutions (φ, ¯ 1 Then (3.1)–(3.2) have a monotonous solution, which is the traveling wave front of (1.1) connecting 0 and 1+a . ¯ φ) of (3.1) in χ Thus our main task is to find a pair of coupled upper and lower solutions (φ, ¯ in the remainder of this section. √ Suppose that τ > 0 and a ∈ (0, 1), choose γˆ < r− 1−a a), choose s > 0 such that 1+a (1 − −1 + s=
1 + 4 cD2 γˆ 2 cD2
<
− ln a . 4τ
(3.9)
¯ Let Since limt→−∞ r(t) = r− , then for any ϵ > 0, there is ξ¯ < 0 such that r(t) < r− + ϵ for t < ξ. c2 − c c2 − 4D(r− + ϵ) − ¯ c ≥ 2 D(r + ϵ), . (3.10) λ1 = 2D 1 Select ξ < ξ¯ and 0 < α < a such that 1 − αesτ > 0 and 0 < 1+a (1 − αe−sξ ) < 1 − a, set B = 1 − a − D 1 −sξ ), 0 < κ1 < κ2 are the roots of the equation c2 κ2 − κ + r− B = 0, κ1 + γ < κ2 , choose δ > 0 1+a (1 − αe 1 δ such that 1+a (1 − αe−sξ ) = 1+a (1 − αeγξ )eκ1 ξ . We define 1 (1 − αe−st ), 1+a φ= δ (1 − αeγt )eκ1 t , 1+a
1 (1 + αe−st ), t > ξ 1 + a ¯ φ= , 1 + α eλ¯ 1 t , t ≤ ξ. 1+a Remark 3.1. If (3.9) is valid, then 1 − aeτ s > 1 −
√
t>ξ (3.11) t ≤ ξ.
a.
¯ ¯ ¯ It is easy to verify that φ(t), φ(t) ∈ χ, ¯ φ(t) ≤ φ(t). Next we verify that (φ(t), φ(t)) are a pair of weak coupled upper and lower solutions of (3.1). ¯ − φ¯ − aφ ) ≤ 0. Step 1: Claim that cD2 φ¯′′ − φ¯′ + r(t)φ(1 τ For t > ξ, (3.11) and direct calculation show φ¯ =
1 (1 + αe−st ), 1+a
φ¯′ =
α (−s)e−st , 1+a
φ¯′′ =
α 2 −st s e , 1+a
φ=
1 (1 − αe−st ), 1+a
then Remark 3.1 implies D ¯′′ ¯′ α aαesτ ¯ − φ¯ − aφ ) = α e−st D s2 + s − r(t)φ¯ φ − φ + r(t) φ(1 − e−st τ c2 1+a c2 1+a 1+a α −st D 2 1 − aesτ α −st e e ≤ s + s − r− 2 1+a c 1+a 1+a √ α −st D 2 α −st D 2 a −1 − = e s + s − r ≤ e s + s − γ ˆ = 0. 1+a c2 1+a 1+a c2 For t ≤ ξ, 1 + α λ¯ 1 t φ¯ = e , 1+a
1 + α ¯ λ¯ 1 t φ¯′1 = λ1 e , 1+a
1 + α ¯ 2 λ¯ 1 t φ¯′′ = λ e , 1+a 1
then from (3.10) D ¯′′ ¯′ ¯ − φ¯ − aφ ) ≤ D φ¯′′ − φ¯′ + (r− + ϵ)φ¯ = 1 + α eλ¯ 1 t D λ ¯2 − λ ¯ 1 + (r− + ϵ) = 0. φ − φ + r(t) φ(1 τ c2 c2 1+a c2 1
Y. Tian, Z. Liu / Applied Mathematics Letters 49 (2015) 91–99
Step 2: Claim For t > ξ + τ ,
D ′′ c2 φ
97
− φ′ + r(t)φ(1 − φ − aφ¯τ ) ≥ 0. 1 α (1 − αe−st ), φ′ = se−st , 1+a 1+a −α 2 −st 1 φ′′ = s e , φ¯τ = (1 + αe−s(t−τ ) ), 1+a 1+a φ=
then from Remark 3.1, D ′′ ′ ¯τ ) = α e−st − D s2 − s + r(t)φ α (1 − aesτ ) e−st φ − φ − a φ − φ + r(t)φ(1 c2 1+a c2 1+a √ α −st D 1−α ≥ e − 2 s2 − s + r− (1 − a) 1+a c 1+a α −st D ≥ e − 2 s2 − s + γˆ = 0. 1+a c For ξ < t ≤ ξ + τ , φ=
1 (1 − αe−st ), 1+a
1 + α λ¯ 1 (t−τ ) , φ¯τ = e 1+a
then D ′′ α −st aα λ¯ 1 (t−τ ) α −st D 2 ′ ¯ φ − φ + r(t)φ(1 − φ − aφτ ) = e − 2 s − s + r(t)φ e − e c2 1+a c 1+a 1+a α ¯ −st D α aαeτ s ≥ e − 2 s2 − s + r− φ − e−st 1+a c 1+a 1+a √ α −st D α 1−α ≥ e − 2 s2 − s + r− (1 − a)e−st 1+a c 1+a 1+a D α −st e − 2 s2 − s + γˆ = 0. ≥ 1+a c For t ≤ ξ, φ =
δ 1+a (1
− αeγt )eκ1 t , then
D ′′ φ − φ′ + r(t)φ(1 − φ − aφ¯τ ) c2 δ D 2 D δ κ1 t (κ1 +γ)t 2 ≥ e κ − κ1 − e (κ1 + γ) − (κ1 + γ) + r− Bφ 1+a c2 1 1+a c2 δ D 2 δ D κ1 t (κ1 +γ)t − 2 − ≥ e κ − κ1 + r B − e (κ1 + γ) − (κ1 + γ) + r B ≥ 0. 1+a c2 1 1+a c2 ¯ φ) which is given by (3.11) are a pair of weak After the above two steps, we have the conclusion that (φ, − coupled √ wave front of (1.1) exists if upper–lower solutions of (3.1) if c > 2 D(r + ϵ). Then the traveling c > 2 D(r− + ϵ). In view of ϵ√is arbitrary, then such solution exists for c > 2 Dr− . Moreover, the following theorem indicates that c∗ = 2 Dr− is the minimal wave speed of (1.1). √ − Theorem 3.2. Assume that a ∈ [0, 1], then for c ≥ 2 Dr √ , condition (R) holds, then (1.1) has a traveling 1 wave front with speed c connected 0 with 1+a . For c < 2 Dr− , there is no such traveling wave fronts. That √ means c∗ = 2 Dr− is the minimal wave speed. √ Proof. For c > 2 Dr−√ , a ∈ [0, 1), the traveling wave front exists from the above detailed discussion and Theorem 3.1. For c = 2 Dr− or a = 1, we can use the similar way in [7] to obtain the existence because the wave solution is monotonous. Thus the first conclusion is valid. We use reduction to absurdity to prove the
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Y. Tian, Z. Liu / Applied Mathematics Letters 49 (2015) 91–99
second conclusion of this theorem. Suppose there is c1 < c∗ such that U (x + c1 t) is the traveling wavefronts 1 connecting 0 with 1+a . Let u0 (0, x) = U (x), then u0 (0, x) > 0 and u(t, x; u0 ) = U (x + c1 t). Since U (s) is increasing for s = x + ct, then u(t − τ, x) = U (x + c1 (t − τ )) < U (x + c1 t) = u(t, x), then ∂u = D∆u + r(t)u(1 − u − au(t − τ )) ≥ D∆u + r− u(1 − u − au). ∂t Choose v(t, x) as the solution of ∂u = D∆u + r− u(1 − (a + 1)u) ∂t with v(t, x) ̸≡ 0, 0 ≤ v(t, x) ≤ U (x + c1 t) = u(t,√x) for t ∈ [0, τ ], then U (x + c1 t) ≥ v(t, x) by using comparison principle. Thus for 0 < c1 < c2 < c∗ = 2 Dr− , from Lemma 2.1, there is 0 = U (−∞) =
lim inf
t→+∞, x=−c2 t
u(t, x) ≥
lim inf
t→+∞, x=−c2 t
v(t, x) =
1 , 1+a
which is a contradiction. The proof is complete. Finally, we give a condition on r(t) for the non-existence of the traveling wave front for (1.1). Theorem 3.3. Assume that a ∈ [0, 1) and limt→+∞ r(t) = +∞ hold, then (1.1) has no traveling wave front 1 connected 0 with 1+a . Proof. We use reduction to absurdity to prove the statement. Suppose that there is a traveling wave front 1 U (x + c˜t) connecting 0 to 1+a with speed c˜. Since limt→+∞ r(t) = +∞, then there is T1 > 0 such that r(t) > M with 2 DM (1 − a) > c˜ for t > T1 . In view of ∂u = D∆u + r(t)u(1 − u − au(t − τ )) ≥ D∆u + M u(1 − a − u), ∂t then choose v(t, x) as the solution of
t > T1 , x ∈ R,
∂u = D∆u + M u(1 − a − u), t > T1 , x ∈ R ∂t with v(T1 , x) ̸≡ 0, 0 ≤ v(T1 , x) ≤ U (x + c˜T1 ) for x ∈ R, then U (x + c˜t) ≥ v(t, x) by using comparison principle. Thus for c˜ < c1 < 2 DM (1 − a), from Lemma 2.1, there is 0 = U (−∞) =
lim inf
t→+∞, x=−c1 t
u(t, x) ≥
lim inf
t→+∞, x=−c1 t
v(t, x) = 1 − a,
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