- Email: [email protected]
Global boundedness of solutions in a reaction–diffusion system of predator–prey model with prey-taxis
Accepted Manuscript Global boundedness of solutions in a reaction–diffusion system of predator–prey model with prey-taxis Xiao He, Sining Zheng PII: DOI: Reference:
S0893-9659(15)00153-6 http://dx.doi.org/10.1016/j.aml.2015.04.017 AML 4784
To appear in:
Applied Mathematics Letters
Received date: 13 January 2015 Accepted date: 29 April 2015 Please cite this article as: X. He, S. Zheng, Global boundedness of solutions in a reaction–diffusion system of predator–prey model with prey-taxis, Appl. Math. Lett. (2015), http://dx.doi.org/10.1016/j.aml.2015.04.017 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Manuscript Click here to view linked References
Global boundedness of solutions in a reaction-diffusion system of predator-prey model with prey-taxis ∗ Xiao He
Sining Zheng†
School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, P. R. China
January 13, 2015
Abstract This paper studies a reaction-diffusion system of a predator-prey model with Holling type II functional response and prey-taxis, proposed by Ainseba, Bendahmane and Noussair [Nonlinear Anal. RWA 9 (2008) 2086–2105], where the prey-taxis means a direct movement of the predator in response to a variation of the prey (which results in the aggregation of the predator). The global existence of classical solutions was established by Tao [Nonlinear Anal. RWA 11 (2010) 2056–2064]. In this paper we prove furthermore that the global classical solutions are globally bounded, by means of the Gagliardo-Nirenberg inequality, the Lp − Lq estimates for the Neumann heat semigroup, and the Lp estimates with Moser’s iteration of parabolic equations. 2010MSC: 92B05, 92D40, 35K57 Keywords: Predator-prey; Prey-taxis; Reaction-diffusion system; Boundedness
1
Introduction
In this paper we study the following reaction-diffusion system of a predator-prey model with prey-taxis: cuv , x ∈ Ω, t ∈ (0, T ), ut − d1 ∆u + ∇ · (uχ(u)∇v) = −au + β m + bv cuv r vt − d2 ∆v = rv − v 2 − , x ∈ Ω, t ∈ (0, T ), K m + bv (1.1) ∂u ∂v = = 0, x ∈ ∂Ω, t ∈ (0, T ), ∂ν ∂ν (u(x, 0), v(x, 0) = (u0 (x), v0 (x)) ≥ 0, x ∈ Ω,
where Ω is a bounded domain in RN (N = 1, 2, 3) with smooth boundary ∂Ω, 0 < T ≤ +∞, initial data u0 (x), v0 (x) ∈ C 2+α (Ω) compatible on ∂Ω, constants d1 , d2 , a, K, r, β, b, c, m > 0, and ν is the normal outer vector on ∂Ω. ∗ †
Supported by the National Natural Science Foundation of China (No. 11171048 and No. 11026168). Corresponding author. E-mail: [email protected] (S. N. Zheng), [email protected] (X. He)
1
There is the Holling type II functional response contained in the model (1.1), where u and v represent the densities of the predator and prey with diffusion rates d1 and d2 , while c b , and m denote the death rate of u, the carrying capacity of v, the intrinsic a, K, r, β, m growth rate of v, the conversion rate of the species, the handing time spent by u to catch and consume v, and the searching efficiency of u, respectively. In addition, the prey-taxis mechanism in the model means a direct movement of the predator u in response to a variation of the prey v (which results in the aggregation of u), and the involved factor is assumed to satisfy χ(u) ∈ C 1 ([0, +∞)), χ(u) ≡ 0 for u ≥ M , and |χ′ (u1 ) − χ′ (u2 )| ≤ L|u1 − u2 | for u1 , u2 ∈ [0, +∞), with L, M > 0. Here the assumption χ(u) ≡ 0 for u ≥ M says that there is a threshold value M for the accumulation of u, over which the prey-tatic cross-diffusion χ(u) vanishes [1]. The model (1.1) was proposed by Ainseba, Bendahmane and Noussair [1]. They established the existence of weak solutions by means of Schauder fixed point theorem and the uniqueness via duality technique. Tao [7] gave the global existence (i.e. T = +∞) and uniqueness of classical solutions to (1.1) by contraction mapping principle together with Lp estimates and Schauder estimates of parabolic equations. Li, Wang, and Shao [5] studied the stability of equilibrium points and the existence of non-constant steady states of (1.1) by eigenvalue theory, Hopf bifurcation and fixed point index theory. Without prey-taxis, the model was studied by Ko and Ryu [3, 4]. In this paper, we will prove that the unique global classical solution is moreover globally bounded. That is the following theorem. Theorem 1 Under the assumptions for χ and initial data described above, the unique nonnegative classical solution of (1.1) is globally bounded. The paper is organized as follows. In Section 2, we introduce some known results as preliminaries. Section 3 gives the proof of Theorem 1.
2
Preliminary results
At first introduce the global existence of classical solutions to (1.1) established in [7]. Lemma 2.1 (Theorem 3.5 in [7]) Under the assumptions for χ and initial data in the paper, α for any given T > 0, there exists a unique solution (u, v) ∈ (C 2+α,1+ 2 (Ω × (0, T )))2 of (1.1). The next is the well-known classical Lp − Lq estimate for the Neumann heat semigroup on bounded domains. Lemma 2.2 (Lemma 2.1 in [2], Lemma 1.3 in [9]) Suppose (et∆ )t>0 is the Neumann heat semigroup in Ω, and λ1 > 0 denotes the first nonzero eigenvalue of −∆ in Ω under Neumann boundary conditions. Then the following Lp −Lq estimates hold with k1 , k2 > 0 only depending on Ω:
2
(i) If 1 ≤ q ≤ p ≤ +∞, then 1
n 1
1
k∇et∆ wkLp (Ω) ≤ k1 (1 + t− 2 − 2 ( q − p ) )e−λ1 t kwkLq (Ω) , t > 0 for all w ∈ Lq (Ω); (ii) If 2 ≤ q ≤ p ≤ +∞, then k∇et∆ wkLp (Ω) ≤ k2 (1 + t
−n ( 1 − p1 ) 2 q
)e−λ1 t k∇wkLq (Ω) , t > 0
for all w ∈ W 1,q (Ω). The third is on the boundedness of v. Lemma 2.3 Let (u, v) be a solution of (1.1). Then u ≥ 0 and 0 ≤ v ≤ K0 = max{maxΩ v0 (x), K}. The proof of the lemma is based on the comparison principle of ODEs. Refer to the proof of Lemma 3.1 in [7] for the details.
3
Proof of Theorem 1
This section is devoted to the proof of Theorem 1. Our approach is motivated by Tao and Winkler [8]. Proof of Theorem 1. The proof consists of four steps. Step 1: Boundedness of kukL1 (Ω) Integrate the sum of the first equation and the β times of the second equation in (1.1) on Ω by parts, Z Z Z Z Z rβ d d v− u+ βv = −a u + rβ v2 dt Ω dt Ω K Ω Ω Ω Z Z βv + Krβ|Ω|, (3.1) ≤ −a u − r Ω
where
R
Ωv
≤
1 2K
R
Ωv
2
+
K|Ω| 2
Ω
is used. Define Z Z βv, t > 0. u+ y(t) = Ω
Ω
Then y ′ (t) + c1 y(t) ≤ c2 for all t > 0 by (3.1) with c1 = min{a, r} and c2 = Krβ|Ω|. This ensures y(t) ≤ A1 = max{y(0), by the comparison principle of ODEs. 3
c2 } for all t > 0 c1
Step 2: Boundedness of kukLp (Ω) with p > 2 Multiply the equation of u in (1.1) by up−1 and integrate on Ω by parts. Since v ≤ K0 by Lemma 2.3, we have Z Z Z Z Z up v 1 d p p−2 2 p u |∇u| = −a u + βc u + d1 (p − 1) + (p − 1) χ(u)up−1 ∇u · ∇v p dt Ω Ω m + bv Ω Ω Z ZΩ βcK0 d (p − 1) 1 ≤ (−a + ) up + up−2 |∇u|2 m + bK0 Ω 2 Ω Z p−1 + χ(u)2 up |∇v|2 . 2d1 Ω Consequently, together with χ(u) ≤ M1 due to χ ∈ C 1 and χ ≡ 0 for u ≥ M , we obtain Z Z Z Z 1 d (p − 1)M12 M p d1 (p − 1) βcK0 |∇v|2 . up + up−2 |∇u|2 ≤ (−a + ) up + p dt Ω 2 m + bK 2d 0 1 Ω Ω Ω (3.2) Multiply the equation for v in (1.1) by −∆v, and integrate on Ω by parts to get Z Z Z Z Z d 4r uv 2 2 2 2 |∇v| − |∆v| = 2r |∇v| + 2d2 v|∇v| + 2c ∆v dt Ω K Ω Ω Ω m + bv ZΩ Z 2cK0 |∇v|2 + u|∆v| ≤ 2r m + bK0 Ω Ω Z Z Z 2c2 K02 ǫ u2 |∆v|2 + |∇v|2 + ≤ 2r 2 2 ǫ(m + bK ) 0 Ω Ω Ω by Young’s inequality. Choosing ǫ = 2d2 , we obtain that Z Z Z Z c2 K02 d 2 2 2 |∇v| + |∆v| ≤ 2r u2 . (3.3) |∇v| + d2 2 dt Ω d (m + bK ) 2 0 Ω Ω Ω R R p−2 p 2 2 Noting d1 (p−1) |∇u|2 = 2d1 (p−1) Ω |∇u | , p > 2, we know from (3.2) and (3.3) 2 Ωu p2 by Young’s inequality that Z Z Z Z p d 2d1 (p − 1) 1 d p 2 2 |∇u 2 | + d2 u + |∇v| + |∆v|2 p dt Ω dt Ω p2 Ω Ω Z Z βcK0 (p − 1)M12 M p ≤ (−a + up + (2r + |∇v|2 + c3 (3.4) + 1) ) m + bK0 2d 1 Ω Ω with c3 > 0. By the Sobolev interpolation inequality with Lemma 2.3, we have for any ǫ1 > 0 that Z Z Z Z 2 2 2 |∆v|2 + c5 (3.5) |v| ≤ ǫ1 |∆v| + c4 |∇v| ≤ ǫ1 Ω
Ω
Ω
Ω
with c4 , c5 > 0 depending on ǫ1 . Applying the Gagliardo-Nirenberg inequality yields Z Z p p p 2(1−θ) p 2 |u 2 |2 ≤ c6 k∇u 2 k2θ up = + c6 ku 2 k22 2 ku k 2 Ω
p
Ω
4
p
with 0 < θ =
N p−N N p−N +2
Z
< 1 and c6 > 0. Thus by Young’s inequality,
Ω
p
p
p
up ≤ ǫ2 k∇u 2 k22 + c7 ku 2 k22 = ǫ2 k∇u 2 k22 + c7 kukp1 p
for any ǫ2 > 0, with c7 > 0 depending on ǫ2 . Since kuk1 ≤ A1 by Step 1, we know that Z p up ≤ ǫ2 k∇u 2 k22 + c8 (3.6) Ω
(p−1)M 2 M p
βcK0 1 . We with c8 > 0. Now fix ǫ1 , ǫ2 with (2r + )ǫ1 = d22 and ( m+bK + 1)ǫ2 = 2d1 (p−1) 2d1 p2 0 have from (3.4)–(3.6) that Z Z Z Z 1 d (p − 1)M12 M p d up + |∇v|2 ≤ −a up − (2r + ) |∇v|2 + c9 p dt Ω dt Ω 2d1 Ω Ω
with c9 > 0. Therefore, the function z(t) =
1 p
Z
up +
Ω
Z
Ω
|∇v|2 , t > 0
satisfies z ′ (t) ≤ −c10 z(t) + c9 for all t > 0 with c10 = min{ap, 2r +
(p−1)M12 M p−2 }, 2d1
and hence
z(t) ≤ max{z(0),
c9 } for all t > 0. c10
Step 3: Boundedness of k∇vkLp (Ω) with p > 2 cuv Define f (u, v) = rv − Kr v 2 − m+bv . It follows from Lemma 2.3 and Step 2 that there is A2 > 0 such that sup kf kLp (Ω) ≤ A2 < +∞. t>0
The variation-of-constants formula for v yields Z t ed2 (t−s)∆ f (u(s), v(s))ds, t > 0. v(·, t) = ed2 t∆ v0 + 0
Because of Lemma 2.2, we conclude that Z t k∇ed2 (t−s)∆ f (u(s), v(s))kLp (Ω) ds k∇vkLp (Ω) ≤ k∇ed2 t∆ v0 kLp (Ω) + 0 Z t 1 −1 −λ1 t (1 + d2 2 (t − s)− 2 )e−λ1 d2 (t−s) kf (s)kLp (Ω) ds ≤ 2k2 e k∇v0 kLp (Ω) + k1 0 Z t 1 −1 (1 + d2 2 s− 2 )e−λ1 d2 s ds ≤ 2k2 e−λ1 t k∇v0 kLp (Ω) + k1 A2 0
5
≤ 2k2 k∇v0 kLp (Ω) + k1 A2 (
1 1 −1 + d2 2 (2 + )) for all t > 0. λ1 d2 λ1 d2
Step 4: Global boundedness Based on Steps 2 and 3, with Lemma A.1 in [8], the global boundedness of solutions can be proved by using of the standard Moser iterative technique. The proof is complete. Remark 1 It is known that without prey-texis, the global boundedness of solutions is an easy result to the corresponding predator-prey model [3, 4]. The existence of prey-taxis in (1.1) makes substantial difficulty to get the global boundedness, and even the global existence of solutions [1, 7]. On the other hand, the prey-taxis term ∇ · (uχ(u)∇v) contained in the model is assumed that χ(u) ≡ 0 whenever u ≥ M , where the maximal density M acts as a switch to repulsion at high densities of the predator population, very similar to the volumefilling effect or prevention of overcrowding for chemotaxis [6]. So, the global boundedness of solutions established via Theorem 1 should be natural and reasonable.
References [1] B.E. Ainseba, M. Bendahmane, A. Noussair, A reaction-diffusion system modeling predator-prey with prey-taxis, Nonlinear Anal. RWA 9 (2008) 2086–2105. [2] X.R. Cao, Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces, Discrete Contin. Dyn. Syst. 35 (5) (2015). [3] W. Ko, K. Ryu, A qualitative study on general Gause-type predator-prey models with constant diffusion rates, J. Math. Anal. Appl. 344 (2008) 217–230. [4] W. Ko, K. Ryu, A qualitative study on general Gause-type predator-prey models with nonmonotonic functional response, Nonlinear Anal. RWA 10 (2009) 2558–2573. [5] C.L. Li, X.H. Wang, Y.F. Shao, Steady states of a predator-prey mdol with prey-taxis, Nonlinear Anal. 97 (2014) 155–168. [6] K.J. Painter, T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q. 10 (2002), 501–543. [7] Y.S. Tao, Global existence of classical solutions to a predator-prey model with nonlinear prey-taxis, Nonlinear Anal. RWA 11 (2010) 2056–2064. [8] Y.S. Tao, M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations 252 (2012) 692-715. [9] M. Winkler, Aggregation vs. global diffusive behavior in the higher–dimensional Keller-Segel model, J. Differential Equations 248 (2010) 2889–2905.
6
S0893-9659(15)00153-6 http://dx.doi.org/10.1016/j.aml.2015.04.017 AML 4784
To appear in:
Applied Mathematics Letters
Received date: 13 January 2015 Accepted date: 29 April 2015 Please cite this article as: X. He, S. Zheng, Global boundedness of solutions in a reaction–diffusion system of predator–prey model with prey-taxis, Appl. Math. Lett. (2015), http://dx.doi.org/10.1016/j.aml.2015.04.017 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Manuscript Click here to view linked References
Global boundedness of solutions in a reaction-diffusion system of predator-prey model with prey-taxis ∗ Xiao He
Sining Zheng†
School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, P. R. China
January 13, 2015
Abstract This paper studies a reaction-diffusion system of a predator-prey model with Holling type II functional response and prey-taxis, proposed by Ainseba, Bendahmane and Noussair [Nonlinear Anal. RWA 9 (2008) 2086–2105], where the prey-taxis means a direct movement of the predator in response to a variation of the prey (which results in the aggregation of the predator). The global existence of classical solutions was established by Tao [Nonlinear Anal. RWA 11 (2010) 2056–2064]. In this paper we prove furthermore that the global classical solutions are globally bounded, by means of the Gagliardo-Nirenberg inequality, the Lp − Lq estimates for the Neumann heat semigroup, and the Lp estimates with Moser’s iteration of parabolic equations. 2010MSC: 92B05, 92D40, 35K57 Keywords: Predator-prey; Prey-taxis; Reaction-diffusion system; Boundedness
1
Introduction
In this paper we study the following reaction-diffusion system of a predator-prey model with prey-taxis: cuv , x ∈ Ω, t ∈ (0, T ), ut − d1 ∆u + ∇ · (uχ(u)∇v) = −au + β m + bv cuv r vt − d2 ∆v = rv − v 2 − , x ∈ Ω, t ∈ (0, T ), K m + bv (1.1) ∂u ∂v = = 0, x ∈ ∂Ω, t ∈ (0, T ), ∂ν ∂ν (u(x, 0), v(x, 0) = (u0 (x), v0 (x)) ≥ 0, x ∈ Ω,
where Ω is a bounded domain in RN (N = 1, 2, 3) with smooth boundary ∂Ω, 0 < T ≤ +∞, initial data u0 (x), v0 (x) ∈ C 2+α (Ω) compatible on ∂Ω, constants d1 , d2 , a, K, r, β, b, c, m > 0, and ν is the normal outer vector on ∂Ω. ∗ †
Supported by the National Natural Science Foundation of China (No. 11171048 and No. 11026168). Corresponding author. E-mail: [email protected] (S. N. Zheng), [email protected] (X. He)
1
There is the Holling type II functional response contained in the model (1.1), where u and v represent the densities of the predator and prey with diffusion rates d1 and d2 , while c b , and m denote the death rate of u, the carrying capacity of v, the intrinsic a, K, r, β, m growth rate of v, the conversion rate of the species, the handing time spent by u to catch and consume v, and the searching efficiency of u, respectively. In addition, the prey-taxis mechanism in the model means a direct movement of the predator u in response to a variation of the prey v (which results in the aggregation of u), and the involved factor is assumed to satisfy χ(u) ∈ C 1 ([0, +∞)), χ(u) ≡ 0 for u ≥ M , and |χ′ (u1 ) − χ′ (u2 )| ≤ L|u1 − u2 | for u1 , u2 ∈ [0, +∞), with L, M > 0. Here the assumption χ(u) ≡ 0 for u ≥ M says that there is a threshold value M for the accumulation of u, over which the prey-tatic cross-diffusion χ(u) vanishes [1]. The model (1.1) was proposed by Ainseba, Bendahmane and Noussair [1]. They established the existence of weak solutions by means of Schauder fixed point theorem and the uniqueness via duality technique. Tao [7] gave the global existence (i.e. T = +∞) and uniqueness of classical solutions to (1.1) by contraction mapping principle together with Lp estimates and Schauder estimates of parabolic equations. Li, Wang, and Shao [5] studied the stability of equilibrium points and the existence of non-constant steady states of (1.1) by eigenvalue theory, Hopf bifurcation and fixed point index theory. Without prey-taxis, the model was studied by Ko and Ryu [3, 4]. In this paper, we will prove that the unique global classical solution is moreover globally bounded. That is the following theorem. Theorem 1 Under the assumptions for χ and initial data described above, the unique nonnegative classical solution of (1.1) is globally bounded. The paper is organized as follows. In Section 2, we introduce some known results as preliminaries. Section 3 gives the proof of Theorem 1.
2
Preliminary results
At first introduce the global existence of classical solutions to (1.1) established in [7]. Lemma 2.1 (Theorem 3.5 in [7]) Under the assumptions for χ and initial data in the paper, α for any given T > 0, there exists a unique solution (u, v) ∈ (C 2+α,1+ 2 (Ω × (0, T )))2 of (1.1). The next is the well-known classical Lp − Lq estimate for the Neumann heat semigroup on bounded domains. Lemma 2.2 (Lemma 2.1 in [2], Lemma 1.3 in [9]) Suppose (et∆ )t>0 is the Neumann heat semigroup in Ω, and λ1 > 0 denotes the first nonzero eigenvalue of −∆ in Ω under Neumann boundary conditions. Then the following Lp −Lq estimates hold with k1 , k2 > 0 only depending on Ω:
2
(i) If 1 ≤ q ≤ p ≤ +∞, then 1
n 1
1
k∇et∆ wkLp (Ω) ≤ k1 (1 + t− 2 − 2 ( q − p ) )e−λ1 t kwkLq (Ω) , t > 0 for all w ∈ Lq (Ω); (ii) If 2 ≤ q ≤ p ≤ +∞, then k∇et∆ wkLp (Ω) ≤ k2 (1 + t
−n ( 1 − p1 ) 2 q
)e−λ1 t k∇wkLq (Ω) , t > 0
for all w ∈ W 1,q (Ω). The third is on the boundedness of v. Lemma 2.3 Let (u, v) be a solution of (1.1). Then u ≥ 0 and 0 ≤ v ≤ K0 = max{maxΩ v0 (x), K}. The proof of the lemma is based on the comparison principle of ODEs. Refer to the proof of Lemma 3.1 in [7] for the details.
3
Proof of Theorem 1
This section is devoted to the proof of Theorem 1. Our approach is motivated by Tao and Winkler [8]. Proof of Theorem 1. The proof consists of four steps. Step 1: Boundedness of kukL1 (Ω) Integrate the sum of the first equation and the β times of the second equation in (1.1) on Ω by parts, Z Z Z Z Z rβ d d v− u+ βv = −a u + rβ v2 dt Ω dt Ω K Ω Ω Ω Z Z βv + Krβ|Ω|, (3.1) ≤ −a u − r Ω
where
R
Ωv
≤
1 2K
R
Ωv
2
+
K|Ω| 2
Ω
is used. Define Z Z βv, t > 0. u+ y(t) = Ω
Ω
Then y ′ (t) + c1 y(t) ≤ c2 for all t > 0 by (3.1) with c1 = min{a, r} and c2 = Krβ|Ω|. This ensures y(t) ≤ A1 = max{y(0), by the comparison principle of ODEs. 3
c2 } for all t > 0 c1
Step 2: Boundedness of kukLp (Ω) with p > 2 Multiply the equation of u in (1.1) by up−1 and integrate on Ω by parts. Since v ≤ K0 by Lemma 2.3, we have Z Z Z Z Z up v 1 d p p−2 2 p u |∇u| = −a u + βc u + d1 (p − 1) + (p − 1) χ(u)up−1 ∇u · ∇v p dt Ω Ω m + bv Ω Ω Z ZΩ βcK0 d (p − 1) 1 ≤ (−a + ) up + up−2 |∇u|2 m + bK0 Ω 2 Ω Z p−1 + χ(u)2 up |∇v|2 . 2d1 Ω Consequently, together with χ(u) ≤ M1 due to χ ∈ C 1 and χ ≡ 0 for u ≥ M , we obtain Z Z Z Z 1 d (p − 1)M12 M p d1 (p − 1) βcK0 |∇v|2 . up + up−2 |∇u|2 ≤ (−a + ) up + p dt Ω 2 m + bK 2d 0 1 Ω Ω Ω (3.2) Multiply the equation for v in (1.1) by −∆v, and integrate on Ω by parts to get Z Z Z Z Z d 4r uv 2 2 2 2 |∇v| − |∆v| = 2r |∇v| + 2d2 v|∇v| + 2c ∆v dt Ω K Ω Ω Ω m + bv ZΩ Z 2cK0 |∇v|2 + u|∆v| ≤ 2r m + bK0 Ω Ω Z Z Z 2c2 K02 ǫ u2 |∆v|2 + |∇v|2 + ≤ 2r 2 2 ǫ(m + bK ) 0 Ω Ω Ω by Young’s inequality. Choosing ǫ = 2d2 , we obtain that Z Z Z Z c2 K02 d 2 2 2 |∇v| + |∆v| ≤ 2r u2 . (3.3) |∇v| + d2 2 dt Ω d (m + bK ) 2 0 Ω Ω Ω R R p−2 p 2 2 Noting d1 (p−1) |∇u|2 = 2d1 (p−1) Ω |∇u | , p > 2, we know from (3.2) and (3.3) 2 Ωu p2 by Young’s inequality that Z Z Z Z p d 2d1 (p − 1) 1 d p 2 2 |∇u 2 | + d2 u + |∇v| + |∆v|2 p dt Ω dt Ω p2 Ω Ω Z Z βcK0 (p − 1)M12 M p ≤ (−a + up + (2r + |∇v|2 + c3 (3.4) + 1) ) m + bK0 2d 1 Ω Ω with c3 > 0. By the Sobolev interpolation inequality with Lemma 2.3, we have for any ǫ1 > 0 that Z Z Z Z 2 2 2 |∆v|2 + c5 (3.5) |v| ≤ ǫ1 |∆v| + c4 |∇v| ≤ ǫ1 Ω
Ω
Ω
Ω
with c4 , c5 > 0 depending on ǫ1 . Applying the Gagliardo-Nirenberg inequality yields Z Z p p p 2(1−θ) p 2 |u 2 |2 ≤ c6 k∇u 2 k2θ up = + c6 ku 2 k22 2 ku k 2 Ω
p
Ω
4
p
with 0 < θ =
N p−N N p−N +2
Z
< 1 and c6 > 0. Thus by Young’s inequality,
Ω
p
p
p
up ≤ ǫ2 k∇u 2 k22 + c7 ku 2 k22 = ǫ2 k∇u 2 k22 + c7 kukp1 p
for any ǫ2 > 0, with c7 > 0 depending on ǫ2 . Since kuk1 ≤ A1 by Step 1, we know that Z p up ≤ ǫ2 k∇u 2 k22 + c8 (3.6) Ω
(p−1)M 2 M p
βcK0 1 . We with c8 > 0. Now fix ǫ1 , ǫ2 with (2r + )ǫ1 = d22 and ( m+bK + 1)ǫ2 = 2d1 (p−1) 2d1 p2 0 have from (3.4)–(3.6) that Z Z Z Z 1 d (p − 1)M12 M p d up + |∇v|2 ≤ −a up − (2r + ) |∇v|2 + c9 p dt Ω dt Ω 2d1 Ω Ω
with c9 > 0. Therefore, the function z(t) =
1 p
Z
up +
Ω
Z
Ω
|∇v|2 , t > 0
satisfies z ′ (t) ≤ −c10 z(t) + c9 for all t > 0 with c10 = min{ap, 2r +
(p−1)M12 M p−2 }, 2d1
and hence
z(t) ≤ max{z(0),
c9 } for all t > 0. c10
Step 3: Boundedness of k∇vkLp (Ω) with p > 2 cuv Define f (u, v) = rv − Kr v 2 − m+bv . It follows from Lemma 2.3 and Step 2 that there is A2 > 0 such that sup kf kLp (Ω) ≤ A2 < +∞. t>0
The variation-of-constants formula for v yields Z t ed2 (t−s)∆ f (u(s), v(s))ds, t > 0. v(·, t) = ed2 t∆ v0 + 0
Because of Lemma 2.2, we conclude that Z t k∇ed2 (t−s)∆ f (u(s), v(s))kLp (Ω) ds k∇vkLp (Ω) ≤ k∇ed2 t∆ v0 kLp (Ω) + 0 Z t 1 −1 −λ1 t (1 + d2 2 (t − s)− 2 )e−λ1 d2 (t−s) kf (s)kLp (Ω) ds ≤ 2k2 e k∇v0 kLp (Ω) + k1 0 Z t 1 −1 (1 + d2 2 s− 2 )e−λ1 d2 s ds ≤ 2k2 e−λ1 t k∇v0 kLp (Ω) + k1 A2 0
5
≤ 2k2 k∇v0 kLp (Ω) + k1 A2 (
1 1 −1 + d2 2 (2 + )) for all t > 0. λ1 d2 λ1 d2
Step 4: Global boundedness Based on Steps 2 and 3, with Lemma A.1 in [8], the global boundedness of solutions can be proved by using of the standard Moser iterative technique. The proof is complete. Remark 1 It is known that without prey-texis, the global boundedness of solutions is an easy result to the corresponding predator-prey model [3, 4]. The existence of prey-taxis in (1.1) makes substantial difficulty to get the global boundedness, and even the global existence of solutions [1, 7]. On the other hand, the prey-taxis term ∇ · (uχ(u)∇v) contained in the model is assumed that χ(u) ≡ 0 whenever u ≥ M , where the maximal density M acts as a switch to repulsion at high densities of the predator population, very similar to the volumefilling effect or prevention of overcrowding for chemotaxis [6]. So, the global boundedness of solutions established via Theorem 1 should be natural and reasonable.
References [1] B.E. Ainseba, M. Bendahmane, A. Noussair, A reaction-diffusion system modeling predator-prey with prey-taxis, Nonlinear Anal. RWA 9 (2008) 2086–2105. [2] X.R. Cao, Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces, Discrete Contin. Dyn. Syst. 35 (5) (2015). [3] W. Ko, K. Ryu, A qualitative study on general Gause-type predator-prey models with constant diffusion rates, J. Math. Anal. Appl. 344 (2008) 217–230. [4] W. Ko, K. Ryu, A qualitative study on general Gause-type predator-prey models with nonmonotonic functional response, Nonlinear Anal. RWA 10 (2009) 2558–2573. [5] C.L. Li, X.H. Wang, Y.F. Shao, Steady states of a predator-prey mdol with prey-taxis, Nonlinear Anal. 97 (2014) 155–168. [6] K.J. Painter, T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q. 10 (2002), 501–543. [7] Y.S. Tao, Global existence of classical solutions to a predator-prey model with nonlinear prey-taxis, Nonlinear Anal. RWA 11 (2010) 2056–2064. [8] Y.S. Tao, M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations 252 (2012) 692-715. [9] M. Winkler, Aggregation vs. global diffusive behavior in the higher–dimensional Keller-Segel model, J. Differential Equations 248 (2010) 2889–2905.
6