Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010, Article ID 764703, 11 pages doi:10.1155/2010/764703 Research Article Asymptotic Behavior of a Periodic Diffusion System Songsong Li1, and Xiaofeng Hui1 School of Management, Harbin Institute of Technology, Harbin 150001, China School of Finance and Economics Management, Harbin University, Harbin 150086, China Correspondence should be addressed to Songsong Li, songsong.li@qq.com Received 26 June 2010; Accepted 25 August 2010 Academic Editor: P J Y Wong Copyright q 2010 S Li and X Hui This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited We study the asymptotic behavior of the nonnegative solutions of a periodic reaction diffusion system By obtaining a priori upper bound of the nonnegative periodic solutions of the corresponding periodic diffusion system, we establish the existence of the maximum periodic solution and the asymptotic boundedness of the nonnegative solutions of the initial boundary value problem Introduction In this paper, we consider the following periodic reaction diffusion system: ∂u ∂t Δum1 b1 uα1 vβ1 , x, t ∈ Ω × R , 1.1 ∂v ∂t Δum2 b2 uα2 vβ2 , x, t ∈ Ω × R , 1.2 with initial boundary conditions u x, t v x, t u x, u0 x , 0, x, t ∈ ∂Ω × R , v x, v0 x , x ∈ Ω, 1.3 1.4 where m1 , m2 > 1, α1 , α2 , β1 , β2 ≥ 1, Ω ⊂ Rn is a bounded domain with a smooth boundary b1 x, t and b2 b2 x, t are nonnegative continuous functions and of T -periodic ∂Ω, b1 T > with respect to t, and u0 and v0 are nonnegative bounded smooth functions 2 Journal of Inequalities and Applications In dynamics of biological groups 1, , the system 1.1 - 1.2 was used to describe the interaction of two biological groups without self-limiting, where the diffusion terms reflect that the speed of the diffusion is slow In addition, the system 1.1 - 1.2 can also be used to describe diffusion processes of heat and burning in mixed media with nonlinear conductivity and volume release, where the functions u, v can be treated as temperatures of interacting components in the combustible mixture For case of m1 m2 1, we get the classical reaction diffusion system of Fujita type ∂u ∂t Δu uα1 vβ1 , ∂v ∂t Δv uα2 vβ2 1.5 This type reaction diffusion system 1.5 models such as heat propagations in a twocomponent combustible mixture , chemical processes , and interaction of two biological groups without self-limiting 6, The problem about system 1.5 includes global existence and global existence numbers, blow-up, blow-up rates, blow-up sets, and uniqueness of weak solutions see 8–10 and references therein In this paper, we will work on the diffusion system 1.1 - 1.2 ; for results about single equation, see 11–16 and so on In the past two decades, the system 1.1 - 1.2 has been deeply investigated by many authors, and there have been much excellent works on the existence, uniqueness, regularity and some other qualitative properties of the weak solutions of the initial boundary value problem see 17–22 and references therein Maddalena 20 especially, established the existence and uniqueness of the solutions of the initial boundary value problem 1.1 – 1.4 , and Wang 22 established the existence of the nonnegative nontrivial periodic solutions of the periodic boundary value problem 1.1 – 1.3 when mi > 1, βi /m2 < 1, i 1, αi , βi ≥ 1, and αi /m1 Our work is to consider the existence and attractivity of the maximal periodic solution of the problem 1.1 – 1.3 It should be remarked that our work is not a simple work The main reason is that the degeneracy of 1.1 , 1.2 makes the work of energy estimates more complicated Since the equations have periodic sources, it is of no meaning to consider the steady state So, we have to seek a new approach to describe the asymptotic behavior of the nonnegative solutions of the initial boundary value problem Our idea is to consider all the nonnegative periodic solutions We fist establish some important estimations on the nonnegative periodic solutions Then by the De Giorgi iteration technique, we provide a priori estimate of the nonnegative periodic solutions from the upper bound according to the maximum norm These estimates are crucial for the proof of the existence of the maximal periodic solution and the asymptotic boundedness of the nonnegative solutions of the initial boundary value problem This paper is organized as follows In Section 2, we introduce some necessary preliminaries and the statement of our main results In Section 3, we give the proof of our main results Preliminary In this section, as preliminaries, we present the definition of weak solutions and some useful principles Since 1.1 and 1.2 are degenerated whenever u v 0, we focus our main efforts on the discussion of weak solutions Journal of Inequalities and Applications Definition 2.1 A vector-valued function u, v is called to be a weak supsolution to the Ω × 0, τ with τ > if |∇um1 |, |∇vm2 | ∈ L2 Qτ , and for any problem 1.1 – 1.4 in Qτ one has nonnegative function ϕ ∈ C1 Qτ with ϕ|∂Ω× 0,τ Ω u x, τ ϕ x, τ dx − Ω u0 x ϕ x, dx − Qτ ∇um1 ∇ϕ dx dt ≥ Qτ Ω u ∂ϕ dxdt ∂t b1 uα1 vβ1 ϕ dx dt, Qτ v x, τ ϕ x, τ dx − Ω v0 x ϕ x, dx − v Qτ ∇vm2 ∇ϕ dx dt ≥ Qτ ∂ϕ dx dt ∂t 2.1 b2 uα2 vβ2 ϕ dx dt, Qτ u x, t ≥ 0, v x, t ≥ 0, u x, ≥ u0 x , x, t ∈ ∂Ω × 0, τ , v x, ≥ v0 x , x ∈ Ω Replacing “≥” by “≤” in the above inequalities follows the definition of a weak subsolution Furthermore, if u, v is a weak supersolution as well as a weak subsolution, then we call it a weak solution of the problem 1.1 – 1.4 Definition 2.2 A vector-valued function u, v is said to be a T -periodic solution of the problem 1.1 – 1.3 if it is a solution such that u ·, u ·, T , v ·, v ·, T a.e in Ω 2.2 A vector-valued function u, v is said to be a T -periodic supersolution of the problem 1.1 – 1.3 if it is a supersolution such that u ·, ≥ u ·, T , v ·, ≥ v ·, T a.e in Ω 2.3 A vector-valued function u, v is said to be a T -periodic subsolution of the problem 1.1 – 1.3 if it is a subsolution such that u ·, ≤ u ·, T , v ·, ≤ v ·, T a.e in Ω 2.4 A pair of supersolution u, v and subsolution u, v are called to be ordered if u ≥ u, v≥v a.e in QT Ω × 0, T 2.5 Several properties of solutions of problem 1.1 – 1.4 are needed in this paper Lemma 2.3 see 17 If αi ≥ 1, βi ≥ 1, αi /m1 βi /m2 < with |Ω| < M0 and M0 is a constant depending on mi , αi , βi , i = 1, 2, then there exist global weak solutions to 1.1 – 1.4 Journal of Inequalities and Applications Lemma 2.4 see 20 Letting u, v be a subsolution of the problem 1.1 – 1.4 with the initial value u0 , v0 , and letting u, v be a supsolution of the problem 1.1 – 1.4 with the initial value u0 , v0 , then u ≤ u, v ≤ v a.e in QT if u0 ≤ u0 , v0 ≤ v0 a.e in Ω Lemma 2.5 regularity 23 Let u x, t be a weak solution of ∂u ∂t Δum f x, t , m > 1, 2.6 subject to the homogeneous Dirichlet condition 1.3 If f ∈ L∞ QT , then there exist positive constants K and β ∈ 0, depending only upon τ ∈ 0, T and f ∞ such that for any x1 , t1 , x2 , t2 ∈ Ω × τ, T , one has |u x1 , t1 − u x2 , t2 | ≤ K |x1 − x2 |β |t1 − t2 |β/2 2.7 The main result of this paper is the following theorem Theorem 2.6 If mi > 1, αi ≥ 1, βi ≥ 1, and αi /m1 βi /m2 < with |Ω| < M0 and M0 is a constant depending on mi , αi , βi , i 1, 2, then problem 1.1 – 1.3 has a maximal periodic solution U, V which is positive in Ω Moreover, if u, v is the solution of the initial boundary value problem 1.1 – 1.4 with nonnegative initial value u0 , v0 , then for any ε > 0, there exists t1 depending on u0 and ε, t2 depending on v0 and ε, such that 0≤u≤U ε, for x ∈ Ω, t ≥ t1 , 0≤v≤V ε, for x ∈ Ω, t ≥ t2 2.8 The Main Results In this section, we first show some important estimates on the solutions of the periodic problem 1.1 – 1.3 Then, by the De Giorgi iteration technique, we establish the a prior upper bound of periodic solutions of 1.1 – 1.3 , which is used to show the existence of the maximal periodic solution of 1.1 – 1.3 and its attractivity with respective to the nonnegative solutions of the initial boundary value problem 1.1 – 1.4 Lemma 3.1 Let u, v be nonnegative solution of 1.1 – 1.3 If αi ≥ 1, βi ≥ 1, αi /m1 βi /m2 < with |Ω| < M0 and M0 is a constant depending on mi , αi , βi , i 1, 2, then there exists positive constants r and s large enough such that α2 m1 < m2 − β2 m2 u Lr QT ≤ C, r − m1 − α1 < , s−1 β1 3.1 ≤ C, 3.2 v Ls QT where C > is a positive constant depending on m1 , m2 , α1 , α2 , β1 , β2 , r, s, and |Ω| Journal of Inequalities and Applications Proof For r > 1, multiplying 1.1 by ur−1 and integrating over QT , by the periodic boundary value condition, we have r − m1 m1 r−1 Ω ∇u m1 r−1 /2 b1 x, t uα1 dx dt r−1 β1 v dx dt, 3.3 QT that is, Ω where Cb ∇u m1 r−1 /2 dx dt ≤ Cb m1 r − r − m1 uα1 r−1 β1 3.4 v dx dt, QT b1 x, t By the Poincar´ inequality, we have e QT Ω um ε r−1 dx ≤ C m1 r−1 /2 Ω ∇uε dx, 3.5 β1 /m2 < where C is a constant depending only on |Ω| and N Notice that α1 /m1 implies α1 < m1 Furthermore, we have α1 r − < m1 r − Then, by Young’s inequality, we obtain uα1 r−1 β1 v ≤ r − m1 CCb 2 r−1 m1 um r−1 C1 vβ1 m1 r−1 / m1 −α1 , 3.6 dx dt, 3.7 where C1 is the constant of Young’s inequality Then, from 3.4 , we have um r−1 dx dt ≤ QT um r−1 dx dt v β1 C1 QT m1 r−1 / m1 −α1 QT that is, um r−1 dx dt ≤ C1 QT v β1 m1 r−1 / m1 −α1 dx dt 3.8 dx dt 3.9 QT Similarly, we get an estimate for vs with s > 1, that is, v m2 s−1 dx dt ≤ C2 QT uα2 m2 s−1 / m2 −β2 QT Hence, um r−1 v m2 dx dt QT s−1 dx dt QT ≤ C1 v QT β1 m1 r−1 / m1 −α1 3.10 dx dt C2 u QT α2 m2 s−1 / m2 −β2 dx dt 6 Journal of Inequalities and Applications βi /m2 < 1, i 1, 2, implies α2 β1 < m1 − α1 m2 − β2 Then there exist Notice that, αi /m1 r ≥ max{2 m1 α1 , 2α2 } and s ≥ max{2 m2 β2 , 2β1 } such that s − m2 − β2 < r−1 α2 β1 m2 < m1 − α1 m1 3.11 By Young’s inequality, we have uα2 m2 s−1 / m2 −β2 dx dt ≤ QT v 2C2 um QT dx dt C|QT |, QT dx dt ≤ 2C1 β1 m1 r−1 / m1 −α1 r−1 3.12 v m2 s−1 dx dt C|QT | QT Together with 3.10 , we obtain um r−1 v m2 dx dt QT s−1 dx dt ≤ C 3.13 QT Thus, we prove the inequality 3.2 Lemma 3.2 Let u, v be nonnegative solution of 1.1 – 1.3 If αi ≥ 1, βi ≥ 1, αi /m1 1, 2, then one has with |Ω| < M0 and M0 is a constant depending on mi , αi , βi , i |∇um1 |2 dx dt ≤ C, βi /m2 < |∇vm2 |2 dx dt ≤ C, QT 3.14 QT where C > is a positive constant depending on m1 , m2 , α1 , α2 , β1 , β2 , r, s, and |Ω| Proof Multiplying 1.1 by um1 and integrating over QT , by Holders equality, we have ă |um1 |2 dx dt uα1 QT m β1 v dx dt QT 1/2 ≤ u2 α1 m1 v2β1 dx dt dx dt QT 3.15 1/2 QT Taking r ≥ max{2 α1 m1 , 2β2 }, s ≥ max{2 β2 m2 , 2α1 }, by Lemma 3.1, we can obtain the first inequality in 3.14 The same is true for the second inequality in 3.14 Before we show the uniform super bound of maximum modulus, we first introduce a lemma as follows see 24 Lemma 3.3 Suppose that a sequence yh , h relation yh 0, 1, 2, of nonnegative numbers satisfies the recursion ≤ cbh yh ε , h 0, 1, , 3.16 Journal of Inequalities and Applications with some positive constants c, ε and b ≥ Then, yh ≤ c ε h −1 /ε b ε h −1 /ε2 −h/ε ε y0 h 3.17 In particular, if c−1/ε b−1/ε , y0 ≤ θ 3.18 b > 1, then, yh ≤ θb−h/ε , 3.19 and consequently yh → for h → ∞ Lemma 3.4 Let u, v be a solution of 1.1 – 1.3 If αi ≥ 1, βi ≥ 1, αi /m1 βi /m2 < with 1, 2, then there is a positive constant C |Ω| < M0 and M0 is a constant depending on mi , αi , βi , i such that u ≤ C, L∞ QT v L∞ QT ≤ C Proof Let k be a positive constant Multiplying 1.1 by u − k have m1 ∂ u−k ∂t QT m1 3.20 m1 and integrating over QT , we ∇ u−k dx dt m1 dx dt QT α1 β1 b1 x, t u v u−k m1 3.21 dx dt, QT where s max{s, 0} Denote that μ k mes{ x, t ∈ QT : u x, t > k} By Lemma 3.1 with r and s large enough and Holder’s inequality, we have ¨ m1 ∂ u−k ∂t QT ≤C m1 ∇ u−k dx dt m1 dx dt QT uα1 vβ1 ξ ξ 1/ξ u−k dx dt QT m1 ξ dx dt QT 1/ξη ≤C u−k QT m1 ξη dx dt μk 1−1/η 1/ξ , 3.22 Journal of Inequalities and Applications where ξ, η > are to be determined Using the Nirenberg-Gagliardo inequality with Lemma 3.1, we have 1/ξη m1 ξη u−k θ/2 ∇ u−k ≤C dx dt QT m1 dx dt 3.23 , QT where 1− θ −1 1 − N ξη ∈ 0, 3.24 Substituting 3.22 and 3.23 in 3.21 , we have θ/2 m1 ∇ u−k ∇ u−k dxdt ≤ C QT m1 dx dt μ k 1−1/η 1/ξ 3.25 QT Setting ∇ u−k w k m1 dx dt, 3.26 QT from 3.25 we obtain w k ≤ Cμ k Take kh M − 2−h , h kh − kh 2/ 2−θ 1−1/η 1/ξ 3.27 0, 1, , and M > is to be determined Then, we have m1 ξη μ kh u − kh ≤ m1 ξη dx dt ≤ Cw kh ξηθ/2 3.28 QT From 3.26 , we have μ kh ≤ C2hm1 ξη μ kh θ η−1 / 2−θ where b 2m1 ξη and γ η − ξη − N/ 2ξη positive constant satisfying η > max 2, Cbh μ kh γ , 3.29 N For any constant ξ > 1, take η to be a 2ξ N −1 , ξN 3.30 then we have γ > By Lemma 3.1, we can select M large enough such that μ k0 μ M ≤ C−1/ γ−1 4−1/ γ−1 3.31 Journal of Inequalities and Applications According to Lemma 3.3, we have μ kh → 0, as h → ∞, which implies that u x, t ≤ 2M in QT The uniform estimate for v x, t L∞ QT may be obtained by a similar method The proof is completed Let μ, ψ be the first eigenvalue and its corresponding eigenfunction to the Laplacian operator −Δ on some domain Ω ⊃⊃ Ω with respect to homogeneous Dirichlet data It is clear that ψ x > for all x ∈ Ω Now we give the proof of the main results of this paper Proof of Theorem 2.6 We first establish the existence of the maximal periodic solution U x, t , V x, t of the problem 1.1 – 1.3 Define the Poincar´ mapping e T1 , T2 : C Ω × C Ω −→ C Ω × C Ω , T T u0 x , v0 x 3.32 u x, T , v x, T , where u x, t , v x, t is the solution of the initial boundary value problem 1.1 – 1.4 with initial value u0 x , v0 x A similar argument as that in 22 shows that the map T is well defined Let un x, t , x, t be the solution of the problem 1.1 – 1.4 with initial value u x ,v x u0 x , v0 x K1 ψ1 , K2 ψ2 , 3.33 where K1 , K2 , ψ1 , and ψ2 are taken as those in 22 Then, by comparison principle, we have Tn u x , v x , un x, T , x, T un x, t ≤ un x, t ≤ u x , x, t ≤ x, t ≤ v x A standard argument shows that there exist u∗ x , v∗ x of {T n u x }, denoted by itself for simplicity, such that u∗ x , v ∗ x 3.34 ∈ C Ω × C Ω and a subsequence lim T n u x , v x 3.35 n→∞ Similar to the proof of Theorem 4.1 in 25 , we can prove that U x, t , V x, t , which is the even extension of the solution of the initial boundary value 1.1 – 1.4 with the initial value u∗ x , v∗ x , is a periodic solution of 1.1 – 1.3 For any nonnegative periodic solution u x, t , v x, t of 1.1 – 1.3 , by Lemma 3.4, we have u x, t ≤ C0 , v x, t ≤ C0 for x, t ∈ QT 3.36 Taking K1 ≥ C0 minx∈Ω ϕ1/m1 x , K2 ≥ C0 minx∈Ω ϕ1/m2 x , 3.37 10 Journal of Inequalities and Applications to be combined with the comparison principle and u∗ x ≥ u x, , v∗ x ≥ v x, , then we obtain U x, t ≥ u x, t , V x, t ≥ v x, t , which implies that U x, t , V x, t is the maximal periodic solution of 1.1 – 1.3 For any given nonnegative initial value u0 x , v0 x , let u x, t , v x, t be the solution of the initial boundary problem 1.1 – 1.4 , and let ω1 x, t , ω2 x, t be the solution R1 ϕ1 x , R2 ϕ2 x , where R1 , R2 satisfy of 1.1 – 1.4 with initial value ω1 x, , ω2 x, the same conditions as K1 , K2 and R1 ≥ For any x, t ∈ QT , k u x, t u0 L∞ 1/m1 minx∈Ω ϕ1 x , R2 ≥ v0 L∞ 1/m2 minx∈Ω ϕ2 x 3.38 0, 1, 2, , we have kT ≤ w1 x, t kT , v x, t kT ≤ w2 x, t kT 3.39 A similar argument as that in 25 shows that ∗ ∗ ω1 x, t , ω2 x, t lim ω1 x, t k→∞ kT , lim ω2 x, t k→∞ kT , 3.40 ∗ ∗ and ω1 x, t , ω2 x, t is a nontrivial nonnegative periodic solution of 1.1 – 1.3 Therefore, for any ε > 0, there exists k0 such that u x, t ∗ kT ≤ ω1 x, t ε ≤ U x, t ε, v x, t ∗ kT ≤ ω2 x, t ε ≤ V x, t ε, 3.41 ∗ ∗ for any k ≥ k0 and x, t ∈ QT Taking the periodicity of ω1 x, t , ω2 x, t , U x, t , and V x, t into account, the proof of the theorem is completed References H Meinhardt, Models of Biological Pattern Formation, Academic Press, London, UK, 1982 Yu M Romanovski˘, N V Stepanova, 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