A Two-Species Cooperative Lotka-Volterra System of Degenerate Parabolic Equations

and Applied Analysis 3 where fM sup{f x, t | x, t ∈ Ω × }, fl inf{f x, t | x, t ∈ Ω × }, we show that the generalized solution is uniformly bounded. At last, by the method of monotone iteration, we establish the existence of the nontrivial periodic solutions of the system 1.1 1.2 , which follows from the existence of a pair of large periodic supersolution and small periodic subsolution. At last, we show the existence and the attractivity of the maximal periodic solution. Our main efforts center on the discussion of generalized solutions, since the regularity follows from a quite standard approach. Hence we give the following definition of generalized solutions of the problem 1.1 – 1.4 . Definition 1.1. A nonnegative and continuous vector-valued function u, v is said to be a generalized solution of the problem 1.1 – 1.4 if, for any 0 ≤ τ < T and any functions φi ∈ C1 Qτ with φi|∂Ω× 0,τ 0 i 1, 2 , ∇um1 ,∇vm2 ∈ L2 Qτ , ∂u1/∂t, ∂v2/∂t ∈ L2 Qτ and ∫∫ Qτ u ∂φ1 ∂t − ∇u1∇φ1 u a − bu cv φ1dx dt ∫ Ω u x, τ φ1 x, τ dx − ∫ Ω u0 x φ1 x, 0 dx, ∫∫ Qτ v ∂φ2 ∂t − ∇v2∇φ2 v ( d eu − fvφ2dxdt ∫ Ω v x, τ φ2 x, τ dx − ∫ Ω v0 x φ2 x, 0 dx, 1.9 where Qτ Ω × 0, τ . Similarly, we can define a weak supersolution u, v subsolution u, v if they satisfy the inequalities obtained by replacing “ ” with “≤” “≥” in 1.3 , 1.4 , and 1.9 and with an additional assumption φi ≥ 0 i 1, 2 . Definition 1.2. 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 in 0, T such that u ·, 0 u ·, T , v ·, 0 v ·, T in Ω. 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 in 0, T such that u ·, 0 ≥ u ·, T , v ·, 0 ≥ v ·, T in Ω. 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 in 0, T such that u ·, 0 ≤ u ·, T , v ·, 0 ≤ v ·, T in Ω. This paper is organized as follows. In Section 2, we show the existence of generalized solutions to the initial boundary value problem and also establish the comparison principle. Section 3 is devoted to the proof of the existence of the nonnegative nontrivial periodic solutions by using the monotone iteration technique. 2. The Initial Boundary Value Problem To solve the problem 1.1 – 1.4 , we consider the following regularized problem: ∂uε ∂t div (( mum1−1 ε ε ) ∇uε ) uε a − buε cvε , x, t ∈ QT, 2.1 ∂vε ∂t div (( mvm2−1 ε ε ) ∇vε ) v ε ( d euε − fvε ) , x, t ∈ QT, 2.2 uε x, t 0, vε x, t 0, x, t ∈ ∂Ω × 0, T , 2.3 uε x, 0 u0ε x , vε x, 0 v0ε x , x ∈ Ω, 2.4 4 Abstract and Applied Analysis whereQT Ω× 0, T , 0 < ε < 1, u0ε, v0ε ∈ C∞ 0 Ω are nonnegative bounded smooth functions and satisfy 0 ≤ u0ε ≤ ‖u0‖L∞ Ω , 0 ≤ v0ε ≤ ‖v0‖L∞ Ω , u1 0ε −→ u1 0 , v2 0ε −→ v2 0 , in W 0 Ω as ε −→ 0. 2.5 The standard parabolic theory cf. 20, 21 shows that 2.1 – 2.4 admits a nonnegative classical solution uε, vε . So, the desired solution of the problem 1.1 – 1.4 will be obtained as a limit point of the solutions uε, vε of the problem 2.1 – 2.4 . In the following, we show some important uniform estimates for uε, vε . Lemma 2.1. Let uε, vε be a solution of the problem 2.1 – 2.4 . 1 If 1 < m1 − α m2 − β , then there exist positive constants r and s large enough such that 1 m2 − β < m1 r − 1 m2 s − 1 < m1 − α, 2.6 ‖uε‖Lr QT ≤ C, ‖vε‖Ls QT ≤ C, 2.7 where C is a positive constant only depending onm1, m2, α, β, r, s, |Ω|, and T . 2 If 1 m1 − α m2 − β , then 2.7 also holds when |Ω| is small enough. Proof. Multiplying 2.1 by ur−1 ε r > 1 and integrating over Ω, we have that ∫ Ω ∂uε ∂t dx − 4r r − 1 m1 m1 r − 1 2 ∫ Ω ∣∣∇u m1 r−1 /2 ε ∣∣ 2 dx r ∫ Ω u r−1 ε a − buε cvε dx. 2.8 By Poincaré’s inequality, we have that