In this paper the existence of global solutions of the parabolic cross-diffusion systems with cooperative reactions is obtained under certain conditions. The uniform boundedness of W _1,2 norms of the local maximal solution is obtained by using interpolation inequalities and comparison results on differential inequalities. This article deals with the following quasilinear parabolic system in population dynamics which is called cooperative cross-diffusion system. where α ₁₂ , α ₂₁ , d, a_i, b_i, c_i are positive constants for i = 1, 2. The initial functions u ₀ , v ₀ are not constantly zero. In the system (1.1) u and v are nonnegative functions which represent the population densities of two species in a cooperative relationship. d ₁ and d ₂ are the diffusion rates of the two species, respectively. a ₁ and a ₂ denote the intrinsic growth rates, b ₁ and c ₂ account for intra-specific cooperative pressures, b ₂ and c ₁ are the coefficients for inter-specific competitions. α ₁₁ and α ₂₂ are usually referred as self-diffusion , and α ₁₂ , α ₂₁ are cross-diffusion pressures. By adopting the coefficients α_ij ( i , j = 1, 2) the system (1.1) takes into account the pressures created by mutually interacting species. For more details on the backgrounds of this model, the readers are refered to Okubo and Levin [7] . Pao [8] in 2005, and Delgado et al. [4] in 2008 have obtained some results on the existence of global solutions of the elliptic cross-diffusion systems with cooperative reactions. In this paper the existence of global solutions of the parabolic cross-diffusion systems with cooperative reactions is obtained under certain conditions. To state results on the system (1.1) we use the following notation throughout this paper. Notations. Let ­ Ω be a region in . The norm in L_p (Ω­) is denoted by |·| _Lp(Ω­) , 1 ≤ p ≤ ∞, where | f | _Lp(­Ω) = ( ∫ _Ω | f ( x )| ^p dx ) ^1/p , if 1 ≤ p ≤ ∞, and | f | _L∞(Ω) = sup {| f ( x )| : x ∈ Ω}. The usual Sobolev spaces of real valued functions in Ω with exponent k ≥ 0 are denoted by , 1 ≤ p ≤ ∞. And represents the norm in the Sobolev space . we shall use the simplified notation ||·|| _k,p for and |·| _p for |·| _Lp(Ω) . ­ The local existence of solutions to (1.1) was established by Amann [1] , [2] , [3] . According to his results the system (1.1) has a unique nonnegative solution u (·, t ), v (·, t ) in , where T ∈ (0,∞] is the maximal existence time for the solution u, v . The following result is also due to Amann [2] . Theorem 1.1. Let u ₀ and v ₀ be in . The system (1.1) possesses a unique nonnegative maximal smooth solution for 0 ≤ t < T , where p > n and 0 < T ≤ ∞. If the solution satisfies the estimates then T = +∞. If, in addition, u ₀ and v ₀ are in then , and Here we state the main results of this paper. Throughout this this paper we assume the condition which means the inter-specific competition pressures are greater than the intra-specific cooperative pressures. Theorem 1.2. Suppose that the initial functions u ₀ , v ₀ are in . Also assume the condition (1.2). Let ( u ( x,t ), v ( x,t )) be the maximal solution to the system (1.1) as in Theorem 1.1 Then there exist positive constant such that For the boundedness results of L ₂ and W _1,2 norms of the maximal solution to the system (1.1) we assume the following condition in Theorem 1.3, Theorem 1.4 Theorem 1.3. Suppose that the initial functions u ₀ , v ₀ are in . Also assume the condition (1.2) and (1.3). Let ( u ( x , t ), v ( x , t )) be the maximal solution to the system (1.1) as in Theorem 1.1. Then there exist a positive constant M ₁ = M ₁ (|| u ₀ || ₁ , || v ₀ || ₁ , d_i , a_i , b_i , c_i , i = 1, 2) such that Theorem 1.4. Suppose that the initial functions u ₀ , v ₀ are in . Also assume the condition (1.2) and (1.3). Let ( u ( x , t ), v ( x , t )) be the maximal solution to the system (1.1) as in Theorem 1.1. Then there exist a positive constant M ₂ = M ₂ (|| u ₀ || ₁ , || v ₀ || ₁ , d_i , α_ij , a_i , b_i , c_i , i = 1, 2) such that From the results of Theorems 1.2, 1.3 and 1.4 and the Sobolev embedding inequality we have positive constants M′ = M′ ( d_i , α_ij , a_i , b_i , c_i , i = 1,2) M = M ( d_i , α_ij , a_i , b_i , c_i , i = 1,2) such that for the maximal solution ( u, v ) of (1.1) with the conditions (1.2), (1.3) We also conclude that T = + ∞ from Theorem 1.1. This paper is organized as follows. Section 2 provides preliminaries on differential equations and a few consequences of Gagliardo-Nirenberg interpolation inequality which are necessary for the proofs of Theorems 1.2, 1.3, and 1.4. And Sections 3, 4, and 5 present the proofs of Theorems 1.2, 1.3, and 1.4, respectively. This section introduce the Gagliardo-Nirenberg interpolation inequality and its consequences. Also some preliminary results on the bounds and comparisons of differential equations and inequalities are provided. Theorem 2.1 (Gagliardo-Nirenberg interpolation inequality). Let be a bounded domain with ∂ Ω in C^m. For every function u in W^m,r (Ω), 1 ≤ q , r ≤ ∞ the derivative D^ju , 0 ≤ j < m , satisfies the inequality where for all a in the interval , provided one of the following three conditions : ( The positive constant C depends only on n, m, j, q, r, a. ) Proof. We refer the reader to A. Friedman [5] or L. Nirenberg [6] for the proof of this well-known calculus inequality. Corollary 2.1. There exist positive constants C, , and such that for every function u in Proof. n = 1, m = 1, r = 2, q = 1 satisfy the condition (ii) in Theorem 2.1. Letting j = 0 in this case the necessary condition on p, a for inequality (2.1) becomes From equation (2.5) if p = 4, then if then then if p = 2, then a = 13. Therefore we have inequalities (2.2), (2.3), (2.4). Corollary 2.2. For every function u in Proof. m = 2, r = 2, q = 1 satisfy the condition (ii) in Theorem 2.1. Theorem 2.2 (Young’s Inequality). If a and b are nonnegative real numbers and p and q are positive real numbers such that then The equality hold if and only if a^p = b^q . Theorem 2.3 (Hölder’s Inequality). If are Lebesgue measurable and p, q ∈ [1,∞] are real numbers such that then Lemma 2.1 below presents a few basic inequalities that will be used for the computations in this paper. Lemma 2.1. Let x ≥ 0, y ≥ 0. Then Proof . Inequalities (2.7), (2.8), (2.9) are simply proved. To show inequalities (2.10), (2.11), let g ( x ) = 2 ^k−1 ( x^k + y^k ) − ( x + y ) ^k . Then Hence the function g ( x ) has the critical value 0 at x = y which is the minimum value if k ≥ 1, and the maximum value if 0 < k ≤ 1. Thus we obtain inequalities (2.10) and (2.11). Theorem 2.4 (Picard’s local existence and uniqueness theorem). If f ( x, t ) is a continuous real-valued function that satisfies the Lipschitz condition in some open rectangle R = {( x, t ) | a < x < b, c < t < d } that contains the point ( x ₀ , t ₀ ), then the initial value problem has a unique solution in some closed interval I = [ t ₀ − ε , t ₀ + ε ] where ε > 0. Theorem 2.5. Let f ( x ) be a real-valued differentiable functions defined on an open interval ( a , b ). Then for every initial point x ₀ in ( a , b ) a solution of the initial value problem is either constant or strictly monotone. Proof . The conclusion follow from the fact that f ( x ( t )) never changes sign for the solution x ( t ) of the given initial value problem. To see why this is so, suppose that x ( t ) is not a constant solution, and f ( x ( t )) changes sign. Then it would have to be f ( x ( t ₁ )) = 0 at some t ₁ > 0 and f ( x ( t )) ≠ 0 for t in the left of t ₁ or right of t ₁ . But it contradict the fact that from Theorem 2.4 the constant solution y ( t ) ≡ x ( t ₁ ) is a unique solution in some closed interval [ t ₁ − ε , t ₁ + ε ], where ε > 0. Corollary 2.3. Let c ₁ > 0, p > 1, and c ₂ , c ₃ be any real numbers. Then there exists a positive constant M = M ( x ₀ , p , c ₁ , c ₂ , c ₃ ) such that the solution of the initial value problem satisfies that Proof. The function f ( x ) = − c ₁ x^p + c ₂ x + c ₃ is differentiable functions on and falls in either of the two cases: In case (a) x′ (0) = f ( x ₀ ) ≤ 0, and thus by Theorem 2.5 x ′( t ) ≤ 0 for all t ≥ 0. Hence x ( t ) ≤ x ₀ for all t ≥ 0. In case (b) if 0 < x ₀ < m then the solution x ( t ) cannot cross the constant solution y ( t ) ≤ m by Theorem 2.5, and thus x ( t ) ≤ m for all t ≥ 0. If x ₀ ≥ m then x′ (0) = f ( x ₀ ) ≤ 0, and thus by Theorem 2.5 x ′( t ) ≤ 0 for all t ≥ 0. Hence x ( t ) ≤ x ₀ for all t ≥ 0. Therefore in any case there exists a positive constant M = M ( x ₀ , p , c ₁ , c ₂ , c ₃ ) such that x ( t ) ≤ M for all t ≥ 0. Lemma 2.2 (Gronwall’s inequality and the Comparison Principle for differential equations). Let a < b ≤ ∞, and ξ ( t ) and β ( t ) be real-valued continuous functions defined on the interval [ a, b ]. If ξ ( t ) is differentiable in ( a, b ) and satisfies the differential inequality then ξ ( t ) is bounded by the solution of the corresponding differential equation y ′( t ) = β ( t ) y ( t ), y ( a ) = ξ ( a ), that is, for all t ∈ [ a, b ]. And it follows that if in addition ξ ( a ) ≤ 0, then ξ ( t ) ≤ 0 for all t ∈ [ a, b ]. Proof. We refer the reader to [2] . Lemma 2.3. Let c ₁ > 0, p > 1, and c ₂ , c ₃ be any real numbers. Suppose that two differentiable functions ϕ ( t ) and x ( t ) satisfy Then And especially, if ϕ ₀ ≥ 0 then there exists a positive constant M = M ( ϕ ₀ , p , c ₁ , c ₂ , c ₃ ) such that Proof . Let ξ = ϕ − x . Then ξ ′ = ϕ ′ − x ′ ≤ −c₁(ϕ^p − x^p) + c₂(ϕ − x) = ξ(−c₁η + c₂), where Here notice that η ( t ) is a continuous function using the mean value theorem and the continuities of ϕ ( t ) and x ( t ). Now, since ξ (0) = 0 we conclude that ξ ( t ) = ϕ ( t ) − x ( t ) ≤ 0 for all t ≥ 0 from Lemma 2.2. And if ϕ ₀ ≥ 0, from Corollary 2.3 there exists a positive constant M = M ( ϕ ₀ , p , c ₁ , c ₂ , c ₃ ) such that x ( t ) ≤ M for all t ≥ 0. Thus ϕ ( t ) ≤ M for all t ≥ 0. Proof of Theorem 1.2. By taking integration over the interval [0, 1] for the first and second equations in (1.1) we have that Let The condition b ₁ c ₂ > b ₂ c ₁ implies δ > 0. It also holds that Thus it is shown that from the facts Using (3.1) we have and thus where From Hölder’s inequality it follows that and thus where Hence by the Gronwall’s type inequailty in Lemma 2.3 there exists positive constant M ₀ = M ₀ (|| u ₀ || ₁ , || v ₀ || ₁ , a ₁ , a ₂ , b ₁ , b ₂ , c ₁ , c ₂ ) satisfying for all t ≥ 0. Proof of Theorem 1.3. Multiplying the first and second equations in (1.1) by u = u ( x, t ) and v = v ( x , t ), respectively, and taking integrations over [0, 1] we have that Using Neumann boundary conditions and similarly Using condition (1.3) that and , we have Thus it follows from (4.1) that By Young’s inequality holds for any ε > 0. And by applying Lemma 2.1 to inequality (2.2) and using the uniform L ₁ -boundedness of u and v from Step 1, we have where C is a positive constant depending only on a _i , b _i , c _i ( i, j = 1, 2). Thus (4.2) becomes where and the constants and are depending on d_i , a_i , b_i , c_i ( i, j = 1, 2). And by applying Lemma 2.1 to inequality (2.4) and using the uniform L ₁ -boundedness of u and v from Step 1, we have where is a positive constant depending only on a_i, b_i, c_i ( i, j = 1, 2). And thus where C′ is a positive constant depending only on a_i, b_i, c_i ( i, j = 1, 2). Thus we have by Lemma 2.1, where C ₀ , C ₁ , C ₂ are positive constants d_i, a_i, b_i, c_i ( i, j = 1, 2). Hence by the Gronwall’s type inequailty in Lemma 2.3 we obtain the following L ₂ -bound of u and v such that where M ₁ is a positive constant depending on || u ₀ || ₂ , || v ₀ || ₂ , d_i, a_i, b_i, c_i ( i, j = 1, 2). Proof of Theorem 1.4. To obtain uniform bounds of | u_x | ₂ and | v_x | ₂ for the solution of (1.1) let us denote that We would show the uniform boundedness of | P_x | ₂ and | P_x | ₂ and then obtain the uniform bounds of | u_x | ₂ and | v_x | ₂ from it. Here we note from Theorem 1.1 that for 0 ≤ t < T , and from the Neumann boundary conditions on u, v . Now, multiplying the first equation in (1.1) by P_t and the second equation by Q_t , we have and thus where C ₁ is a positive constant depending on d_i, α_ij, a_i, b_i, c_i ( i, j = 1, 2). Here we notice from the condition (1.3) that there exists a positive constant λ = λ ( α_i,j , i, j = 1, 2) satisfying since for all u ≥ 0, v ≥ 0, if λ = λ ( α_ij , i , j = 1, 2) > 0 is small enough. The terms in (5.1) ae estimated in terms of P and Q from inequality (2.7) in lemma 2.1. Now we observe using Young’s inequality that hold for any ε > 0. Similar estimates are applied to the terms , , , and so on. Using these inequalities and inequalities (2.8), (2.9) in lemma 2.1 we obtain that where C ₂ , C ₃ , C ₄ are positive constant depending on d_i, α_ij, a_i, b_i, c_i ( i, j = 1, 2). Thus we have for any ε > 0. Here we choose a small ε > 0 so that and thus where C ₅ is a positive constants depending on d_i, α_ij, a_i, b_i, c_i ( i, j = 1, 2). Now we observe that and thus where C ₆ is a positive constant depending only on d_i, α_ij , a_i, b_i, c_i ( i, j = 1, 2). Applying the inequalities (2.6) and (2.3) to the function P = d ₁ u + α ₁₁ u ² + α ₁₂ uv we have Here using the uniform boundedness of the L ₁ norm of P , we have where C ₇ , C ₈ , C ₉ , C ₁₀ are positive constants depending on d_i, α_ij, a_i, b_i, c_i ( i, j = 1, 2). Similar estimates are obtained also for Q . Hence we have where C ₁₁ , C ₁₂ , C ₁₃ are positive constants depending on d_i, α_ij, a_i, b_i, c_i ( i, j = 1, 2). Hence by the Gronwall’s type inequailty in Lemma 2.3 we obtain the following W _1,2 -bound of P and Q such that where is a positive constant depending on || u ₀ || ₂ , || v ₀ || ₂ , d_i, α_ij, a_i, b_i, c_i ( i, j = 1, 2). In order to obtain estimates for u_x and v_x , we notice that where Here we note that | A |, the determinant of A , is bounded below by the positive constant d ₁ d ₂ , and | A | is of class O ( u ² + v ² ) as u → ∞ and v → ∞, we have the inequality for some constant C ₁₄ depending only on d_i, α_ij , ( i, j = 1, 2). Therefore we obtain the following W _1,2 -bound of u and v such that where M ₂ is a positive constant depending on || u ₀ || ₂ , || v ₀ || ₂ , d_i, α_ij, a_i, b_i, c_i ( i, j = 1, 2).