1. INTRODUCTION
This article deals with the following quasilinear parabolic system in population dynamics which is called cooperative cross-diffusion system.
where α12, α21, d, ai, bi, ci are positive constants for i = 1, 2. The initial functions u0, v0 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. d1 and d2 are the diffusion rates of the two species, respectively. a1 and a2 denote the intrinsic growth rates, b1 and c2 account for intra-specific cooperative pressures, b2 and c1 are the coefficients for inter-specific competitions. α11 and α22 are usually referred as self-diffusion, and α12, α21 are cross-diffusion pressures. By adopting the coefficients αij (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 Lp(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 u0 and v0 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, u0 and v0 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 u0, v0 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
M0 = M0(||u0||1, ||v0||1, a1,a2,b1,b2,c1,c2)
such that
sup{||u(·, t)||1, ||v(·, t)||1 : t ∈ [0, T)} ≤ M0
For the boundedness results of L2 and W1,2 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 u0, v0 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 M1 = M1(||u0||1, ||v0||1, di,ai,bi,ci,i = 1, 2) such that
sup{||u(·, t)||2, ||v(·, t)||2 : t ∈ [0, T)} ≤ M1.
Theorem 1.4. Suppose that the initial functions u0, v0 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 M2 = M2(||u0||1, ||v0||1, di,αij,ai,bi,ci,i = 1, 2) such that
sup{||u(·, t)||1,2, ||v(·, t)||1,2 : t ∈ [0, T)} ≤ M2.
From the results of Theorems 1.2, 1.3 and 1.4 and the Sobolev embedding inequality we have positive constants M′ = M′(di,αij,ai,bi,ci,i = 1,2) M = M(di,αij,ai,bi,ci,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.
2. PRELIMINARIES
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 Cm. For every function u in Wm,r (Ω), 1 ≤ q, r ≤ ∞ the derivative Dju, 0 ≤ j < m, satisfies the inequality
where for all a in the interval , provided one of the following three conditions :
(i) r ≤ q, (ii) , or (iii) and is not a nonnegative integer.(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 ap = bq.
Theorem 2.3 (Hölder’s Inequality). If are Lebesgue measurable and p, q ∈ [1,∞] are real numbers such that then
|fg|1 ≤ |f|p|g|q.
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) = 2k−1(xk + yk) − (x + y)k. Then
g′(x) = k2k−1xk−1 − k(x + y)k−1.
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
|f(x, t) − f(y, t)| ≤ L|x − y|
in some open rectangle R = {(x, t) | a < x < b, c < t < d} that contains the point (x0, t0), then the initial value problem
x′ = f(x, t), x(t0) = x0
has a unique solution in some closed interval I = [t0 − ε, t0 + 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 x0 in (a, b) a solution of the initial value problem
x′ = f(x), x(0) = x0
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(t1)) = 0 at some t1 > 0 and f(x(t)) ≠ 0 for t in the left of t1 or right of t1. But it contradict the fact that from Theorem 2.4 the constant solution y(t) ≡ x(t1) is a unique solution in some closed interval [t1− ε, t1 + ε], where ε > 0.
Corollary 2.3. Let c1 > 0, p > 1, and c2, c3 be any real numbers. Then there exists a positive constant M = M(x0, p, c1, c2, c3) such that the solution of the initial value problem
x′ = −c1xp + c2x + c3, x(0) = x0 ≥ 0
satisfies that
x(t) ≤ M for all t ≥ 0.
Proof. The function f(x) = −c1xp + c2x + c3 is differentiable functions on and falls in either of the two cases:
case(a) f(x) ≤ 0 for all x ≥ 0 case(b) there exist a positive constant m = m(p, c1, c2, c3) such that f(m) = 0, f(x) > 0 for x in some interval on the left of m, and f(x) < 0 for all x > M.
In case (a) x′(0) = f(x0) ≤ 0, and thus by Theorem 2.5 x′(t) ≤ 0 for all t ≥ 0. Hence x(t) ≤ x0 for all t ≥ 0. In case (b) if 0 < x0 < 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 x0 ≥ m then x′(0) = f(x0) ≤ 0, and thus by Theorem 2.5 x′(t) ≤ 0 for all t ≥ 0. Hence x(t) ≤ x0 for all t ≥ 0. Therefore in any case there exists a positive constant M = M(x0, p, c1, c2, c3) 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
ξ′(t) ≤ β(t)ξ(t), t ∈ (a, b),
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 c1 > 0, p > 1, and c2, c3 be any real numbers. Suppose that two differentiable functions ϕ(t) and x(t) satisfy
ϕ′ ≤ −c1ϕp + c2ϕ + c3, ϕ(0) = ϕ0 x′ = −c1xp + c2x + c3, x(0) = ϕ0.
Then
ϕ(t) ≤ x(t) for all t ≥ 0.
And especially, if ϕ0 ≥ 0 then there exists a positive constant M = M(ϕ0, p, c1, c2, c3) such that
ϕ(t) ≤ M for all t ≥ 0.
Proof. Let ξ = ϕ − x. Then
ξ′ = ϕ′ − x′ ≤ −c1(ϕp − xp) + 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 ϕ0 ≥ 0, from Corollary 2.3 there exists a positive constant M = M(ϕ0, p, c1, c2, c3) such that x(t) ≤ M for all t ≥ 0. Thus ϕ(t) ≤ M for all t ≥ 0.
3. L1-BOUND OF SOLUTIONS TO (1.1)
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 b1c2 > b2c1 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 M0 = M0(||u0||1, ||v0||1, a1,a2,b1,b2,c1,c2) satisfying
for all t ≥ 0.
4. L2-BOUND OF SOLUTIONS TO (1.1)
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 L1-boundedness of u and v from Step 1, we have
where C is a positive constant depending only on ai, bi, ci (i, j = 1, 2). Thus (4.2) becomes
where and the constants and are depending on di, ai, bi, ci (i, j = 1, 2). And by applying Lemma 2.1 to inequality (2.4) and using the uniform L1-boundedness of u and v from Step 1, we have
where is a positive constant depending only on ai, bi, ci (i, j = 1, 2). And thus
where C′ is a positive constant depending only on ai, bi, ci (i, j = 1, 2). Thus we have
by Lemma 2.1, where C0, C1, C2 are positive constants di, ai, bi, ci (i, j = 1, 2). Hence by the Gronwall’s type inequailty in Lemma 2.3 we obtain the following L2-bound of u and v such that
where M1 is a positive constant depending on ||u0||2, ||v0||2, di, ai, bi, ci (i, j = 1, 2).
5. W1,2-BOUND OF SOLUTIONS TO (1.1)
Proof of Theorem 1.4. To obtain uniform bounds of |ux|2 and |vx|2 for the solution of (1.1) let us denote that
P = d1u + α11u2 + α12uv, Q = d2v + α21uv + α22v2.
We would show the uniform boundedness of |Px|2 and |Px|2 and then obtain the uniform bounds of |ux|2 and |vx|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 Pt and the second equation by Qt, we have
and thus
where C1 is a positive constant depending on di, αij, ai, bi, ci (i, j = 1, 2). Here we notice from the condition (1.3) that there exists a positive constant λ = λ(αi,j, i, j = 1, 2) satisfying
since
for all u ≥ 0, v ≥ 0, if λ = λ(αij, 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 C2, C3, C4 are positive constant depending on di, αij, ai, bi, ci (i, j = 1, 2). Thus we have
for any ε > 0. Here we choose a small ε > 0 so that and thus
where C5 is a positive constants depending on di, αij, ai, bi, ci (i, j = 1, 2). Now we observe that
P = d1u + α11u2 + α12uv ≥ α11u2, Q = d2v + α21uv + α22v2 ≥ α22v2,
and thus
where C6 is a positive constant depending only on di, αij , ai, bi, ci (i, j = 1, 2). Applying the inequalities (2.6) and (2.3) to the function P = d1u + α11u2 + α12uv we have
Here using the uniform boundedness of the L1 norm of P, we have
where C7, C8, C9, C10 are positive constants depending on di, αij, ai, bi, ci (i, j = 1, 2). Similar estimates are obtained also for Q. Hence we have
where C11, C12, C13 are positive constants depending on di, αij, ai, bi, ci (i, j = 1, 2). Hence by the Gronwall’s type inequailty in Lemma 2.3 we obtain the following W1,2-bound of P and Q such that
where is a positive constant depending on ||u0||2, ||v0||2, di, αij, ai, bi, ci (i, j = 1, 2).
In order to obtain estimates for ux and vx, we notice that
where
Here we note that |A|, the determinant of A, is bounded below by the positive constant d1d2, and |A| is of class O(u2 + v2) as u → ∞ and v → ∞, we have the inequality
|ux| + |vx| ≤ C14 (|Px| + |Qx|) for every x ∈ [0, 1] × [0,∞)
for some constant C14 depending only on di, αij, (i, j = 1, 2). Therefore we obtain the following W1,2-bound of u and v such that
where M2 is a positive constant depending on ||u0||2, ||v0||2, di, αij, ai, bi, ci (i, j = 1, 2).
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