A NECESSARY AND SUFFICIENT CONDITION FOR GLOBAL EXISTENCE FOR A QUASILINEAR REACTION-DIFFUSION SYSTEM

We show that the reaction-diffusion system u t = Δ φ ( u ) + f ( v ) , v t = Δ ψ ( v ) + g ( u ) , with homogeneous Neumann boundary conditions, has a positive global solution on Ω × [ 0 , ∞ ) if and only if ∫ ∞ d s / f ( F − 1 ( G ( s ) ) ) = ∞ (or, equivalently, ∫ ∞ d s / g ( G − 1 ( F ( s ) ) ) = ∞ ), where F ( s ) = ∫ 0 s f ( r ) d r and G ( s ) = ∫ 0 s g ( r ) d r . The domain Ω ⊆ ℝ N ( N ≥ 1 ) is bounded with smooth boundary. The functions φ , ψ , f , and g are nondecreasing, nonnegative C ( [ 0 , ∞ ) ) functions satisfying φ ( s ) ψ ( s ) f ( s ) g ( s ) > 0 for s > 0 and φ ( 0 ) = ψ ( 0 ) = 0 . Applied to the special case f ( s ) = s p and g ( s ) = s q , p > 0 , q > 0 , our result proves that the system has a global solution if and only if p q ≤ 1 .

Reaction-diffusion systems have been studied for decades (see, e.g., [4,5,10,11], and their references). The particular problem of determining conditions under which such systems have global solutions has been the object of studies for almost as long. (See [6,7,8,9,10,11,15,16,17,18,19] and their references.) For the system (p > 0, q > 0) (1.5) it is well known that the existence of global solutions in general depends on more than just the values of the exponents p and q. In particular, when homogeneous Dirichlet boundary conditions are imposed, it is well known [6,8] that for pq ≤ 1, the system has only global solutions, but if pq > 1, the system will have a global solution for "small" initial data but not for "large" initial data. A similar phenomenon occurs for the Cauchy problem [5,7]. However, our results (see Theorems 2.1 and 3.2) show that this cannot occur with homogeneous Neumann boundary data, where blowup (i.e., no global solution) depends exclusively on the reaction terms and occurs (for (1.5)) precisely if pq > 1. We show that this is true also in the presence of nonlinear diffusion. Thus the existence of a global solution is also independent of the diffusion term, although the diffusion rate may well determine the nature of blowup as in the scalar case (see [12,14]). On the other hand, with homogenous Dirichlet boundary data, Galaktionov et al. [10,11] have shown that the quasilinear system has only global solutions if pq < (1 + µ)(1 + ν), but for pq ≥ (1 + µ)(1 + ν), the existence of global solutions depends on the initial data and the size of the domain. In the present case, this does not occur. Indeed applying Theorem 3.2 to the system (1.6) with homogeneous Neumann boundary conditions, we find that a global solution exists if and only if pq ≤ 1. We note also that some authors (e.g., [9,16]) have been concerned with whether a diffusion-free system can have a global solution while the corresponding diffusive system does not. Obviously, this cannot occur with the present system.

Smooth constitutive functions
Before establishing the general case, we first consider the case where the constitutive functions and the initial and boundary data are smooth. Thus we prove the following theorem. Before proving this, we establish a preliminary lemma. Proof. Necessity. Without loss of generality, assume that a > 0, and problem (1.4) has solution (y,z). Then y g(y) = z f (z), which gives d/dt[G(y) − F(z)] = 0. Thus there is a constant K so that G(y) = F(z) + K, and clearly from the initial values of y and z, we get Letting t → ∞, we establish condition (1.3). If b = 0, then for every ε > 0, we get d dt which, after integrating from δ > 0 to t, gives Letting δ → 0 gives which implies that the integral on the right converges, and hence letting ε → 0, we establish that (2.1) holds for b = 0. The proof now continues as in the case b > 0.
We need to prove that the problem (1.4) has a classical solution. Once again, we assume that a > 0. Define Clearly, H(a) = 0, H (s) > 0 for s > 0. Thus H is one-to-one and from (1.3), which holds for F replaced by F and G replaced by G, . We now show that 1812 Reaction-diffusion system y, z satisfy (1.4). Clearly, y(0) = a and H(y(t)) = t. Thus H (y(t))y (t) = 1 so that This completes the proof.
Proof of Theorem 2.1. Necessity. Suppose that problem (1.1) has a nonnegative classical solution (u,v). From [13, Theorem 5.1], there exist T 0 > 0 and a > 0 such that . We now consider the system (2.7) We will show that this system has a solution, and then invoke Lemma 2.2 to yield that (1.3) holds, which will complete the proof of necessity. Clearly, the system (2.7) has a solution on some, perhaps small, interval. Let t 0 > T 0 be the supremum of all values τ such that a solution exists on [T 0 ,τ). If t 0 = ∞, then the system (2.7) has a solution and (1.3) holds as a result of Lemma 2.2. Thus suppose that t 0 < ∞. We will first show that Thus suppose that there exists ( x,T) ∈ Ω × [T 0 ,t 0 ), where at least one of the two inequalities (2.8) fails to hold. Let ζ ∈ C 2 (Ω) such that ∂ ν ζ < 0 on ∂Ω and ζ ≥ 1 on Ω. Clearly, inequalities (2.8) hold for t near T 0 since they hold for Clearly t 1 > T 0 and at t = t 1 , either W − α or Z − β is zero for some x 0 ∈ Ω. Without loss of generality, we assume that it is , we must have x 0 ∈ Ω, and hence ∇Z(x 0 ,t 1 ) = 0 and ∆Z(x 0 ,t 1 ) ≥ 0. Thus, at (x 0 ,t 1 ), we have the following: (2.11) We thus arrive at a contradiction. Therefore inequalities (2.8) hold. Hence, the solution of (2.7) can be extended to an interval [0,t * 0 ), where t * 0 > t 0 . This contradicts the fact that t 0 is the supremum of all such values. Therefore, our assumption that t 0 < ∞ cannot hold. Thus (2.7) has a global solution, and therefore Lemma 2.2 implies that (1.3) holds.

Nonsmooth constitutive functions
We now consider the case where the data and constitutive functions are not smooth. In this case, it is well known that the system (1.1) does not, in general, have a classical solution even in the case of a single equation (see, e.g., [1]). Therefore, we will consider a weak formulation of a solution motivated by Bénilan et al. [3] and similar to that of [14].
Definition 3.1. The sequence of problems ∂u n ∂t = ∆ϕ n u n + f n v n , ∂v n ∂t = ∆ψ n v n + g n u n in Q ∞ , is called a sequence of approximating problems for (1.1) if We prove the following theorem. Since much of the proof that follows is like that of Theorem 2.1 above, we merely point out important differences.
Proof. Necessity. Suppose that problem (1.1) has a generalized solution (u,v). Let (u n , v n ) be a sequence of approximating solutions, thus satisfying (3.1). Since u 0 and v 0 are strictly positive on Ω and the sequence {(u 0,n ,v 0,n )} converges uniformly on Ω, there exists a subsequence, which, for convenience, we will assume is the sequence itself, for which there exists a positive constant a such that min{u n (x,t),v n (x,t)} > a on Ω × [0,∞). (We note that in the smooth case (Theorem 2.1), the initial data did not need to be strictly positive. However, for nonsmooth data and constitutive functions, it is unknown whether a generalized solution with nonnegative, nontrivial initial data ever becomes strictly positive at a later time.) The proof may now proceed as with Theorem 2.1 (with T 0 = 0) to prove α n (t) < u n (x,t), β n (t) < v n (x,t) on Ω × [0,∞), (3.4) where (α n ,β n ) is the solution to system (2.7) with f and g replaced with f n and g n , respectively, and hence for all T > 0, We can now use this inequality to show that (2.7) has a solution on [0,∞). To do this, we note that there is an interval, perhaps small, on which a solution (α,β) to (2.7) exists. In fact, from the proof of Lemma 2. . Therefore, (α n ,β n ) → (α,β) as n → ∞ uniformly on compact subsets of [0, t 0 ), and hence (α,β) must satisfy (3.7) Therefore, the functions α and β must be defined on [0,∞). Indeed, the only way that α and β can fail to exist at t 0 is for lim t→t − 0 α(t) = ∞ (similarly for β), which is impossible because of (3.7). Therefore α and β exist on [0, t 0 ] and can be extended to a larger interval [0,t 0 + ε), which contradicts the fact that t 0 was the extent of the existence. Therefore, we must have t 0 = ∞ so that (1.3) holds.
Sufficiency. Now suppose that (1.3) holds. We show that problem (1.1) has a nonnegative generalized solution. We choose sequences { f n }, {g n }, {ϕ n }, {ψ n }, {u 0,n }, and {v 0,n } as specified in the definition of a generalized solution. Such sequences are not difficult to construct using mollifiers and the properties of the functions f , g, ϕ, ψ, u 0 , and v 0 . Furthermore, the sequences { f n }, {g n } may be (and are) chosen so that for each n, they satisfy (1.3) with f and g replaced by f n and g n , respectively. Let (u n ,v n ) be the smooth solution of (3.1), and let (y n ,z n ) be the solution of y n (t) = 2 f n z n (t) , z n (t) = 2g n y n (t) , 0< t < ∞, y n (0) = z n (0) = M + 1, (3.8) where M = sup n ( u 0,n ∞,Ω + v 0,n ∞,Ω ). It is then clear that, as in the proof of (2.12), Also, since (y n ,z n ) converges locally uniformly to (y,z), we know that (y n ,z n ) is locally Alan V. Lair 1817 bounded so that (3.3) holds. To complete the proof, we will prove that the sequence {(u n ,v n )} has a subsequence {(U n ,V n )} defined on Q ∞ and obviously satisfying (3.3), which converges weakly in L 1 (Q T ) to a function pair (u,v) for all T > 0. To do this, we note that (3.3) implies that {(u n ,v n )} is pointwise bounded on Q T which, in turn, implies that for each k ∈ N, the L 2 (Q k ) norm (and every L p (Q k ) norm for p ≥ 1) of the sequence {(u n ,v n )} is bounded. In particular, the L 2 (Q 1 ) norm is bounded independent of n so the sequence {(u n ,v n )} has a weakly convergent subsequence in L 2 (Q 1 ). We denote this subsequence by {(u n,1 ,v n,1 )}, and we let (P 1 ,R 1 ) be its weak L 2 (Q 1 ) limit. Likewise, the sequence {(u n,1 ,v n,1 )} is bounded in the L 2 (Q 2 ) norm, and hence has a subsequence {(u n,2 ,v n,2 )} which is weakly convergent to a function pair (P 2 ,R 2 ) in L 2 (Q 2 ). Clearly (P 1 ,R 1 ) = (P 2 ,R 2 ) on Q 1 . We continue the process to produce for each k ∈ N the sequence {(u n,k ,v n,k )}, a subsequence of {(u n,k−1 ,v n,k−1 )}, which is weakly convergent to (P k ,R k ) in L 2 (Q k ) and (P k ,R k ) = (P k−1 ,R k−1 ) on Q k−1 . Clearly the sequence (P k ,R k ) converges weakly in L 2 (Q T ) for all T > 0 to the function pair (u,v) defined on Q ∞ by (u,v) = (P j ,R j ) on Q j , j ∈ N. In addition, it is easy to prove that the sequence of diagonal entries of the double-indexed sequence {(u n,k ,v n,k )}, namely {(u n,n ,v n,n )}, converges weakly in L 2 (Q T ), and hence weakly in L 1 (Q T ) to (u,v) for all T > 0. Thus the desired sequence {(U n ,V n )} of approximating solutions which converges to (u,v) is {(u n,n ,v n,n )}, and therefore (u,v) is a generalized solution of (1.1). This completes the proof.
An open problem. We note that there is an important difference regarding the initial data in the hypothesis of Theorem 2.1, the smooth case, and Theorem 3.2, the nonsmooth case. In the latter, the initial data is required to be strictly positive, whereas in the former it needs only to be nonnegative and nontrivial. This leaves open the problem: can Theorem 3.2 be extended to the case where u 0 and v 0 are merely nonnegative with at least one of them nontrivial? With smooth constitutive functions, the solution will, in time, become strictly positive with only nonnegative nontrivial initial data. It is unknown whether this will ever occur in the nonsmooth case. However, it may be possible that a different proof can be devised, as in the scalar case [14], where Theorem 3.2 can be extended.