A LIOUVILLE-TYPE THEOREM FOR COOPERATIVE PARABOLIC SYSTEMS

. We prove Liouville-type theorem for semilinear parabolic system of the form u t − ∆ u = a 11 u p + a 12 u r v s +1 ,v t − ∆ v = a 21 u r +1 v s + a 22 v p where r,s > 0, p = r + s + 1. The real matrix A = ( a ij ) satisﬁes conditions a 12 ,a 21 ≥ 0 and a 11 ,a 22 > 0. This paper is a continuation of Phan-Souplet ( Math. Ann. , 366, 1561-1585, 2016) where the authors considered the special case s = r for the system of m components. Our tool for the proof of Liouville- type theorem is a reﬁnement of Phan-Souplet, which is based on Gidas-Spruck (Commun. Pure Appl.Math. 34, 525–598 1981) and Bidaut-V´eron (´Equations aux d´eriv´ees partielles et applications. Elsevier, Paris, pp 189–198, 1998).


1.
Introduction. In this article, we consider the semilinear parabolic system of the form u t − ∆u = a 11 u p + a 12 u r v s+1 , (x, t) ∈ Ω × I, v t − ∆v = a 21 u r+1 v s + a 22 v p , (x, t) ∈ Ω × I, where r, s > 0, p = r + s + 1, Ω is a domain of R N , and I is an interval of R. Throughout this paper, the real matrix A = (a ij ) is assumed to satisfy the following conditions a 12 , a 21 ≥ 0 and a 11 , a 22 > 0.
System (1) arises in different mathematical models in physics, chemistry and biology. It has been used to describe heat propagations in a two-component combustible mixture [3,9]. In this case, u and v stand for the temperatures of the interacting components. In dynamics of biological groups [6,14], the system (1) models the interaction of two biological groups where the speed of the diffusion is slow. Furthermore, it can be used to describe some models of Bose-Einstein condensation [7], or of chemical processes [12]. So far, system (1) has attracted much attention in various mathematical directions. For instance, the local and global existence was obtained via theory of abstract evolution equations (see e.g. [1]). The results on the regularity, symmetry property, or blow-up phenomena were considered in e.g. [7,10,9]. Recently, the singularity estimates and the Liouville-type theorems have been studied in [15,18,16].
The aim of this paper is to prove the nonexistence of nontrivial solution of problem (1) in the entire space, such a result is called the Liouville-type theorem. This is a continuation of [16] where the author prove the Liouville-type theorem in the special case s = r for the system of m components under the range of p < p B . Here, p B is the Bidaut-Véron exponent We propose to study the system of two components (1) and look for the difficulty arising on the nonexistence of solutions when the exponents r = s. Before stating the main result, let us first recall the elliptic counterpart which arises in mathematical models of physical phenomena, such as nonlinear optics and Bose-Einstein condensation (see e.g. [2,8,21]). It is well known that the Liouville-type result for (3) plays an important role in the study of both elliptic and parabolic problems. The optimal Liouville-type theorem for problem (3) was completely proved in [20] (see also [13]) via moving sphere techniques under the optimal range p < p S , where p S is Sobolev exponent For the corresponding parabolic problem (1), the Liouville property is less understood. First, by adding up the two equations and using Young's inequality, one can easily reduce to a scalar parabolic inequality and deduce the Fujita-type result of problem (1), which asserts that, there is no positive solution in R N × R + if 1 < p ≤ 1 + 2 N . The Liouville-type theorem for problem (1) under the restrictions a 11 = a 22 = 1, a 12 = a 21 , r = s has been proved in [15] for the class of radial solutions in any dimension. Recently, Quittner [18] has provided a new important technique to prove the Liouville-type theorem for parabolic systems with gradient structure, under the condition (N − 2)p < N . More precisely, the result of Quittner [18,Theorem 3] is formulated for parabolic system of the form U t −∆U = F (U ), where U = (u 1 , u 2 , ..., u m ) and F satisfies the conditions: there exists ξ ∈ (0, ∞) m such that ξ · F (U ) > 0 for U = 0.
Then the system U t − ∆U = F (U ) does not possess any nontrivial nonnegative classical solution in R N × R.
In particular, the Liouville-type result of [18] is optimal in dimensions N ≤ 2, and there is an additional condition p < N N −2 in dimension N ≥ 3. The main tools in [18] are scaling argument and energy estimates. We note that, by a simple scaling, one can reduce the system (1) to a parabolic system with gradient structure as in Theorem A. More recently, Phan and Souplet [16] have used a different approach to establish a Liouville-type theorem for problem (1) in a larger range of p in dimension N ≥ 3, under the restrictions r = s and a 12 = a 21 .
In this paper, we establish a Liouville-type theorem for problem (1) in general case which allows r = s and/or a 12 = a 21 . Our main result is as follows.
Then, under the assumption (2), the system (1) does not possess any nontrivial nonnegative classical solution in R N × R.
Due to the fact that N N −2 < p B when N ≥ 3, our result is a partial improvement of that in Quittner [18] in high dimensions (as seen for system of two components). Our tool for the proof of Theorem 1.1 is a refinement of [16], which is essentially based on Gidas-Spruck technique [11] developed by Bidaut-Véron [4] (see also [5] for elliptic system). This technique consists of nonlinear integral estimates and Bochner formula. We remark that, for the Liouville-type theorem, the condition p < p B is the best up-to-date one for parabolic problem in dimension N ≥ 3, even for the simplest model u t − ∆u = u p .
We stress that the proof of Theorem 1.1 is not straightforward in comparison with that in [16]. The only main difficulty arising in this paper is the presence of different exponents r and s. This makes the Gidas-Spruck and Bidaut-Véron techniques become more delicate. In addition, it requires a suitable combination of nonlinear intergral estimates on each component, see the Lemma 2.1. The rest of the proof is similar to that in [16].
From Theorems 1.1, one can deduce the singularity estimates by rescaling and doubling arguments. We just give here the result without proof since it is totally similar to that in [17,Theorem 3.1].
We note that the condition p < p B in Proposition 1 can be removed if the solution is radial and Ω is a symmetric domain (see [15,Section 2]). By a symmetric domain, we mean either the whole space R N , a ball B R = B(0, R), an annulus {x ∈ R N : R 1 < |x| < R 2 }, or an exterior domain {x ∈ R N : |x| > R}.
The rest of this paper is devoted to the proof of Theorem 1.1.
Then there holds where C = C(N, r, s, A).
Using (19) and the Young inequality, for any ε > 0, we can control the RHS of (5) by the following estimates.
By taking η = ε 2 and choosing ε sufficiently small, we obtain I, L ≤ C and the Lemma follows.
Proof of Theorem 1.1. We recall that, by the same argument of Quittner [18], Theorem is true for N ≤ 2. Moreover, Theorem is straightforward if a 12 = 0 or a 21 = 0 since it is reduced to scalar equation. We then assume that a 12 > 0, a 21 > 0 and N ≥ 3. We first consider the case of bounded solutions. Assume for contradiction that (u, v) is a nontrivial, bounded, nonnegative solution of (1) in R N × R. Since the components u and v are supersolutions of the heat equation, it follows from the strong maximum principle that either (u, v) is positive in R N × R, or there exists t 0 ∈ R such that u = v = 0 in R N × (−∞, t 0 ]. In the later case, since (u, v) is bounded, we have ∂ t u − ∆u ≤ C(u + v), ∂ t v − ∆v ≤ C(u + v) for some constant C > 0. The maximum principle (see e.g. [16, Proposition 6.1]) then guarantees that u = v = 0 in R N × (t 0 , ∞), hence u ≡ v ≡ 0. Consequently, we may assume without loss of generality that (u, v) is positive in R N × R.

ANH TUAN DUONG AND QUOC HUNG PHAN
Then (u R , v R ) is also a solution to (1). By Lemma 2.2, we have |y|<R/2 |s|<R 2 /2 (u 2p + v 2p )(y, s)dyds Letting R → ∞ and noting p < p B ≤ p S , we deduce that u ≡ v ≡ 0. This is a contradiction.
Finally, for the general case, we recall that the Liouville-type property of Theorem 1.1 for bounded solutions is sufficient for the proof of Proposition 1. But after a time shift, formula (4) in Proposition 1 guarantees that any solution of (1) in R N × (−T, T ) always satisfies u(x, t) + v(x, t) ≤ CT −1/(p−1) in R N × (−T /2, T /2). The conclusion then follows by letting T → ∞.