A LIOUVILLE-TYPE THEOREM FOR COOPERATIVE PARABOLIC SYSTEMS

We prove Liouville-type theorem for semilinear parabolic system of the form u(t) - Delta u = a(11)u(p) + a(12)u(r)v(s+1) , v(t) - Delta 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-Veron (Equations aux derivees partielles et applications. Elsevier, Paris, pp 189-198, 1998).