A Liouville-type theorem for cooperative parabolic systems

Abstract

We prove Liouville-type theorem for semilinear parabolic system of the form ut − ∆u = a 11 u p + a 12 u r v s+1 , vt − ∆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 Liouvilletype theorem is a refinement of Phan-Souplet, which is based on Gidas-Spruck (Commun.