A Liouville type theorem for a scaling invariant parabolic system with no gradient structure

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We prove a Liouville-type theorem for semilinear parabolic systems of the form∂tui−Δui=∑j=1mβijuirujp−r,i=1,2,...,m in the whole space RN×R, where r>0, p>r+1 and (βij) is a real (m,m) symmetric matrix with nonnegative entries, positive on the diagonal. Due to the lack of gradient structure, a nontrivial modifications of the techniques of Gidas and Spruck and of Bidaut-Véron is introduced. We expect to see more applications of this method to other problems with no gradient structure.