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In this paper, we are concerned with the problem for a periodic parabolic systemwhere  p ,  q &gt;0, Ω is a bounded and smooth domain in,andare properly smooth, bounded and positive periodic functions with periodicity  ω . It is known that for the elliptic systemin, there exists a Sobolev hyperbola, whereis defined by (3), which characterizes the existence and nonexistence of nontrivial nonnegative solutions. By using the classical blowing-up method, Liouville-type theorem, Leray–Schauder fixed point theory and Pohozaev identity, we will prove that the Sobolev hyperbola is also a critical curve for the existence and non-existence of the positive periodic solutions for the periodic parabolic system.

Liouville-type theorem of positive periodic solutions for the periodic parabolic system