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Applicable Analysis
An International Journal
Volume 102, 2023 - Issue 15
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Articles

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

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Pages 4286-4305 | Received 19 Dec 2021, Accepted 15 Jul 2022, Published online: 08 Aug 2022
 

ABSTRACT

In this paper, we are concerned with the problem for a periodic parabolic system {utΔu=a(t)upvq+1,xΩ,t>0,vtΔv=b(t)up+1vq,xΩ,t>0,u(x,t)=0,v(x,t)=0,xΩ,t>0,u(x,t+ω)=u(x,t),v(x,t+ω)=v(x,t),xΩ,t0,where p, q>0, Ω is a bounded and smooth domain in RN, a(t) and b(t) are properly smooth, bounded and positive periodic functions with periodicity ω. It is known that for the elliptic system Δu=upvq+1,Δv=up+1vq in RN, there exists a Sobolev hyperbola p+q+1=2, where 2 is 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.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This research was supported by National Natural Science Foundation of China [grant number 11871278], [grant number 11571093].

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