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

ABSTRACT

In this paper, we are concerned with the problem for a periodic parabolic system $\{\begin{array}{ll}{\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }t}}-\mathrm{\Delta }u=a(t)u^p{v}^{q+1},& x\in \mathrm{\Omega },t>0,\\ {\displaystyle \frac{\mathrm{\partial }v}{\mathrm{\partial }t}}-\mathrm{\Delta }v=b(t)u^{p+1}{v}^q,& x\in \mathrm{\Omega },t>0,\\ u(x,t)=0,v(x,t)=0,& x\in \mathrm{\partial }\mathrm{\Omega },t>0,\\ u(x,t+\omega )=u(x,t),v(x,t+\omega )=v(x,t),& x\in \mathrm{\Omega },t\ge 0,\end{array}$where p, q>0, Ω is a bounded and smooth domain in ${\mathbb{R}}^N$, $a(t)$ and $b(t)$ are properly smooth, bounded and positive periodic functions with periodicity ω. It is known that for the elliptic system $-\mathrm{\Delta }u=u^p{v}^{q+1},-\mathrm{\Delta }v=u^{p+1}{v}^q$ in ${\mathbb{R}}^N$, there exists a Sobolev hyperbola $p+q+1=2^{\ast }$, where $2^{\ast }$ 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.

KEYWORDS:

• Periodic parabolic system
• Sobolev hyperbola
• blowing-up method
• Leray–Schauder fixed point theory
• Pohozaev identity

MSC2020:

• 35A01
• 35A16
• 35B53
• 35B09

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].