The well-known Lane–Emden conjecture indicates that, for the elliptic Lane–Emden system $-\Delta u=v^p$, $-\Delta v=u^q$ in ${\mathbb{R}}^N$, the Sobolev hyperbola $\frac{1}{p+1}+\frac{1}{q+1}=\frac{N-2}{N}$ is expected as the critical curve for the existence and nonexistence of entire solutions. In this paper, we study the periodic Lane–Emden heat flow system $u_t-\Delta u=a\left(t\right)v^p$, $v_t-\Delta v=b\left(t\right)u^q$ in a bounded domain $\Omega$ of ${\mathbb{R}}^N$, subject to homogeneous Dirichlet boundary condition. We will show that the Sobolev hyperbola is also a critical curve for the existence and nonexistence of periodic solutions. Moreover, if $pq=1$, the nontrivial periodic solutions may exist or not exist.

著名的 Lane-Emden 猜想表明，对于椭圆的 Lane-Emden 系统 $-\Delta 你=v^{磷}$, $-\Delta v={你}^q$ 在 ${\mathbb{电阻}}^N$, 索博列夫双曲线 $\frac{1}{磷+1}+\frac{1}{q+1}=\frac{N-2}{N}$预期为整个解存在和不存在的临界曲线。在本文中，我们研究了周期 Lane-Emden 热流系统${你}_{吨}-\Delta 你=\mathrm{一种}\left(吨\right)v^{磷}$, $v_{吨}-\Delta v=乙\left(吨\right){你}^q$ 在有界域中 $\Omega$ 的 ${\mathbb{电阻}}^N$，服从齐次狄利克雷边界条件。我们将证明 Sobolev 双曲线也是周期解存在和不存在的临界曲线。此外，如果$磷q=1$，非平凡周期解可能存在也可能不存在。