In this paper, we establish some sharp Liouville type theorems for nonnegative weak supersolutions in ${\mathbb{R}}^N\times \mathbb{R}$ of porous medium equation $u_t-\mathrm{\Delta }{u}^m=u^p,$ and of porous medium system $\{\begin{array}{l}{u}_t-\mathrm{\Delta }{u}^m=v^p\\ v_t-\mathrm{\Delta }{v}^m=u^q\end{array},$ where p, q>m>1. More precisely, we show that the sharp conditions for nonexistence of supersolutions to the scalar equation and the system are respectively given by $N\le \frac{2}{p-m}$ and $N\le max(\frac{2(p+1)}{p(q+1)-m(p+1)},\frac{2(q+1)}{q(p+1)-m(q+1)}).$

中文翻译：

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有源多孔介质系统的最优刘维尔型定理

在本文中，我们建立了一些关于非负弱超解的尖锐刘维尔型定理${\mathbb{R}}^{否}\times \mathbb{R}$多孔介质方程${你}_{吨}-\mathrm{△}{你}^{米}={你}^p,$和多孔介质系统$\{\begin{array}{l}{你}_{吨}-\mathrm{△}{你}^{米}=v^p\\ v_{吨}-\mathrm{△}{v}^{米}={你}^q\end{array},$其中p , q > m >1。更准确地说，我们表明标量方程​​和系统不存在超解的尖锐条件分别由下式给出$否\le \frac{\mathrm{2个}}{p-米}$和$否\le 最大限度(\frac{\mathrm{2个}(p+\mathrm{1个})}{p(q+\mathrm{1个})-米(p+\mathrm{1个})},\frac{\mathrm{2个}(q+\mathrm{1个})}{q(p+\mathrm{1个})-米(q+\mathrm{1个})}).$