We study a generalized class of supersolutions, so-called p-supercaloric functions, to the parabolic p-Laplace equation. This class of functions is defined as lower semicontinuous functions that are finite in a dense set and satisfy the parabolic comparison principle. Their properties are relatively well understood for \(p\ge 2\), but little is known in the fast diffusion case \(1<p<2\). Every bounded p-supercaloric function belongs to the natural Sobolev space and is a weak supersolution to the parabolic p-Laplace equation for the entire range \(1<p<\infty \). Our main result shows that unbounded p-supercaloric functions are divided into two mutually exclusive classes with sharp local integrability estimates for the function and its weak gradient in the supercritical case \(\frac{2n}{n+1}<p<2\). The Barenblatt solution and the infinite point source solution show that both alternatives occur. Barenblatt solutions do not exist in the subcritical case \(1<p\le \frac{2n}{n+1}\) and the theory is not yet well understood.

我们研究了抛物线p -Laplace方程的一类广义超解，即所谓的p-超热函数。此类函数定义为在密集集中有限且满足抛物线比较原理的下半连续函数。对于\（p \ ge 2 \），它们的性质相对比较好理解，但在快速扩散情况\（1 <p <2 \）中知之甚少。每个有界p-超热函数都属于自然Sobolev空间，并且对于整个范围\（1 <p <\ infty \）是抛物线p -Laplace方程的弱超解。我们的主要结果表明，无界p-超热函数被分为两个互斥的类，它们在超临界情况\（\ frac {2n} {n + 1} <p <2 \）中具有清晰的局部可积估计值及其弱梯度。Barenblatt解和无限点源解表明这两种选择都发生了。在次临界情况\（1 <p \ le \ frac {2n} {n + 1} \）中不存在Barenblatt解，并且该理论尚未得到很好的理解。