We consider the natural time-dependent fractional p-Laplacian equation posed in the whole Euclidean space, with parameter \(1<p<2\) and fractional exponent \(s\in (0,1)\). Rather standard theory shows that the Cauchy Problem for data in the Lebesgue \(L^q\) spaces is well posed, and the solutions form a family of non-expansive semigroups with regularity and other interesting properties. The superlinear case \(p>2\) has been dealt with in a recent paper. We study here the “fast” regime \(1<p<2\) which is more complex. As main results, we construct the self-similar fundamental solution for every mass value M and any p in the subrange \(p_c=2N/(N+s)<p<2\), and we show that this is the precise range where they can exist. We also prove that general finite-mass solutions converge towards the fundamental solution having the same mass, and convergence holds in all \(L^q\) spaces. Fine bounds in the form of global Harnack inequalities are obtained. Another main topic of the paper is the study of solutions having strong singularities. We find a type of singular solution called Very Singular Solution that exists for \(p_c<p<p_1\), where \(p_1\) is a new critical number that we introduce, \(p_1\in (p_c,2)\). We extend this type of singular solutions to the “very fast range” \(1<p<p_c\). They represent examples of weak solutions having finite-time extinction in that lower p range. We briefly examine the situation in the limit case \(p=p_c\). Finally, we show that very singular solutions are related to fractional elliptic problems of the nonlinear eigenvalue type, of interest in their own right.

我们考虑在整个欧几里得空间中提出的自然时间相关分数p -拉普拉斯方程，参数为\(1<p<2\)和分数指数\(s\in (0,1)\)。相当标准的理论表明，Lebesgue \(L^q\)空间中数据的柯西问题是适定的，并且解决方案形成了一系列具有正则性和其他有趣特性的非膨胀半群。超线性情况\(p>2\)已在最近的一篇论文中讨论过。我们在这里研究更复杂的“快速”机制\(1<p<2\)。作为主要结果，我们为每个质量值M和任何p构造自相似基本解在子范围\(p_c=2N/(N+s)<p<2\) 中，我们证明这是它们可以存在的精确范围。我们还证明了一般有限质量解向具有相同质量的基本解收敛，并且收敛在所有\(L^q\)空间中成立。获得了全局 Harnack 不等式形式的细边界。该论文的另一个主题是研究具有强奇异性的解。我们找到了一种称为非常奇异解的奇异解，它存在于\(p_c<p<p_1\) 中，其中\(p_1\)是我们引入的新临界数\(p_1\in (p_c,2)\ )。我们将这种奇异解扩展到“非常快的范围” \(1<p<p_c\). 它们代表了在较低p范围内具有有限时间消光的弱解的例子。我们简要检查极限情况\(p=p_c\) 中的情况。最后，我们证明了非常奇异的解与非线性特征值类型的分数椭圆问题有关，它们本身就很有趣。