We consider the natural time-dependent fractional $p$-Laplacian equation posed in the whole Euclidean space, with parameters $p>2$ and $s\in (0,1)$ (fractional exponent). We show that the Cauchy Problem for data in the Lebesgue $L^q$ spaces is well posed, and show that the solutions form a family of non-expansive semigroups with regularity and other interesting properties. As main results, we construct the self-similar fundamental solution for every mass value $M,$ and prove that general finite-mass solutions converge towards that fundamental solution having the same mass, and convergence holds in all $L^q$ spaces. A number of additional properties and estimates complete the picture.

我们考虑自然时间相关的分数 $p$-拉普拉斯方程在整个欧氏空间中构成，其参数 $p>2$ 和 $s\in （0，\mathrm{1个}）$（分数指数）。我们证明了Lebesgue中数据的柯西问题${\mathrm{大号}}^q$空间的位置很好，表明这些解形成了一个具有规则性和其他有趣性质的非扩展半群族。作为主要结果，我们为每个质量值构造了自相似的基本解$\mathrm{中号}，$ 并证明一般的有限质量解收敛于具有相同质量的基本解，并且收敛在所有 ${\mathrm{大号}}^q$空格。许多其他属性和估计值可以完成整个过程。