Following the methodology of Brasco (Adv Math 394:108029, 2022), we study the long-time behavior for the signed fractional porous medium equation in open bounded sets with smooth boundary. Homogeneous exterior Dirichlet boundary conditions are considered. We prove that if the initial datum has sufficiently small energy, then the solution, once suitably rescaled, converges to a nontrivial constant sign solution of a sublinear fractional Lane–Emden equation. Furthermore, we give a nonlocal sufficient energetic criterion on the initial datum, which is important to identify the exact limit profile, namely the positive solution or the negative one.

遵循 Brasco (Adv Math 394:108029, 2022) 的方法，我们研究了具有光滑边界的开有界集合中带符号分数多孔介质方程的长期行为。考虑均匀外部狄利克雷边界条件。我们证明，如果初始数据具有足够小的能量，那么解一旦适当重新调整，就会收敛到次线性分数 Lane-Emden 方程的非平凡常数符号解。此外，我们在初始数据上给出了非局部足够能量准则，这对于识别精确的极限轮廓（即正解或负解）非常重要。