Complexity of asymptotic behavior of solutions for the fractional porous medium equation

Abstract

In this paper, we study the large time behavior of solutions to the fractional porous medium equation $ u_{t} = \nabla\cdot (u\nabla^{\alpha-1}u) $ in $ \mathbb{R}^{N} $ with $ 0<\alpha<2 $. More precisely, we reveal that for any given $ 0<\mu<\frac{2N}{N+\alpha} $ and $ \beta>\frac{2-\mu}{2\alpha} $, there exists an initial-value $ u_{0}(x) $ such that the complexity of asymptotic behavior for the rescaled solutions $ t^{\frac{\mu}{2}}u(t^{\beta}\cdot,t) $ occurs in $ C_{0}^{+}(\mathbb{R}^{N}) $. For this purpose, we apply the $ L^{1} $-$ L^{\infty} $ smoothing effect and establish the propagation estimates for the solutions.