Longtime behavior and weak-strong uniqueness for a nonlocal porous media equation
Longtime behavior and weak-strong uniqueness for a nonlocal porous media equation
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非局部多孔介质方程的长期行为和弱强唯一性
DOI：
10.1016/j.jde.2019.09.029
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发表时间：
2018
影响因子：
2.4
通讯作者：
Nicola Zamponi
中科院分区：
文献类型：
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作者：
Esther S. Daus;M. Gualdani;Nicola Zamponi
在这篇手稿中，我们考虑由分数热算子 {∂ t u= div (u∇ p),∂ t p=−(− Δ) s p+ u 2 给出的具有非局部扩散效应的非局部多孔介质方程，在三个空间维度中 3/4≤ s< 1 并分析长时间渐近。该证明基于能量方法，并导致 L 2 (R 3)-范数中的稳态解 u= 0 和∇ p= 0 的代数衰减。衰减率取决于指数 s。我们还展示了解决方案的弱-强唯一性以及对初始数据的持续依赖性。作为我们分析的副产品，我们还表明，如果我们考虑环面中的问题，弱解的存在性（之前在 [4] 中显示的 3/4≤ s≤ 1 的情况）也适用于 1/2< s≤ 1。
In this manuscript we consider a non-local porous medium equation with non-local diffusion effects given by a fractional heat operator {∂ t u= div (u∇ p),∂ t p=−(− Δ) s p+ u 2, in three space dimensions for 3/4≤ s< 1 and analyze the long time asymptotics. The proof is based on energy methods and leads to algebraic decay towards the stationary solution u= 0 and∇ p= 0 in the L 2 (R 3)-norm. The decay rate depends on the exponent s. We also show weak-strong uniqueness of solutions and continuous dependence from the initial data. As a side product of our analysis we also show that existence of weak solutions, previously shown in [4] for 3/4≤ s≤ 1, holds for 1/2< s≤ 1 if we consider our problem in the torus.