Daus, E., Gualdani, M. P., & Zamponi, N. (2020). Longtime behavior and weak-strong uniqueness for a nonlocal porous media equation. Journal of Differential Equations, 268(4), 1820–1839. https://doi.org/10.1016/j.jde.2019.09.029
Entropy method; Fractional diffusion; Long time behavior; Nonlocal porous media equation; Weak-strong uniqueness
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Abstract:
In this manuscript we consider a non-local porous medium equation with non-local diffusion effects given by a fractional heat operator {∂tu=div(u∇p),∂tp=−(−Δ)sp+u2, 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 L2(R3)-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.
