Longtime behavior and weak-strong uniqueness for a nonlocal porous media equation

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.