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

## TL;DR

This paper studies the long-term behavior and uniqueness of solutions for a nonlocal porous media equation involving fractional diffusion, establishing decay rates, weak-strong uniqueness, and extending existence results to a broader range of parameters.

## Contribution

It provides new insights into the asymptotic decay, uniqueness, and existence of solutions for a fractional nonlocal porous media equation in three dimensions.

## Key findings

- Algebraic decay towards zero in the $L^2$-norm depending on the fractional exponent s.
- Weak-strong uniqueness of solutions and continuous dependence on initial data.
- Existence of weak solutions extended to the torus for a wider range of s.

## Abstract

In this manuscript we consider a non-local porous medium equation with non-local diffusion effects given by a fractional heat operator \begin{equation*} \partial_t u = \mbox{div}(u\nabla p),\qquad \partial_t p = -(-\Delta)^s p + u^2, \end{equation*} in three space dimensions for $3/4\le s < 1$ 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 $\nabla p=0$ in the $L^2(\mathbb{R}^3)$-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 [Caffarelli, Gualdani, Zamponi 2018] for $3/4\le s \le 1$, holds for $1/2 < s\le 1$ if we consider our problem in the torus.