Decay estimates for time-fractional porous medium flow with nonlocal pressure

Abstract

The main purpose of this paper is to study weak solutions of time-fractional of porous medium equation with nonlocal pressure: \[ \partial^α_t u=\operatorname{div}\left( |u|^{m}\nabla (-Δ)^{-s} u\right) \,\, \text{in } \mathbb{R}^N\times (0,T) \,, \] with $m\ge 1$, $N\ge 2$, $\frac{1}{2}\le s<1$, and $\alpha \in (0,1)$. We first prove an existence of weak solutions to the equation with initial data in $L^1({\mathbb{R}}^N)\cap L^{\infty }({\mathbb{R}}^N)$ (possibly mixed sign). After that, we establish the $L^q-L^{\infty }$ decay estimate of weak solutions: \[ \|u(t)\|_{L^\infty(\mathbb{R}^N)} \leq C t^{-\fracα{q(1-λ_0)+ m}} \|u_0\|_{L^{q}(\mathbb{R}^N)}^{\frac{q(1-λ_0)}{q(1-λ_0) + m}} ,\quad \text{for } t\in(0,\infty), \] with ${\lambda }_0=\frac{N-2(1-s)}{N}$.

26 pages