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

The main purpose of this paper is to study weak solutions of time-fractional of porous medium equation with nonlocal pressure: $${\mathrm{\partial }}_t^{\alpha }u=\mathrm{div}(\mathrm{\mid }u{\mathrm{\mid }}^m\mathrm{\nabla }(-\mathrm{\Delta }{)}^{-s}u)\text{in }{\mathbb{R}}^N\times (0,T),$$ 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 $L^1({\mathbb{R}}^N)\cap L^{\mathrm{\infty }}({\mathbb{R}}^N)$ (possibly mixed sign). After that, we establish the $L^q-L^{\mathrm{\infty }}$ decay estimate of weak solutions: $$\mathrm{\parallel }u(t){\mathrm{\parallel }}_{L^{\mathrm{\infty }}({\mathbb{R}}^N)}\le C{t}^{-\frac{\alpha }{q(1-{\lambda }_0)+m}}\mathrm{\parallel }{u}_0{\mathrm{\parallel }}_{L^q({\mathbb{R}}^N)}^{\frac{q(1-{\lambda }_0)}{q(1-{\lambda }_0)+m}},\text{for }t\in (0,\mathrm{\infty }),$$ with ${\lambda }_0=\frac{N-2(1-s)}{N}$.