  The main purpose of this paper is to study weak solutions of time-fractional
of porous medium equation with nonlocal pressure: \[ \partial^\alpha_t
u=\operatorname{div}\left( |u|^{m}\nabla (-\Delta)^{-s} u\right) \,\, \text{in
} \mathbb{R}^N\times (0,T) \,, \] with $m\geq 1$, $N\geq 2$, $\frac{1}{2}\leq
s<1$, 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: \[
\|u(t)\|_{L^\infty(\mathbb{R}^N)} \leq C t^{-\frac{\alpha}{q(1-\lambda_0)+ m}}
\|u_0\|_{L^{q}(\mathbb{R}^N)}^{\frac{q(1-\lambda_0)}{q(1-\lambda_0) + m}}
,\quad \text{for } t\in(0,\infty), \] with $\lambda_0=\frac{N-2(1-s)}{N}$.
