We apply capacities to explore the space–time fractional dissipative equation:
(0.1)
$$ \begin{align} \left\{\begin{aligned} &\partial^{\beta}_{t}u(t,x)=-\nu(-\Delta)^{\alpha/2}u(t,x)+f(t,x),\quad (t,x)\in\mathbb R^{1+n}_{+},\\ &u(0,x)=\varphi(x),\ x\in\mathbb R^{n}, \end{aligned}\right. \end{align} $$
where
$\alpha>n$
and
$\beta \in (0,1)$
. In this paper, we focus on the regularity and the blow-up set of mild solutions to (0.1). First, we establish the Strichartz-type estimates for the homogeneous term
$R_{\alpha ,\beta }(\varphi )$
and inhomogeneous term
$G_{\alpha ,\beta }(g)$
, respectively. Second, we obtain some space–time estimates for
$G_{\alpha ,\beta }(g).$
Based on these estimates, we prove that the continuity of
$R_{\alpha ,\beta }(\varphi )(t,x)$
and the Hölder continuity of
$G_{\alpha ,\beta }(g)(t,x)$
on
$\mathbb {R}^{1+n}_+,$
which implies a Moser–Trudinger-type estimate for
$G_{\alpha ,\beta }.$
Then, for a newly introduced
$L^{q}_{t}L^p_{x}$
-capacity related to the space–time fractional dissipative operator
$\partial ^{\beta }_{t}+(-\Delta )^{\alpha /2},$
we perform the geometric-measure-theoretic analysis and establish its basic properties. Especially, we estimate the capacity of fractional parabolic balls in
$\mathbb {R}^{1+n}_+$
by using the Strichartz estimates and the Moser–Trudinger-type estimate for
$G_{\alpha ,\beta }.$
A strong-type estimate of the
$L^{q}_{t}L^p_{x}$
-capacity and an embedding of Lorentz spaces are also derived. Based on these results, especially the Strichartz-type estimates and the
$L^{q}_{t}L^p_{x}$
-capacity of fractional parabolic balls, we deduce the size, i.e., the Hausdorff dimension, of the blow-up set of solutions to (0.1).