Gradient estimates for singular parabolic $p$-Laplace type equations with measure data

Abstract

We are concerned with gradient estimates for solutions to a class of singular quasilinear parabolic equations with measure data, whose prototype is given by the parabolic $p$-Laplace equation $u_t-\Delta_p u=\mu$ with $p\in (1,2)$. The case when $p\in \big(2-\frac{1}{n+1},2\big)$ were studied in [15]. In this paper, we extend the results in [15] to the open case when $p\in \big(\frac{2n}{n+1},2-\frac{1}{n+1}\big]$ if $n\geq 2$ and $p\in(\frac{5}{4}, \frac{3}{2}]$ if $n=1$. More specifically, in a more singular range of $p$ as above, we establish pointwise gradient estimates via linear parabolic Riesz potential and gradient continuity results via certain assumptions on parabolic Riesz potential.