Time derivative estimates for normalized parabolic $p$-Laplace equations

Abstract

We establish the local boundedness of time derivatives of bounded viscosity solutions to the normalized parabolic $p$-Laplace equation for every $p\in(1,\infty)$. The proof combines a logarithmic Bernstein argument with quantitative stability to derive a recursive estimate between different regularizations. For $p=1$ and $n\geq2$, we construct a bounded viscosity solution which is not H\"older continuous in time with any positive exponent.