Equivalence of viscosity and weak solutions for a $p$-parabolic equation

Abstract

We study the relationship of viscosity and weak solutions to the equation \[ \smash{\partial_{t}u-Δ_{p}u=f(Du)} \] where $p>1$ and $f\in C({\mathbb{R}}^N)$ satisfies suitable assumptions. Our main result is that bounded viscosity supersolutions coincide with bounded lower semicontinuous weak supersolutions. Moreover, we prove the lower semicontinuity of weak supersolutions when $p\ge 2$.