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

We study the relationship of viscosity and weak solutions to the equation $$\smash{\partial_{t}u-\Delta_{p}u=f(Du)}$$ 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 $p\geq2$.