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

## TL;DR

This paper establishes the equivalence between viscosity and weak solutions for a class of p-parabolic equations, showing that bounded supersolutions in both frameworks coincide and proving lower semicontinuity of weak supersolutions for p≥2.

## Contribution

It proves the equivalence of viscosity and weak solutions for p-parabolic equations under certain conditions, and demonstrates lower semicontinuity of weak supersolutions when p≥2.

## Key findings

- Bounded viscosity supersolutions coincide with bounded lower semicontinuous weak supersolutions.
- Lower semicontinuity of weak supersolutions is established for p≥2.
- The relationship between viscosity and weak solutions is clarified for a class of nonlinear parabolic equations.

## Abstract

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 $p\geq2$.