H\"older continuity to Hamilton-Jacobi equations with superquadratic growth in the gradient and unbounded right-hand side

TL;DR

This paper proves local Hölder continuity for solutions of certain degenerate parabolic equations with superquadratic gradient growth and unbounded right-hand sides, using simplified methods.

Contribution

It introduces a new approach relying on sub- and supersolutions and oscillation improvement to establish regularity, avoiding complex existing proofs.

Findings

Solutions are locally Hölder continuous.

Method applies to equations with unbounded right-hand side.

Simplifies proof techniques for regularity results.

Abstract

We show that solutions of time-dependent degenerate parabolic equations with super-quadratic growth in the gradient variable and possibly unbounded right-hand side are locally ${\mathcal C}^{0,\alpha}$. Unlike the existing (and more involved) proofs for equations with bounded right-hand side, our arguments rely on constructions of sub- and supersolutions combined with improvement of oscillation techniques.