$C^{1,\alpha}$ regularity for quasilinear parabolic equations with nonstandard growth

TL;DR

This paper establishes $C^{1,eta}$ regularity for weak solutions of certain quasilinear parabolic equations with nonstandard growth, using advanced scaling techniques under minimal regularity assumptions.

Contribution

It extends regularity results to equations with nonstandard growth, including variable exponents and phase switching factors, with minimal regularity assumptions.

Findings

Proves $C^{1,eta}$ regularity for solutions with bounded gradient.

Handles singular and degenerate cases uniformly.

Allows for variable exponents and phase switching factors.

Abstract

In this paper, we obtain $C^{1,\alpha}$ estimates for weak solutions of certain quasilinear parabolic equations satisfying nonstandard growth conditions, the prototype examples being $$u_t - \text{div} (|\nabla u|^{p-2} \nabla u + a(t)|\nabla u|^{q-2} \nabla u) = 0,$$ $$u_t - \text{div} (|\nabla u|^{p(t)-2} \nabla u) = 0.$$ under the assumption that the solutions a priori have bounded gradient. We build on the recently developed scaling and covering argument which allows us to consider the singular and degenerate cases in a uniform manner and with minimal regularity requirements on the phase switching factor $a(t)$ and the variable exponent $p(t)$. Moreover, we are able to take any $p \leq q < \infty$ to obtain the desired regularity.