Gradient higher integrability for degenerate/ singular parabolic multi-phase problems

TL;DR

This paper proves an interior gradient higher integrability result for solutions to complex parabolic multi-phase problems, unifying degenerate and singular cases through a novel intrinsic scaling and advanced inequalities.

Contribution

It introduces a unified intrinsic scaling method for degenerate and singular regimes, establishing higher integrability for a broad class of parabolic multi-phase equations.

Findings

Established uniform parabolic Sobolev-Poincaré inequalities

Proved reverse Hölder inequalities across multiple phases

Unified treatment of degenerate and singular regimes

Abstract

This article establishes an interior gradient higher integrability result for weak solutions to parabolic multi-phase problems. The prototype equation for the parabolic multi-phase problem of $p$-Laplace type is given by \[ u_t - \operatorname{div} \left(|\nabla u|^{p-2} \nabla u + a(z) |\nabla u|^{q-2} \nabla u + b(z) |\nabla u|^{s-2} \nabla u \right) = 0, \] where $\frac{2n}{n+2} < p \leq q \leq s < \infty$, and the coefficients $a(z)$ and $b(z)$ are non-negative H\"older continuous functions on $\Omega_T = \Omega \times (0, T)$, with $\Omega \subset \mathbb{R}^n$. We introduce a novel intrinsic scaling to address the problem in both the degenerate regime ($p \geq 2$) and the singular regime $\left(\frac{2n}{n+2} < p < 2\right),$ providing a unified framework. Our approach involves proving uniform parabolic Sobolev-Poincar\'e inequalities, which are key to establishing reverse H\"older type inequalities, along with covering lemmas for the $p$, $(p,q)$, $(p,s)$, and $(p,q,s)$-phases without distinguishing between the regimes of $p$, $q$, and $s$. In the end, we also discuss the gradient higher integrability for general parabolic multi-phase problem involving a finite number of phases.