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

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 $\frac{2n}{n+2}<p\le q\le s<\infty$, and the coefficients $a(z)$ and $b(z)$ are non-negative Hölder 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 ($p\ge 2$) and the singular regime $\left(\frac{2n}{n+2}<p<2\right),$ providing a unified framework. Our approach involves proving uniform parabolic Sobolev-Poincaré inequalities, which are key to establishing reverse Hölder 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.

arXiv comment67 pages. The paper is fully revised and the main result of this version is now extended to the multi-phase setting. Several new estimates are derived to achieve this, and minor typographical errors are corrected. Feedback is welcome