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

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 $$u_t-\mathrm{div}(\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{p-2}\mathrm{\nabla }u+a(z)\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{q-2}\mathrm{\nabla }u+b(z)\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{s-2}\mathrm{\nabla }u)=0,$$ where $\frac{2n}{n+2}<p\le q\le s<\mathrm{\infty }$, and the coefficients $a(z)$ and $b(z)$ are non-negative Hölder continuous functions on ${\mathrm{\Omega }}_T=\mathrm{\Omega }\times (0,T)$, with $\mathrm{\Omega }\subset {\mathbb{R}}^n$. We introduce a novel intrinsic scaling to address the problem in both the degenerate regime ($p\ge 2$) and the singular regime $(\frac{2n}{n+2}<p<2),$ 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.