Gradient higher integrability for degenerate parabolic double phase systems with two modulating coefficients

Abstract

We establish an interior gradient higher integrability result for weak solutions to degenerate parabolic double phase systems involving two modulating coefficients. To be more precise, we study systems of the form \[ u_t-\operatorname{div} \left(a(z)|Du|^{p-2}Du+ b(z)|Du|^{q-2}Du\right)=-\operatorname{div} \left(a(z)|F|^{p-2}F+ b(z)|F|^{q-2}F\right), \] where $2\leq p\leq q < \infty$ and the modulating coefficients $a(z)$ and $b(z)$ are non-negative, with $a(z)$ being uniformly continuous and $b(z)$ being H\"{o}lder continuous. We further assume that the sum of two modulating coefficients is bounded from below by some positive constant. To establish the gradient higher integrability result, we introduce a suitable intrinsic geometry and develop a delicate comparison scheme to separate and analyze the different phases--namely, the $p$-phase, $q$-phase and $(p,q)$-phase. To the best of our knowledge, this is the first regularity result in the parabolic setting that addresses general double phase systems within the framework of weak solutions.