Gradient higher integrability of bounded solutions to parabolic double-phase systems

TL;DR

This paper establishes higher integrability of gradients for bounded solutions to a class of degenerate parabolic double-phase systems, extending regularity results under specific conditions on the exponents and the weight function.

Contribution

It proves local higher integrability of gradients for solutions to double-phase parabolic systems with Hölder continuous weights, covering the sharp exponent range.

Findings

Gradients of solutions are locally higher integrable.

Results hold for the sharp exponent range p<q≤p+α.

The regularity depends on the Hölder continuity of the weight a.

Abstract

We prove that bounded solutions to degenerate parabolic double-phase problem modelled upon \[u_t-\dv(|\na u|^{p-2}\na u+a(x,t)|\na u|^{q-2}\na u)=-\dv(|F|^{p-2}F+a(x,t)|F|^{q-2}F)\,, \] where a nonnegative weight $a$ is $\alpha$-H\"older continuous in space and $\tfrac \alpha 2$-H\"older continuous in time, have locally higher integrable gradients for the sharp range of exponents $p<q\le p+\alpha$.