Higher integrability for weak solutions to parabolic multi-phase equations

In this paper, we establish a local higher integrability result for the gradient of a weak solution to a parabolic multi-phase equation. To achieve this, we prove parabolic Poincaré type inequalities and reverse Hölder type inequalities for the gradient of a weak solution in each of difference types of intrinsic cylinders. In particular, we formulate a delicate plan of alternatives and stopping time arguments to address the presence of two different transitions.