The Hölder interpolative gap bound for degenerate parabolic double phase problems

Abstract: We study weak solutions to degenerate parabolic double phase equations with growth exponents $2\le p<q$ and a modulating coefficient that is Hölder continuous with exponent $α$. If the solution itself is Hölder continuous with exponent $γ$, we prove higher integrability of its gradient under the gap condition $q\le p+α/(1-γ)$ together with $q<p+1$. This is the first gap bound of Hölder interpolative type for parabolic double phase problems, and it is the parabolic counterpart of the corresponding bound for elliptic problems. For $α<1$ it allows exponents beyond the range $q\le p+α$ known for bounded solutions.