Higher fractional differentiability for solutions to parabolic equations with double-phase growth

Abstract

We devote this paper to a higher fractional differentiability of solutions for a class of parabolic double-phase equations ${\partial }_{t}u-\text{div}\left({\left|Du\right|}^{p-2}Du+a(x,t){\left|Du\right|}^{q-2}Du\right)=-\text{div}\left({\left|F\right|}^{p-2}F+a(x,t){\left|F\right|}^{q-2}F\right)\text{in}{\Omega }_T.$A higher fractional differentiability of spatial gradients is established by way of the finite difference quotient, under assumptions that $0\le a(x,t)\in C^{\alpha ,\frac{\alpha }{2}}\left({\Omega }_T\right)$ for $\alpha \in (0,1]$, the exponents $p,q$ satisfies $2\le p\le q\le p+\frac{2\alpha }{n+2}$, and $F(x,t)$ belongs to $L_{loc}^{\vartheta }\left(0,T;B_{\Phi ,\infty ;loc}^{\beta }\left(\Omega \right)\right)$ for $0<\beta <1$ and $\vartheta ≔max\{\frac{q(2q-p)}{p},q+1\}$, where $B_{\Phi ,\infty ;loc}^{\beta }$ is the local Besov-Orlicz space.