Interpolative Refinement of Gap Bound Conditions for Singular Parabolic Double Phase Problems

Abstract

We consider inhomogeneous singular parabolic double phase equations of type $$u_t-\mathrm{div}(|Du{|}^{p-2}Du+a(x,t)|Du{|}^{q-2}Du)=-\mathrm{div}(|F{|}^{p-2}F+a(x,t)|F{|}^{q-2}F)$$ in $\hat{I}{\text{c◯}}_T:=\hat{I}\text{c◯}\times (0,T)\subset {\mathbb{R}}^n\times \mathbb{R}$, where $\frac{2n}{n+2}<p\le 2$, $p<q$ and $0\le a(\cdot )\in C^{\hat{I}\pm ,\frac{\hat{I}}{\pm }2}(\hat{I}{\text{c◯}}_T)$. We establish gradient higher integrability results for weak solutions to the above problems under one of the following two assumptions: $$u\in L^{\infty }(\hat{I}{\text{c◯}}_T)\text{and}q\le p+\frac{\hat{I}\pm (p(n+2)-2n)}{4},$$ or $$u\in C(0,T;L^s(\hat{I}\text{c◯})),s\ge 2\text{and}q\le p+\frac{\hat{I}\pm \hat{I}{¼}_s}{n+s},$$ where $\hat{I}{¼}_s:=\frac{(p(n+2)-2n)s}{4}$. These results yield an interpolation refinement of gap bounds in the singular parabolic double phase setting.