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

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