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

Abstract

We consider inhomogeneous singular parabolic double phase equations of type $$ u_t-\operatorname{div}(|Du|^{p-2}Du + a(x,t)|Du|^{q-2}Du)=-\operatorname{div} (|F|^{p-2}F + a(x,t)|F|^{q-2}F) $$ in $\Omega_T := \Omega \times (0,T)\subset \mathbb{R}^n\times \mathbb{R}$, where $\frac{2n}{n+2}<p\leq 2$, $p<q$ and $0\leq a(\cdot)\in C^{\alpha,\frac{\alpha}{2}}(\Omega_T)$. We establish gradient higher integrability results for weak solutions to the above problems under one of the following two assumptions: $$ u\in L^\infty (\Omega_T) \quad\text{and}\quad q\leq p +\frac{\alpha(p(n+2)-2n)}{4}, $$ or $$ u\in C(0,T;L^s(\Omega)),\quad s\geq 2 \quad\text{and}\quad q\leq p+\frac{\alpha \mu_s}{n+s}, $$ 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.