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

TL;DR

This paper proves gradient higher integrability for solutions to singular parabolic double phase equations, refining the conditions under which these solutions exhibit improved regularity.

Contribution

It introduces interpolative methods to refine gap bounds for regularity conditions in singular parabolic double phase problems.

Findings

Established gradient higher integrability under new assumptions.

Refined the gap bounds for regularity conditions.

Provided interpolation techniques for 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.