Interpolative Refinement of Gap Bound Conditions for Singular Parabolic Double Phase Problems
Bogi Kim, Jehan Oh
TL;DR
This work addresses gradient higher integrability for singular parabolic double-phase problems with a Hölder-modulating coefficient $a(z)$. By constructing $p$- and $(p,q)$-intrinsic cylinders and employing a stopping-time argument together with reverse Hölder inequalities, the authors obtain a robust higher-integrability result for $H(z,|Du|)$. They present two interpolation frameworks: (i) when $u\in L^\infty$ and $q$ is tightly controlled by a gap bound, and (ii) when $u\in C(0,T;L^s(\Omega))$ with an $s$-dependent gap condition; both yield $H(z,|Du|)^{1+\varepsilon}$-type estimates with data-dependent constants and source-term integrability. The methodology combines a Vitali covering of intrinsic cylinders and localization-decay arguments to produce a global estimate, thus refining the classical gap bounds for singular parabolic double-phase problems and enabling interpolation between degenerate and bounded-solution regimes.
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 $Ω_T := Ω\times (0,T)\subset \mathbb{R}^n\times \mathbb{R}$, where $\frac{2n}{n+2}<p\leq 2$, $p<q$ and $0\leq a(\cdot)\in C^{α,\fracα{2}}(Ω_T)$. We establish gradient higher integrability results for weak solutions to the above problems under one of the following two assumptions: $$ u\in L^\infty (Ω_T) \quad\text{and}\quad q\leq p +\frac{α(p(n+2)-2n)}{4}, $$ or $$ u\in C(0,T;L^s(Ω)),\quad s\geq 2 \quad\text{and}\quad q\leq p+\frac{αμ_s}{n+s}, $$ where $μ_s := \frac{(p(n+2)-2n)s}{4}$. These results yield an interpolation refinement of gap bounds in the singular parabolic double phase setting.