Gradient higher integrability for a general class of degenerate parabolic double phase systems
We establish quantitative local higher integrability estimates for weak solutions of degenerate parabolic systems with generalized double phase growth, whose prototype is $$u_t-\operatorname{div}\left(\frac{G'(|Du|)}{|Du|}Du+a(x,t)\frac{H'(|Du|)}{|Du|}Du\right)=0\qquad\text{in }Ω_T.$$ Here $G$ and $H$ are Young functions satisfying suitable structural assumptions. The nonnegative coefficient $a$ is allowed to vanish and has a concave modulus of continuity $ω$ with respect to the parabolic distance. For $Ψ(z,\varkappa)=G(\varkappa)+a(z)H(\varkappa)$, we prove that solutions in $L^\infty(0,T;L^s(Ω,\mathbb{R}^N))$, $2\le s<\infty$, satisfy $Ψ(\cdot,|Du|)\in L^{1+\varepsilon}_{\mathrm{loc}}$ for some $\varepsilon>0$, provided $$\limsup_{ρ\to 0+}ω(ρ)\frac{H(ρ^{-\frac{n+s}{s}})}{G(ρ^{-\frac{n+s}{s}})}<\infty.$$ For bounded solutions, the corresponding condition involves $H(ρ^{-1})/G(ρ^{-1})$. These conditions extend the known interpolative gap bounds from power growth to general growth functions and continuity moduli. In particular, for $G(\varkappa)=\varkappa^p$, $H(\varkappa)=\varkappa^q$, and $ω(ρ)\simeqρ^α$, they recover $q\le p+sα/(n+s)$ and $q\le p+α$, respectively. The proof combines an intrinsic stopping-time argument with inverse-function estimates to derive Poincaré and reverse Hölder inequalities adapted to the general growth phases.