Gradient higher integrability for a general class of degenerate parabolic double phase systems

Abstract

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.