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 ut−div(G'(|Du|)/|Du|Du+a(x,t)H'(|Du|)/|Du|Du)=0 in ΩT. 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∞(0,T;Ls(Ω,ℝN)), 2≤ s<∞, satisfy Ψ(·,|Du|)∈ L1+εloc for some ε>0, provided limsupρ→ 0+ω(ρ)(H(ρ−(n+s)/s))/(G(ρ−(n+s)/s))<∞. 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)=varkappap, H(varkappa)=varkappaq, and ω(ρ)≃ρα, they recover q≤ p+sα/(n+s) and q≤ 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.
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BibTeX
@misc{kim2026gradient,
title = {{Gradient higher integrability for a general class of degenerate parabolic double phase systems}},
author = {Bogi Kim and Jehan Oh and Abhrojyoti Sen},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.08291},
eprint = {2611.08291},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.08291},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Gradient higher integrability for a general class of degenerate parabolic double phase systems
AU - Bogi Kim
AU - Jehan Oh
AU - Abhrojyoti Sen
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.08291
UR - https://arxiv.org/abs/2611.08291
AB - We establish quantitative local higher integrability estimates for weak solutions of degenerate parabolic systems with generalized double phase growth, whose prototype is u_t−div(G'(|Du|)/|Du|Du+a(x,t)H'(|Du|)/|Du|Du)=0 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^∞(0,T;L^s(Ω,ℝ^N)), 2≤ s<∞, satisfy Ψ(·,|Du|)∈ L^1+ε_loc for some ε>0, provided limsup_ρ→ 0+ω(ρ)(H(ρ^−(n+s)/s))/(G(ρ^−(n+s)/s))<∞. 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)≃ρ^α, they recover q≤ p+sα/(n+s) and q≤ 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.
ER -