Mixed Grushin Heat Equations with Riesz-Potential Nonlinearities in Marcinkiewicz Spaces: Subcritical, Critical, and Supercritical Regimes
Abstract
We study the nonlinear evolution equation ∂t u+(G+Gδ)u=Iα(|u|ρ) on ℝN+k, where G is the nonnegative self adjoint realization of Grushin operator, 0<δ<1, and Iα is the potential operator with kernel |z|−α, 0<α<N+k. We develop a well-posedness theory for initial data in Marcinkiewicz spaces, allowing singular profiles outside the corresponding Lebesgue spaces. Using spectral calculus, subordination, and interpolation, we establish Lebesgue and Lorentz smoothing estimates for the mixed semigroup e−t(G+Gδ), which distinguish the second-order behaviour at short times from the fractional decay at large times. In the subcritical regime, considering 1/β=2/Q+1−α/d, 1/βδ=2δ/Q+1−α/d, where Q is homogenous dimension, we prove local existence, uniqueness, Lipschitz dependence on the initial data, and a blow-up alternative in Lp,∞ when ρ<1+p/β. At the critical relation ρ=1+p/β, a Yamazaki-type estimate and Lorentz duality yield global mild solutions for sufficiently small data in Lp,∞. In the supercritical regime ρ>1+p/β, well-posedness is recovered in higher-integrability Marcinkiewicz spaces Lq,∞ satisfying β(ρ−1)<q<βδ(ρ−1), with global existence for sufficiently small initial data. In all cases, the solutions attain their initial data in the weak-* sense. These results extend the Marcinkiewicz-space theory to mixed Grushin diffusion with a spatially nonlocal potential source and reveal the role of the two competing diffusion scales in determining the admissible integrability regimes.
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BibTeX
@misc{dasgupta2026mixed,
title = {{Mixed Grushin Heat Equations with Riesz-Potential Nonlinearities in Marcinkiewicz Spaces: Subcritical, Critical, and Supercritical Regimes}},
author = {Aparajita Dasgupta and Uttam Kumar Dolai},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23481},
eprint = {2609.23481},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23481},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Mixed Grushin Heat Equations with Riesz-Potential Nonlinearities in Marcinkiewicz Spaces: Subcritical, Critical, and Supercritical Regimes
AU - Aparajita Dasgupta
AU - Uttam Kumar Dolai
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2609.23481
UR - https://arxiv.org/abs/2609.23481
AB - We study the nonlinear evolution equation ∂_t u+(G+G^δ)u=I_α(|u|^ρ) on ℝ^N+k, where G is the nonnegative self adjoint realization of Grushin operator, 0<δ<1, and I_α is the potential operator with kernel |z|^−α, 0<α<N+k. We develop a well-posedness theory for initial data in Marcinkiewicz spaces, allowing singular profiles outside the corresponding Lebesgue spaces. Using spectral calculus, subordination, and interpolation, we establish Lebesgue and Lorentz smoothing estimates for the mixed semigroup e^−t(G+G^δ), which distinguish the second-order behaviour at short times from the fractional decay at large times. In the subcritical regime, considering 1/β=2/Q+1−α/d, 1/β_δ=2δ/Q+1−α/d, where Q is homogenous dimension, we prove local existence, uniqueness, Lipschitz dependence on the initial data, and a blow-up alternative in L^p,∞ when ρ<1+p/β. At the critical relation ρ=1+p/β, a Yamazaki-type estimate and Lorentz duality yield global mild solutions for sufficiently small data in L^p,∞. In the supercritical regime ρ>1+p/β, well-posedness is recovered in higher-integrability Marcinkiewicz spaces L^q,∞ satisfying β(ρ−1)<q<β_δ(ρ−1), with global existence for sufficiently small initial data. In all cases, the solutions attain their initial data in the weak-* sense. These results extend the Marcinkiewicz-space theory to mixed Grushin diffusion with a spatially nonlocal potential source and reveal the role of the two competing diffusion scales in determining the admissible integrability regimes.
ER -