Mixed Grushin Heat Equations with Riesz-Potential Nonlinearities in Marcinkiewicz Spaces: Subcritical, Critical, and Supercritical Regimes

Abstract

We study the nonlinear evolution equation $$ \partial_t u+(G+G^δ)u=I_α(|u|^ρ) \quad\text{on }\mathbb{R}^{N+k}, $$ where $G$ is the nonnegative self adjoint realization of Grushin operator, $0<δ<1$, and $I_α$ is the potential operator with kernel $|z|^{-α}$, $0<α 1+p/β$, well-posedness is recovered in higher-integrability Marcinkiewicz spaces $L^{q,\infty}$ 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.