发表机构
Indian Institute of Technology, Delhi（德里印度理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究Marcinkiewicz空间中混合Grushin热方程在Riesz位势非线性下的适定性，通过谱演算建立光滑性估计，并分别在次临界、临界与超临界情形证明局部或全局解的存在唯一性。
AI 中文摘要
我们研究非线性演化方程 $$ \partial_t u+(G+G^\delta)u=I_\alpha(|u|^\rho) \quad\text{在 }\mathbb{R}^{N+k} \text{上}, $$ 其中 $G$ 是Grushin算子的非负自伴实现，$0<\delta<1$，$I_\alpha$ 是核为 $|z|^{-\alpha}$ 的位势算子，$0<\alpha<N+k$。我们为Marcinkiewicz空间中的初值发展了适定性理论，允许奇异轮廓位于相应的Lebesgue空间之外。利用谱演算、从属与插值，我们建立了混合半群 $e^{-t(G+G^\delta)}$ 的Lebesgue与Lorentz光滑性估计，这些估计区分了短时间内的二阶行为与长时间内的分数阶衰减。在次临界情形下，考虑 $$ \frac1\beta=\frac2Q+1-\frac{\alpha}{d}, \qquad \frac1{\beta_\delta}=\frac{2\delta}{Q}+1-\frac{\alpha}{d}, $$ 其中 $Q$ 是齐次维数，我们证明了当 $$ \rho<1+\frac{p}{\beta} $$ 时，在 $L^{p,\infty}$ 中的局部存在性、唯一性、对初值的Lipschitz依赖性以及爆破替代。在临界关系 $\rho=1+p/\beta$ 下，Yamazaki型估计与Lorentz对偶性给出了 $L^{p,\infty}$ 中足够小数据的全局温和解。在超临界情形 $\rho>1+p/\beta$ 下，在满足 $$ \beta(\rho-1)<q<\beta_\delta(\rho-1) $$ 的更高可积性Marcinkiewicz空间 $L^{q,\infty}$ 中恢复了适定性，且对足够小的初值具有全局存在性。在所有情形中，解在弱-$*$意义下取得其初值。这些结果将Marcinkiewicz空间理论推广到带有空间非局部位势源的混合Grushin扩散，并揭示了两个竞争扩散尺度在确定允许可积性区间中的作用。
英文摘要
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 $|z|^{-α}$, $0<α<N+k$. 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 $$ \frac1β=\frac2Q+1-\fracα{d}, \qquad \frac1{β_δ}=\frac{2δ}{Q}+1-\fracα{d}, $$ where $Q$ is homogenous dimension, we prove local existence, uniqueness, Lipschitz dependence on the initial data, and a blow-up alternative in $L^{p,\infty}$ when $$ ρ<1+\frac{p}β. $$ At the critical relation $ρ=1+p/β$, a Yamazaki-type estimate and Lorentz duality yield global mild solutions for sufficiently small data in $L^{p,\infty}$. In the supercritical regime $ρ>1+p/β$, 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.
