In the Heisenberg group ${\mathbb{H}}^n$, we obtain the local second-order $H{W}_{\mathrm{loc}}^{2,2}$-regularity for the weak solution u to a class of degenerate parabolic quasi-linear equations ${\partial }_{t}u={\displaystyle \sum _{i=1}^{2n}}{X}_i{A}_i(Xu)$ modeled on the parabolic p-Laplacian equation. Specifically, when $2\le p\le 4$, we demonstrate the integrability of ${({\partial }_{t}u)}^2$, namely, ${\partial }_{t}u\in L_{\mathrm{loc}}^2$; when $2\le p<3$, we demonstrate the $H{W}_{\mathrm{loc}}^{2,2}$-regularity of u, namely, $XXu\in L_{\mathrm{loc}}^2$. For the $H{W}_{\mathrm{loc}}^{2,2}$-regularity, when $p\ge 2$, the range of p is optimal compared to the Euclidean case.

Forest fires pose escalating threats to ecological security and public safety in Guangdong Province. This study presents a novel machine learning framework for fire occurrence prediction by synergistically integrating multi-source geospatial data. Utilizing Moderate-resolution Imaging Spectroradiometer (MODIS) active fire detections from 2014 to 2023, we quantified historical fire patterns and incorporated four categories of predisposing factors: meteorological variables, topographic attributes, vegetation characteristics, and anthropogenic activities. Spatiotemporal clustering dynamics were characterized via kernel density estimation and spatial autocorrelation analysis. An XGBoost classifier, hyperparameter-optimized through the Sparrow Search Algorithm (SSA), achieved a predictive accuracy of 90.4%, with performance evaluated through precision, recall, and $F_1$-score. Risk zoning maps generated from predicted probabilities were validated against independent fire records from 2019 to 2024. Results reveal pronounced spatial heterogeneity, with high-risk zones concentrated in northern and western mountainous areas, constituting 29% of the provincial territory. Critical driving factors include slope gradient, proximity to roads and rivers, temperature, population density, and elevation. This robust predictive framework furnishes a scientific foundation for spatially-explicit fire prevention strategies and optimized resource allocation in key high-risk jurisdictions, notably Qingyuan, Shaoguan, Zhanjiang, and Zhaoqing.

In the Heisenberg group ${\mathbb{H}}^n$ and for exponents $2\le q\le 4$, the local Lipschitz ($C^{0,1}$) regularity is demonstrated for weak solutions to a non-homogeneous highly degenerate q-Laplace equation of the form $-{\sum }_{i=1}^{2n}{X}_i\left({\lambda }_i(Xu)X_{i}u\right)=g(x)$, where ${\lambda }_i(z)={(z_i^2+z_{i+n}^2)}^{{\displaystyle \frac{q-2}{2}}}$ for $i=1,\dots ,n$ and ${\lambda }_i(z)={(z_i^2+z_{i-n}^2)}^{{\displaystyle \frac{q-2}{2}}}$ for $i=n+1,\dots ,2n$. Our approach draws heavily on the work of X. Zhong. However, owing to technical limitations, the regularity of weak solutions for $1<q<2$ and $4<q<\infty$ remains unclear.

In the Heisenberg group ${\mathbb{H}}^n$, we establish the local regularity theory for weak solutions to non-homogeneous degenerate nonlinear parabolic equations of the form ${\partial }_{t}u-{\displaystyle \sum _{i=1}^{2n}}{X}_i{A}_i\left(Xu\right)=K(x,t,u,Xu)$, where the nonlinear structure is modeled on non-homogeneous parabolic p-Laplacian-type operators. Specifically, we prove two main local regularities: (i) For $2\le p\le 4$, we establish the local Lipschitz regularity ($u\in C_{\mathrm{loc}}^{0,1}$), with the horizontal gradient satisfying $Xu\in L_{\mathrm{loc}}^{\infty }$; (ii) For $2\le p<3$, we establish the local second-order horizontal Sobolev regularity ($u\in H{W}_{\mathrm{loc}}^{2,2}$), with the second-order horizontal derivative satisfying $XXu\in L_{\mathrm{loc}}^2$. These results solve an open problem proposed by Capogna et al.