| 摘要: |
| 本文研究非线性抛物型方程 $(\Delta -\partial_t)u=q(x,t)u(x,t)+au(x,t)A((\log u(x,t))^\alpha)$ 正解的梯度估计问题.
通过利用文献\cite{azami2024differential}中引入的辅助函数和极值原理,我们建立了该方程的Li-Yau型、Hamilton型、Li-Xu型及线性Li-Xu型四类梯度估计.
此外，通过对对数结构的深入分析，我们刻画了梯度估计在奇点附近的爆破行为(参见注记 1.3)，从而对 \cite{azami2024differential} 中的定理 1.3 给出了一个更精准、更聚焦的推广. |
| 关键词: 梯度估计 Harnack不等式 抛物型方程 |
| DOI： |
| 分类号:O175.29;O186.12 |
| 基金项目: |
|
| Differential gradient estimates for a nonlinear parabolic equation under integral Ricci curvature bounds |
|
Huang Chenlin, Yangfei
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| Abstract: |
| This paper investigates gradient estimates for positive solutions of
the nonlinear parabolic equation $(\Delta -\partial_t)u=q(x,t)u(x,t)+au(x,t)A((\log u(x,t))^\alpha)$.
By employing the auxiliary function from \cite{azami2024differential} and maximum principle, we establish four types of gradient
estimates: Li-Yau type, Hamilton type, Li-Xu type, and linear Li-Xu type.
Moreover, our analysis of the logarithmic structure characterizes how gradient estimates blow up near singularities (see Remarks 1.3), giving a sharper, more focused extension of Theorem 1.3 in \cite{azami2024differential}. |
| Key words: Gradient estimates Harnack inequalities Parabolic equations |
