发表机构
School of Mathematics; Statistics Wuhan University\ 430072, China（; ）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文在Ricci曲率下有界的完备黎曼流形上，对所有指数$m>0$建立了多孔介质与快速扩散方程正解的局部Hamilton型梯度估计，并由此推出非负Ricci曲率流形上一致有界正古老解必为常数。
AI 中文摘要
我们在Ricci曲率下有下界的完备黎曼流形上，对正$C^{2,1}$解$u_t=\Delta u^m$建立了局部Hamilton型梯度估计。对于每个固定的$m>1$和每个固定的$0<m<1$，存在$\beta=\beta(m,n)>0$和$C=C(m,n)>0$，使得在$B_{2R}(x_0)\times(t_0-T,t_0]$中满足$0<u\leqslant A$的解具有如下局部梯度估计：\begin{equation*} \sup_{B_R(x_0)\times(t_0-T/2,t_0]}|\nabla u^\beta| \leqslant C A^\beta \left(\frac1R+\sqrt{k}+\frac{A^{(1-m)/2}}{\sqrt T}\right), \end{equation*} 其中$\mathrm{Ric}_M\geqslant-k$。证明使用了一种内在的定量替代方法，在每个给定的正点附近定位一个柱体，在该柱体上解具有受控的上-下比。在该柱体上的局部梯度估计与一个停止论证相结合。作为推论，在具有非负Ricci曲率的连通完备流形上，每个一致有界的正古老解都是常数。
英文摘要
We establish local Hamilton-type gradient estimates for positive $C^{2,1}$ solutions of $u_t=Δu^m$ on complete Riemannian manifolds whose Ricci curvature is bounded from below. For every fixed $m>1$ and every fixed $0<m<1$, there exist $β=β(m,n)>0$ and $C=C(m,n)>0$ such that a solution $0<u\leqslant A$ in $B_{2R}(x_0)\times(t_0-T,t_0]$ satisfies the following local gradient estimate \begin{equation*} \sup_{B_R(x_0)\times(t_0-T/2,t_0]}|\nabla u^β| \leqslant C A^β\left(\frac1R+\sqrt{k}+\frac{A^{(1-m)/2}}{\sqrt T}\right), \end{equation*} where $\mathrm{Ric}_M\geqslant-k$. The proof uses an intrinsic quantitative alternative to locate, around each prescribed positive point, a cylinder on which the solution has a controlled upper-to-lower ratio. A local gradient estimate on this cylinder is combined with a stopping argument. As a consequence, every uniformly bounded positive ancient solution on a connected complete manifold with nonnegative Ricci curvature is constant.
