论文标题
Riemannian流形的一类椭圆形和抛物线方程的梯度估计值
Gradient estimates for a class of elliptic and parabolic equations on Riemannian manifolds
论文作者
论文摘要
令$（n，g）$为完整的非政策riemannian歧管，从下方界定RICCI曲率。在本文中，我们研究了一类非线性椭圆方程$$Δu（x）+a（x）+a（x）u（x）\ log u（x）+b（x）+b（x）u（x）u（x）u（x）= 0 $ n $ on $ a（x）$ a（x）$ as $ c^{2} $ b（x $ c counter $ b（x）$ c^$ c^y是$ c^） $（δ-\ frac {\ partial} {\ partial t}）到$ x \ in n $，而相对于时间$ t $的$ c^{1} $。与许多相似的结果相反，在这里我们不假定方程系数是恒定的，因此我们的结果可以视为对几个经典估计的扩展。
Let $(N, g)$ be a complete noncompact Riemannian manifold with Ricci curvature bounded from below. In this paper, we study the gradient estimates of positive solutions to a class of nonlinear elliptic equations $$Δu(x)+a(x)u(x)\log u(x)+b(x)u(x)=0$$ on $N$ where $a(x)$ is $C^{2}$-smooth while $b(x)$ is $C^{1}$ and its parabolic counterparts $$(Δ-\frac{\partial}{\partial t})u(x,t)+a(x,t)u(x,t)\log u(x,t) + b(x,t)u(x,t)=0$$ on $N\times[0, \infty)$ where $a(x,t)$ and $b(x,t)$ are $C^{2} $ with respect to $x\in N$ while are $C^{1}$ with respect to the time $t$. In contrast with lots of similar results, here we do not assume the coefficients of equations are constant, so our results can be viewed as extensions to several classical estimates.
