Gradient Estimates and Harnack Inequality for a Nonlinear Parabolic Equation on Complete Manifolds
Gradient Estimates and Harnack Inequality for a Nonlinear Parabolic Equation on Complete Manifolds
复制标题
DOI：
10.1007/s40304-014-0026-x
复制
发表时间：
2013-12
影响因子：
0.9
通讯作者：
Jiaxian Wu;Yihu Yang
中科院分区：
文献类型：
--
作者：
Jiaxian Wu;Yihu Yang
设是一个非紧完备黎曼流形。本文考虑$$\开始{aligned} u_t（x，t）=\Delta u（x，t）+au（x，t）\ln u（x，t）+ bu^{\alpha }（x，t）上的非线性抛物方程.证明了上述方程正解的Li-Yau型梯度估计，作为应用，我们还得到了相应的Harnack不等式.这些结果推广了Li（J Funct Anal 100：233-256，1991）所证明的相应结果.
Letbe a noncompact complete Riemannian manifold. In this paper, we consider the following nonlinear parabolic equation on$$\begin{aligned} u_t(x,t)=\Delta u(x,t) + a u(x,t)\ln u(x,t) + bu^{\alpha }(x,t). \end{aligned}$$We prove a Li–Yau type gradient estimate for positive solutions to the above equation; as an application, we also derive the corresponding Harnack inequality. These results generalize the corresponding ones proved by Li (J Funct Anal 100:233–256, 1991).