Gradient estimates for a nonlinear parabolic equation on Riemannian manifolds

Yang, Yunyan

doi:10.1090/S0002-9939-08-09398-2

Gradient estimates for a nonlinear parabolic equation on Riemannian manifolds
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Proc. Amer. Math. Soc. 136 (2008), 4095-4102 Request permission

Abstract:

Let $(M,g)$ be a complete noncompact Riemannian manifold. In this paper, we derive a local gradient estimate for positive solutions to a simple nonlinear parabolic equation \[ \frac {\partial u}{\partial t}=\Delta u+au\log u+bu\] on $M\times [0,+\infty )$, where $a$, $b$ are two real constants. This equation is closely related to the gradient Ricci soliton. We extend the result of L. Ma (Journal of Functional Analysis 241 (2006) 374-382).

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