Differential and Integral Equations

Gradient estimates for a nonlinear parabolic equation under Ricci flow

Shu-Yu Hsu

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Let $(M,g(t))$, $0\le t\le T$, be an n-dimensional complete noncompact manifold, $n\ge 2$, with bounded curvatures and metric $g(t)$ evolving by the Ricci flow $\frac{\partial g_{ij}}{\partial t}=-2R_{ij}$. We will extend the result of L. Ma and Y. Yang and prove a local gradient estimate for positive solutions of the nonlinear parabolic equation $\frac{{\partial} u}{{\partial} t}=\Delta u-au\log u-qu,$ where $a\in\mathbb R$ is a constant and $q$ is a smooth function on $M\times [0,T]$.

Article information

Differential Integral Equations, Volume 24, Number 7/8 (2011), 645-652.

First available in Project Euclid: 27 December 2012

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 58J05: Elliptic equations on manifolds, general theory [See also 35-XX] 58J35: Heat and other parabolic equation methods 35K55: Nonlinear parabolic equations


Hsu, Shu-Yu. Gradient estimates for a nonlinear parabolic equation under Ricci flow. Differential Integral Equations 24 (2011), no. 7/8, 645--652. https://projecteuclid.org/euclid.die/1356628827

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