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Gradient estimates for a nonlinear parabolic equation under Ricci flow. (English) Zbl 1249.53083

Summary: Let \((M,g(t))\), \(0\leq t\leq T\), be an \(n\)-dimensional complete noncompact manifold, \(n\geq 2\), 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 [J. Funct. Anal. 241, No. 1, 374–382 (2006; Zbl 1112.58023)] and Y. Yang [Proc. Am. Math. Soc. 136, No. 11, 4095–4102 (2008; Zbl 1151.58013)] 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 \(M\times [0,T]\).

MSC:

53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
35K55 Nonlinear parabolic equations
35R01 PDEs on manifolds
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