Abstract
Let M be a complete noncompact Riemannian manifold. We consider gradient estimates for the positive solutions to the following nonlinear parabolic equation
on \({M \times [0, + \infty)}\), where a, b are two real constants, f is a smooth real-valued function on M and \({\Delta_f = \Delta - \nabla f \nabla}\). Under the assumption that the N-Bakry-Emery Ricci tensor is bounded from below by a negative constant, we obtain a gradient estimate for positive solutions of the above equation. As an application, we obtain a Harnack inequality and a Gaussian lower bound of the heat kernel of such an equation.
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This research is supported by NSF Project (No. 10926172) of China, Project of Henan Provincial department of Sciences and Technology (No. 092300410143), and NSF of Henan Provincial Education department (No. 2009A110010).
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Huang, G., Ma, B. Gradient estimates for a nonlinear parabolic equation on Riemannian manifolds. Arch. Math. 94, 265–275 (2010). https://doi.org/10.1007/s00013-009-0091-7
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DOI: https://doi.org/10.1007/s00013-009-0091-7