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arXiv · 1409.0933

Gradient estimates for a nonlinear parabolic equation with potential under geometric flow

Abimbola Abolarinwa
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Abstract

Let $(M, g)$ be an dimensional complete Riemannian manifold. In this paper we prove local Li-Yau type gradient estimates for all positive solutions to the following nonlinear parabolic equation \begin{equation*} (\partial_t - \Delta_g + \mathcal{R}) u(x, t) = - a u(x, t) \log u(x, t) \end{equation*} along the generalised geometric flow. Here $ \mathcal{R} = \mathcal{R} (x, t)$ is a smooth potential function and $a$ is a constant. As an application we derived a global estimate and a space-time Harnack inequality.

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Abimbola Abolarinwa. 2014-09-03. Gradient estimates for a nonlinear parabolic equation with potential under geometric flow. https://arxiv.org/abs/1409.0933

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