Gradient estimates for a nonlinear parabolic equation with potential under geometric flow
Let (M,g) be an n dimensional complete Riemannian manifold. In this article we prove local Li-Yau type gradient estimates for all positive solutions to the nonlinear parabolic equation $$ (\partial_t - \Delta_g + \mathcal{R}) u( x, t) = - a u( x, t) \log u( x, t) $$ along the generalised geom...
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| Format: | Article |
| Language: | English |
| Published: |
Texas State University
2015-01-01
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| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2015/12/abstr.html |
| Summary: | Let (M,g) be an n dimensional complete Riemannian manifold.
In this article we prove local Li-Yau type gradient estimates for all positive
solutions to the nonlinear parabolic equation
$$
(\partial_t - \Delta_g + \mathcal{R}) u( x, t) = - a u( x, t) \log u( x, t)
$$
along the generalised geometric flow on M. Here
$\mathcal{R} = \mathcal{R} (x, t)$ is a smooth potential function and a
is an arbitrary constant. As an application we derive a global estimate and
a space-time Harnack inequality. |
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| ISSN: | 1072-6691 |