Electron. J. Diff. Equ., Vol. 2015 (2015), No. 12, pp. 1-11.

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

Abimbola Abolarinwa

Abstract:
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.

Submitted December 8, 2014. Published January 8, 2015.
Math Subject Classifications: 35K55, 53C21, 53C44, 58J35.
Key Words: Gradient estimates; Harnack inequalities; parabolic equations; geometric flows.

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Abimbola Abolarinwa
Department of Mathematics, University of Sussex
Brighton, BN1 9QH, United Kingdom
email: A.Abolarinwa@sussex.ac.uk

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