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Differential Harnack Estimates for a Nonlinear Evolution Equation of Allen–Cahn Type

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Abstract

We discuss local gradient estimates of Li and Yau type on the smooth bounded positive solutions \(w: \mathcal {M} \times [0,\infty ) \rightarrow \mathbb {R}\) to a nonlinear evolution equation \(w_t=\Delta w+a(w-w^3)\), where \(a>0\) is a constant, on a complete Riemannian manifold \(\mathcal {M}\). Global estimates are obtained from the local ones, the consequence of which will eventually yield classical Harnack inequalities for Parabolic Allen–Cahn equation and a Liouville type result for steady state solutions under the hypothesis of nonnegative Ricci curvature tensor.

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Acknowledgements

The author would like to thank the anonymous reviewers for their useful suggestions which have helped to improve the initial version of this paper.

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  1. Department of Mathematics, University of Lagos, Akoka, Lagos, Nigeria

    Abimbola Abolarinwa

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Abolarinwa, A. Differential Harnack Estimates for a Nonlinear Evolution Equation of Allen–Cahn Type. Mediterr. J. Math. 18, 200 (2021). https://doi.org/10.1007/s00009-021-01864-9

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