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Gradient Estimates for a Class of Semilinear Parabolic Equations and Their Applications

Abstract

In this paper we address the following parabolic equation @@ on a smooth metric measure space with Bakry–Émery curvature bounded from below for F being a differentiable function defined on \(\mathbb {R}\). Our motivation is originally inspired by gradient estimates of Allen–Cahn and Fisher-KKP equations (Bǎileşteanu, M., Ann. Glob. Anal. Geom. 51, 367–378, 2017; Cao et al., Pac. J. Math. 290, 273–300, 2017). We show new gradient estimates for these equations. As their applications, we obtain Liouville type theorems for positive or bounded solutions to the above equation when either F = cu(1 − u) (the Fisher-KKP equation) or; F = −u3 + u (the Allen–Cahn equation); or \(F=au\log u\) (the equation involving gradient Ricci solitons).

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Acknowledgements

The authors would like to thank anonymous referees for their useful comments to improve the presentation of our paper. The first author thanks Ha Tuan Dung for useful comments to correct a coefficient in Theorem 1.1. This work was partially supported by NAFOSTED under grant number 101.02-2017.313.

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Correspondence to Nguyen Thac Dung.

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Dung, N.T., Khanh, N.N. Gradient Estimates for a Class of Semilinear Parabolic Equations and Their Applications. Vietnam J. Math. 50, 249–259 (2022). https://doi.org/10.1007/s10013-021-00492-w

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  • DOI: https://doi.org/10.1007/s10013-021-00492-w

Keywords

  • Gradient estimates
  • Bakry–Émery curvature
  • Smooth metric measure space
  • Harnack-type inequalities
  • Liouville-type theorems

Mathematics Subject Classification (2010)

  • Primary 32M05
  • Secondary 32H02