In this paper, we study the nonlinear parabolic equation with two exponents on complete noncompact Riemannian maniflods. The special types of such equation include the Fisher-KPP equation, the parabolic Allen-Cahn equation and the Newell-Whitehead equation. We get the Souplet-Zhang's gradient estimates for the positive solutions to such equation. We also obtain the Liouville theorem for positive ancient solutions. Our results extend those of Souplet-Zhang (Bull. London. Math. Soc. 38 (2006), 1045--1053) and Zhu (Acta Math. Sci. Ser. B 36 (2016), no. 2, 514--526).
"Gradient Estimates for the Nonlinear Parabolic Equation with Two Exponents on Riemannian Manifolds." Taiwanese J. Math. 24 (6) 1439 - 1448, December, 2020. https://doi.org/10.11650/tjm/200401