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In this paper, we study the nonlinear parabolic equation with two exponentson complete noncompact Riemannian maniflods. The special types of such equationinclude the Fisher-KPP equation, the parabolic Allen-Cahn equation and theNewell-Whitehead equation. We get the Souplet-Zhang's gradeint estimates forthe positive solutions to such equation. We also obtain the Liouville theoremfor positive ancient solutions. Our results extend those of Souplet-Zhang(Bull. London. Math. Soc. 38:1045-1053, 2006) and Zhu (Acta MathematicaScientia 36B(2): 514-526, 2016).

Gradient estimates for the nonlinear parabolic equation with two exponents on Riemannian manifolds