  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 gradeint 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. 38:1045-1053, 2006) and Zhu (Acta Mathematica
Scientia 36B(2): 514-526, 2016).

Differential Geometry