Abstract
In this note, we study parabolic equations of the type (△- λ^{2}q(x,t)- d/dt)u(x,t)=0 on an n-dimensional Riemannian manifold M. By studying the parabolic equation of this type, we discuss the gradient estimates and the Harnack inequality for positive solutions. In some case, we utilize the Harnack inequality to estimate the upper bound and the lower bound for positive solutions. Applications of these estimates for the equation are also discussed. Finally, we study the asymptotic behaviour for the fundamental solution of the operator △- λ^{2}q(x,t)- d/dt as λ--> ∞.