  In this paper, we would like to give an answer to \textbf{Problem 1} below
issued firstly in [J. Mao, Eigenvalue estimation and some results on finite
topological type, Ph.D. thesis, IST-UTL, 2013]. In fact, by imposing some
conditions on the mean curvature of the initial hypersurface and the
coefficient function of the forcing term of a forced mean curvature flow
considered here, we can obtain that the first eigenvalues of the Laplace and
the $p$-Laplace operators are monotonic under this flow. Surprisingly, during
this process, we get an interesting byproduct, that is, without any complicated
constraint, we can give lower bounds for the first nonzero closed eigenvalue of
the Laplacian provided additionally the second fundamental form of the initial
hypersurface satisfies a pinching condition.
