  We prove constrained trace, matrix and constrained matrix Harnack
inequalities for the nonlinear heat equation $\omega_t=\Delta\omega+a\omega\ln
\omega$ on closed manifolds. We also derive a new interpolated Harnack
inequality for the equation $\omega_t=\Delta\omega-\omega\ln\omega+\varepsilon
R\omega$ on closed surfaces under the $\varepsilon$-Ricci flow. Finally we
prove a new differential Harnack inequality for the equation
$\omega_t=\Delta\omega-\omega\ln\omega$ under the Ricci flow without any
curvature condition. Among these Harnack inequalities, the correction terms are
all time-exponential functions, which are superior to time-polynomial
functions.
