New Differential Harnack Inequalities for Nonlinear Heat Equations

This paper deals with constrained trace, matrix and constrainedmatrix Harnack inequalities for the nonlinear heat equation$omega_t=Deltaomega+aomegaln omega$ on closed manifolds. A newinterpolated Harnack inequality for$omega_t=Deltaomega-omegalnomega+varepsilon Romega$ onclosed surfaces under $varepsilon$-Ricci f/low is also derived.Finally, the author proves a new differential Harnack inequality for$omega_t=Deltaomega-omegalnomega$ under Ricci f/low withoutany curvature condition. Among these Harnack inequalities, thecorrection terms are all time-exponential functions, which aresuperior to time-polynomial functions.