New differential Harnack inequalities for nonlinear heat equations

Abstract

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 ${\omega }_t=\Delta \omega -\omega \ln \omega +\epsilon R\omega$ on closed surfaces under the $\epsilon$-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.

19 pages, to appear in Chin. Ann. Math. Ser. B