A Matrix Differential Harnack Estimate for a Class of Ultraparabolic Equations

Abstract

Let u be a positive solution of the ultraparabolic equation

$$\partial _{t} u=\sum\limits_{i=1}^{n} \partial _{x_{i}}^{2} u+\sum\limits_{i=1}^{k} x_{i}\partial _{x_{n+i}}u \hspace {8mm} \text {on} \hspace {4mm} \mathbb {R}^{n+k}\times (0,T),$$

where 1 ≤ k ≤ n and 0 < T ≤ + ∞. Assume that u and its derivatives (w.r.t. the space variables) up to the second order are bounded on any compact subinterval of (0, T). Then the difference H(log u) − H (log f) of the Hessian matrices of log u and of log f (both w.r.t. the space variables) is non-negatively definite, where f is the fundamental solution of the above equation with pole at the origin (0, 0). The estimate in the case n = k = 1 is due to Hamilton. As a corollary we get that $\Delta l+\frac {n+3k}{2t}+\frac {6k}{t^{3}}\geq 0$, where l = log u, and $\Delta =\sum _{i=1}^{n+k} \partial _{x_{i}}^{2}$.