This question is from my question on [mathematics][1].

Picture below is from [231 page][2] . For to prove $Q\ge 0$ on $M \times (0,T)$, $(\partial_t -\Delta)Q \ge 0$ and $Q\ge 0$ are needed to prove.But I can't see $Q\ge 0$ when $t=0$.

Besides, how to ensure $$ \left\{ \begin{aligned} & (\partial_t -\Delta) W_a =\frac{1}{t}W_a \\ & \nabla_aW_b=0 \end{aligned} \right.~~~~~~~~~~~~~ \left\{ \begin{aligned} & (\partial_t-\Delta)U_{ab} = 0 \\ & \nabla_aU_{bc} = \frac{1}{2}(R_{ab}W_c-R_{ac}W_b)+\frac{1}{4t}(g_{ab}W_c-g_{ac}W_b) \end{aligned} \right. $$ have solutions ? And $W_a$ and $U_{ab}$ are arbitrary , how to explain it ? (It is enough that proving the basis meets the equations ?)

[![enter image description here][3]][3]

[![enter image description here][4]][4]

[![enter image description here][5]][5]


  [1]: http://math.stackexchange.com/questions/1790329/two-questions-about-li-yau-hamilton-estimate/1796194#1796194
  [2]: http://projecteuclid.org/download/pdf_1/euclid.ajm/1154098947
  [3]: https://i.sstatic.net/8nEYI.png
  [4]: https://i.sstatic.net/KaaIe.png
  [5]: https://i.sstatic.net/EKbV5.png

This question is from my question on [mathematics][1].

Picture below is from [231 page][2] . For to prove $Q\ge 0$ on $M \times (0,T)$, $(\partial_t -\Delta)Q \ge 0$ and $Q\ge 0$ are needed to prove.But I can't see $Q\ge 0$ when $t=0$.

Besides, how to ensure $$ \left\{ \begin{aligned} & (\partial_t -\Delta) W_a =\frac{1}{t}W_a \\ & \nabla_aW_b=0 \end{aligned} \right.~~~~~~~~~~~~~ \left\{ \begin{aligned} & (\partial_t-\Delta)U_{ab} = 0 \\ & \nabla_aU_{bc} = \frac{1}{2}(R_{ab}W_c-R_{ac}W_b)+\frac{1}{4t}(g_{ab}W_c-g_{ac}W_b) \end{aligned} \right. $$ have solutions ? And $W_a$ and $U_{ab}$ are arbitrary , how to explain it ? (It is enough that proving the basis meets the equations ?)

[![enter image description here][3]][3]

[![enter image description here][4]][4]

[![enter image description here][5]][5]


  [1]: https://math.stackexchange.com/questions/1790329/two-questions-about-li-yau-hamilton-estimate/1796194#1796194
  [2]: http://projecteuclid.org/download/pdf_1/euclid.ajm/1154098947
  [3]: https://i.sstatic.net/8nEYI.png
  [4]: https://i.sstatic.net/KaaIe.png
  [5]: https://i.sstatic.net/EKbV5.png