11. 933.

Brief solution: Given $a_{n+2} a_{n+1}-a_{n+1} a_{n}=-2$, let $b_{n}=a_{n} a_{n+1}$, then $b_{n+1}-b_{n}=-2$. Since $b_{1}=19 \times 98=1862$, we have $b_{931}=2, b_{932}=0$, thus $a_{931} a_{932}=2, a_{932} a_{933}=0$. Therefore, $a_{933}=0, m=$ 933.
11. The sequence $\left\{a_{n}\right\}$ satisfies $a_{1}=19, a_{2}=98, a_{n+2}=a_{n}-\frac{2}{a_{n}+1}$, then when $a_{m}=0$, $m$ is