Kuhn-Tucker and optimization (continue)
I tried this for a=0.5, T=10, w=2 and I get the following real solutions for t12 and t21:
(1) $t_{12}=t_{21}=2.8165$
(2) $t_{12}=t_{21}=1.1835$

For (1) the function is valued at 3.78821 and for (2) the function is valued at 3.68444. 

So it seems that (1) is the solution for the maximum. 

If it helps, here is the function and the contour sets at those parameter values. 

Function


Contour Set ($t_{12}$ on X axis):  



By the way, are your FOCs correct? For the FOC with t12, I have:
$-\frac{(1 - a)}{(T - t_{12})} + \frac{a (-1 + t_{21})}{(T - t_{12} + t_{12} t_{21})} + \frac{a t_{21}}{(
  T - t_{21} + t_{12} t_{21})} = 0$