Optimization problem with Kuhn-Tucker conditions
Player $1$'s utility maximization problem is the following :

$$\max_{0 \leq t_{12} \leq T} \ \ \left(T-t_{12} + t_{12}t_{21}\right)^a \left(w(T-t_{12})\right)^{1-a}$$

An equivalent way to solve the above problem is maximizing the $\log$ of the objective i.e. one can solve 

$$\max_{0 \leq t_{12} \leq T} \ \ a\ln\left(T-t_{12} + t_{12}t_{21}\right) +(1-a)\ln  \left(w\right) + (1-a)\ln  \left(T-t_{12}\right) $$

Solving it we get the best response function of player 1 as

\begin{eqnarray*} t_{12} = \begin{cases} 0 &amp; \text{if } t_{21} \leq\frac{1}{a} \\ \frac{T(at_{21}-1)}{(t_{21}-1)} &amp; \text{if } t_{21} &gt; \frac{1}{a} \end{cases} \end{eqnarray*}

Likewise, the best response function of player 2 is

\begin{eqnarray*} t_{21} = \begin{cases} 0 &amp; \text{if } t_{12} \leq  \frac{1}{a} \\ \frac{T(at_{12}-1)}{(t_{12}-1)} &amp; \text{if } t_{12} \geq  \frac{1}{a} \end{cases} \end{eqnarray*}

Now you can solve the best responses to get the Nash Equilibria.