Introduce three binary variables indicating which region you are in and the function value.

$$\begin{align}
\delta_1 &\rightarrow x_2\geq c, ~x_3\geq d, ~x_1 = a\\ 
\delta_2 &\rightarrow x_2\geq c, ~0\leq x_3 \leq d, ~x_1 = bx_3\\
\delta_3 &\rightarrow x_2\leq c, ~x_1 = 0
\end{align}
$$

You are in one of the regions so $\delta_1+\delta_2 +\delta_3=1$, and all the implications are standard big-M representable such as $x_2-c\geq -M(1-\delta_1), -M(1-\delta_1) \leq x_1-a\leq M(1-\delta_1)$ etc.