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

\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\\
\delta_4 &\rightarrow x_3\leq 0, ~x_1 = 0

You are in one of the regions so $\delta_1+\delta_2 +\delta_3+\delta_4=1$, 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. By exloiting structure and linearity in some terms this can be reduced and simplified, but this is the basic model.