I am considering the following semilinear parabolic PDE: $$ \partial_tu+\theta(x-y)\partial_yu+\frac12x^2\partial_{xx}u+\frac12y^2\partial_{yy}u + y^{-3/2}u^2+a=0$$ with terminal condition $$u(T,x,y) = b\,. $$
$u$ is defined over $\Omega:=[0,T]\times[x_0,x_1]\times[y_0,y_1] $. The PDE comes from a stochastic control problem and I managed to prove that $u$ is bounded on $\Omega$. I had a hard time finding an existence result for the solution and I was wondering if you could pointing me the literature that could help me to solve this problem